An Exp erime - Freenguyen.hong.hai.free.fr/EBOOKS/SCIENCE AND ENGINEERING/MECANIQUE... · xii A ma jor goal of this pro ject is to sho who wto in tegrate to ols from these di eren

An Experimental Approach to Nonlinear Dynamicsand Chaos

by

Nicholas B� Tu�llaro� Tyler Abbott� and Jeremiah P� ReillyINTERNET ADDRESSES� nbt�reed�edu

COPYRIGHT c�� TUFILLARO� ABBOTT� AND REILLYPublished by Addison�Wesley� ��

v� ��b November ��

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Contents

Preface xi

Introduction �What is nonlinear dynamics� � � � � � � � � � � � � � � � � � � � � � � � � � �What is in this book� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �Some Terminology� Maps� Flows� and Fractals � � � � � � � � � � � � � � � �References and Notes � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Bouncing Ball �� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Model � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

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� �� Motivation and Geometric Description � � � � � � � � � � � � � �� Algebraic Description � � � � � � � � � � � � � � � � � � � � � � � � �� Location of Knots � � � � � � � � � � � � � � � � � � � � � � � � �� Calculation of Relative Rotation Rates � � � � � � � � � � � � � �

�� Intertwining Matrices � � � � � � � � � � � � � � � � � � � � � � � � � � �� Horseshoe � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Lorenz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Du�ng Template � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��References and Notes � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��Problems � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Appendix A� Bouncing Ball Code ��

Appendix B� Exact Solutions for a Cubic Oscillator ��

Appendix C� Ode Overview ��

Appendix D� Discrete Fourier Transform ��

Appendix E� Henon�s Trick ��

Appendix F� Periodic Orbit Extraction Code ��

CONTENTS ix

Appendix G� Relative Rotation Rate Package ��

Appendix H� Historical Comments ��

Appendix I� Projects ��

x CONTENTS

��

Preface

An Experimental Approach to Nonlinear Dynamics and Chaos is a textbook and areference work designed for advanced undergraduate and beginning graduate stu�dents� This book provides an elementary introduction to the basic theoretical andexperimental tools necessary to begin research into the nonlinear behavior of me�chanical� electrical� optical� and other systems� A focus of the text is the descriptionof several desktop experiments� such as the nonlinear vibrations of a current�carryingwire placed between the poles of an electromagnet and the chaotic patterns of a ballbouncing on a vibrating table� Each of these experiments is ideally suited for thesmall�scale environment of an undergraduate science laboratory�

In addition� the book includes software that simulates several systems describedin this text� The software provides the student with the opportunity to immedi�ately explore nonlinear phenomena outside of the laboratory� The feedback of theinteractive computer simulations enhances the learning process by promoting theformation and testing of experimental hypotheses� Taken together� the text andassociated software provide a hands�on introduction to recent theoretical and ex�perimental discoveries in nonlinear dynamics�

Studies of nonlinear systems are truly interdisciplinary� ranging from experimen�tal analyses of the rhythms of the human heart and brain to attempts at weatherprediction� Similarly� the tools needed to analyze nonlinear systems are also inter�disciplinary and include techniques and methodologies from all the sciences� Thetools presented in the text include those of�

theoretical and applied mathematics �dynamical systems theory and perturba�tion theory��

theoretical physics �development of models for physical phenomena� applicationof physical laws to explain the dynamics� and the topological characterizationof chaotic motions��

experimental physics �circuit diagrams and desktop experiments��

engineering �instabilities in mechanical� electrical� and optical systems�� and

computer science �numerical algorithms in C and symbolic computations withMathematica��

xi

xii

A major goal of this project is to show how to integrate tools from these di�erentdisciplines when studying nonlinear systems�

Many sections of this book develop one speci�c �tool� needed in the analysisof a nonlinear system� Some of these tools are mathematical� such as the appli�cation of symbolic dynamics to nonlinear equations� some are experimental� suchas the necessary circuit elements required to construct an experimental surface ofsection� and some are computational� such as the algorithms needed for calculat�ing fractal dimensions from an experimental time series� We encourage students totry out these tools on a system or experiment of their own design� To help withthis� Appendix I provides an overview of possible projects suitable for research byan advanced undergraduate� Some of these projects are in acoustics �oscillationsin gas columns�� hydrodynamics �convective loop�Lorenz equations� Hele�Shawcell� surface waves�� mechanics �oscillations of beams� stability of bicycles� forcedpendulum� compass needle in oscillating B��eld� impact�oscillators� chaotic art mo�biles� ball in a swinging track�� optics �semiconductor laser instabilities� laser rateequations�� and other systems showing complex behavior in both space and time�video�feedback� ferrohydrodynamics� capillary ripples��

This book can be used as a primary or reference text for both experimentaland theoretical courses� For instance� it can be used in a junior level mathematicscourse that covers dynamical systems or as a reference or lab manual for juniorand senior level physics labs� In addition� it can serve as a reference manual fordemonstrations and� perhaps more importantly� as a source book for undergraduateresearch projects� Finally� it could also be the basis for a new interdisciplinarycourse in nonlinear dynamics� This new course would contain an equal mixture ofmathematics� physics� computing� and laboratory work� The primary goal of thisnew course is to give students the desire� skills� and con�dence needed to begin theirown research into nonlinear systems�

Regardless of her �eld of study� a student pursuing the material in this bookshould have a �rm grounding in Newtonian physics and a course in di�erentialequations that introduces the qualitative theory of ordinary di�erential equations�For the latter chapters� a good dose of mathematical maturity is also helpful�

To assist with this new course we are currently designing labs and software�including complementary descriptions of the theory� for the bouncing ball system�the double scroll LRC circuit� and a nonlinear string vibrations apparatus� Thebouncing ball package has been completed and consists of a mechanical apparatus�a loudspeaker driven by a function generator and a ball bearing�� the Bouncing Ballsimulation system for the Macintosh computer� and a lab manual� This packagehas been used in the Bryn Mawr College Physics Laboratory since ��

This text is the �rst step in our attempt to integrate nonlinear theory with easilyaccessible experiments and software� It makes use of numerical algorithms� symbolicpackages� and simple experiments in showing how to attack and unravel nonlinearproblems� Because nonlinear e�ects are commonly observed in everyday phenomena�avalanches in sandpiles� a dripping faucet� frost on a window pane�� they easilycapture the imagination and� more importantly� fall within the research capabilities

xiii

of a young scientist� Many experiments in nonlinear dynamics are individual orsmall group projects in which it is common for a student to follow an experimentfrom conception to completion in an academic year�

In our opinion nonlinear dynamics research illustrates the �nest aspects of smallscience� It can be the e�ort of a few individuals� requiring modest funding� and oftendeals with �homemade� experiments which are intriguing and accessible to studentsat all levels� We hope that this book helps its readers in making the transition fromstudying science to doing science�

We thank Neal Abraham� Al Albano� and Paul Melvin for providing detailedcomments on an early version of this manuscript� We also thank the Departmentof Physics at Bryn Mawr College for encouraging and supporting our e�orts in thisdirection over several years� We would also like to thank the text and softwarereviewers who gave us detailed comments and suggestions� Their corrections andquestions guided our revisions and the text and software are better for their scrutiny�

One of the best parts about writing this book is getting the chance to thank allour co�workers in nonlinear dynamics� We thank Neal Abraham� Kit Adams� Al Al�bano� Greg Alman� Ditza Auerbach� Remo Badii� Richard Bagley� Paul Blanchard�Reggie Brown� Paul Bryant� Gregory Buck� Jose�na Casasayas� Lee Casperson� R�Crandall� Predrag Cvitanovi�c� Josh Degani� Bob Devaney� Andy Dougherty� BonnieDuncan� Brian Fenny� Neil Gershenfeld� Bob Gilmore� Bob Gioggia� Jerry Gollub�David Gri�ths� G� Gunaratne� Dick Hall� Kath Hartnett� Doug Hayden� Gina Lucaand Lois Ho�er�Lippi� Phil Holmes� Reto Holzner� Xin�Jun Hou� Tony Hughes�Bob Jantzen� Raymond Kapral� Kelly and Jimmy Kenison�Falkner� Tim Kerwin�Greg King� Eric Kostelich� Pat Langhorne� D� Lathrop� Wentian Li� Barbara Litt�Mark Levi� Pat Locke� Amy Lorentz� Takashi Matsumoto� Bruce McNamara� TinaMello� Paul Melvin� Gabriel Mindlin� Tim Molteno� Ana Nunes� Oliver O�Reilly�Norman Packard� R� Ramshankar� Peter Rosenthal� Graham Ross� Miguel Rubio�Melora and Roger Samelson� Wes Sandle� Peter Saunders� Frank Selker� John Selker�Bill Sharpf� Tom Shieber� Francesco Simonelli� Lenny Smith� Hern�an Solari� TomSolomon� Vernon Squire� John Stehle� Rich Super�ne� M� Tarroja� Mark Taylor� J�R� Tredicce� Hans Troger� Jim Valerio� Don Warrington� Kurt Wiesenfeld� StephenWolfram� and Kornelija Zgonc�

We would like to express a special word of thanks to Bob Gilmore� GabrielMindlin� Hern�an Solari� and the Drexel nonlinear dynamics group for freely sharingand explaining their ideas about the topological characterization of strange sets� Wealso thank Tina Mello for experimental expertise with the bouncing ball system�Tim Molteno for the design and construction of the string apparatus� and AmyLorentz for programing expertise with Mathematica�

Nick would like to acknowledge agencies supporting his research in nonlineardynamics� which have included Sigma Xi� the Fulbright Foundation� Bryn MawrCollege� Otago University Research Council� the Beverly Fund� and the NationalScience Foundation�

Nick would like to o�er a special thanks to Mom and Dad for lots of homecooked meals during the writing of this book� and to Mary Ellen and Tracy for

xiv

encouraging him with his chaotic endeavors from an early age�Tyler would like to thank his own family and the Duncan and Dill families for

their support during the writing of the book�Jeremy and Tyler would like to thank the exciting teachers in our lives� Jeremy

and Tyler dedicate this book to all inspiring teachers�

Introduction

What is nonlinear dynamics�

A dynamical system consists of two ingredients� a rule or �dynamic� which speci�es how a system evolves� and an initial condition or �state from which the system starts� The most successful class of rules for de�scribing natural phenomena are di�erential equations� All the majortheories of physics are stated in terms of di�erential equations� Thisobservation led the mathematician V� I� Arnold to comment� �conse�quently� di�erential equations lie at the basis of scienti�c mathematicalphilosophy� our scienti�c world view� This scienti�c philosophy beganwith the discovery of the calculus by Newton and Leibniz and continuesto the present day�

Dynamical systems theory and nonlinear dynamics grew out of thequalitative study of di�erential equations� which in turn began as anattempt to understand and predict the motions that surround us� theorbits of the planets� the vibrations of a string� the ripples on thesurface of a pond� the forever evolving patterns of the weather� The �rsttwo hundred years of this scienti�c philosophy� from Newton and Eulerthrough to Hamilton and Maxwell� produced many stunning successesin formulating the �rules of the world� but only limited results in�nding their solutions� Some of the motions around us�such as theswinging of a clock pendulum�are regular and easily explained� whileothers�such as the shifting patterns of a waterfall�are irregular andinitially appear to defy any rule�

The mathematician Henri Poincar�e �� was the �rst to appreci�ate the true source of the problem� the di�culty lay not in the rules�but rather in specifying the initial conditions� At the beginning of this

�

Introduction

century� in his essay Science and Method� Poincar�e wrote�

A very small cause which escapes our notice determinesa considerable e�ect that we cannot fail to see� and thenwe say that that e�ect is due to chance� If we knew ex�actly the laws of nature and the situation of the universe atthe initial moment� we could predict exactly the situationof that same universe at a succeeding moment� But evenif it were the case that the natural laws had no longer anysecret for us� we could still only know the initial situationapproximately� If that enabled us to predict the succeed�ing situation with the same approximation� that is all werequire� and we should say that the phenomenon had beenpredicted� that it is governed by laws� But it is not alwaysso� it may happen that small di�erences in the initial con�ditions produce very great ones in the �nal phenomena� Asmall error in the former will produce an enormous error inthe latter� Prediction becomes impossible� and we have thefortuitous phenomenon�

Poincar�e�s discovery of sensitive dependence on initial conditions inwhat are now termed chaotic dynamical systems has only been fullyappreciated by the larger scienti�c community during the past threedecades� Mathematicians� physicists� chemists� biologists� engineers�meteorologists�indeed� individuals from all �elds have� with the helpof computer simulations and new experiments� discovered for them�selves the cornucopia of chaotic phenomena existing in the simplestnonlinear systems�

Before we proceed� we should distinguish nonlinear dynamics fromdynamical systems theory�� The latter is a well�de�ned branch ofmathematics� while nonlinear dynamics is an interdisciplinary �eld thatdraws on all the sciences� especially mathematics and the physical sci�ences�

�For an outline of the mathematical theory of dynamical systems see D� V�Anosov� I� U� Bronshtein� S�Kh� Aranson� and V� Z� Grines� Smooth dynamicalsystems� in Encyclopaedia of Mathematical Sciences� Vol� �� edited by D� V� Anosovand V� I� Arnold �Springer�Verlag� New York� ��

What is nonlinear dynamics� �

Scientists in all �elds are united by their need to solve nonlinearequations� and each di�erent discipline has made valuable contribu�tions to the analysis of nonlinear systems� A meteorologist discov�ered the �rst strange attractor in an attempt to understand the un�predictability of the weather�� A biologist promoted the study of thequadratic map in an attempt to understand population dynamics�� Andengineers� computer scientists� and applied mathematicians gave us awealth of problems along with the computers and programs neededto bring nonlinear systems alive on our computer screens� Nonlineardynamics is interdisciplinary� and nonlinear dynamicists rely on theircolleagues throughout all the sciences�

To de�ne a nonlinear dynamical system we �rst look at an exampleof a linear dynamical system� A linear dynamical system is one inwhich the dynamic rule is linearly proportional to the system variables�Linear systems can be analyzed by breaking the problem into piecesand then adding these pieces together to build a complete solution�For example� consider the second�order linear di�erential equation

d�x

dt�� x�

The dynamical system de�ned by this di�erential equation is linear be�cause all the terms are linear functions of x� The second derivative of x�the acceleration� is proportional to �x� To solve this linear di�erentialequation we must �nd some function x�t� with the following property�the second derivative of x �with respect to the independent variable t�is equal to �x� Two possible solutions immediately come to mind�

x��t� � sin�t� and x��t� � cos�t��

sinced�

dt�x��t� � � sin�t� � �x��t�

andd�

dt�x��t� � � cos�t� � �x��t��

�E� N� Lorenz� Deterministic nonperiodic �ow� J� Atmos� Sci� �� R� M� May� Simple mathematical models with very complicated dynamics� Na�

ture ��

Introduction

that is� both x� and x� satisfy the linear di�erential equation� Becausethe di�erential equation is linear� the sum of these two solutions de�nedby

x�t� � x��t� � x��t�

is also a solution�� This can be veri�ed by calculating

d�

dt�x�t� �

d�

dt�x��t� �

d�

dt�x��t�

� ��x��t� � x��t��

� �x�t��

Any number of solutions can be added together in this way to form anew solution� this property of linear di�erential equations is called theprinciple of superposition� It is the cornerstone from which all lineartheory is built�

Now let�s see what happens when we apply the same method toa nonlinear system� For example� consider the second�order nonlineardi�erential equation

d�x

dt�� x��

Let�s assume we can �nd two di�erent solutions to this nonlinear dif�ferential equation� which we will again call x��t� and x��t�� A quickcalculation�

d�x

dt��

d�x�dt�

�d�x�dt�

� ��x�� x��

�� x�� x�� x�x��

� ��x� � x��

� �x��shows that the solutions of a nonlinear equation cannot usually beadded together to build a larger solution because of the �cross�terms �x�x�� The principle of superposition fails to hold for nonlinear sys�tems�

�The full de�nition of a linear system also requires that the sum of scalar productsx�t� � a�x��t� � a�x��t�� where a� and a� are constants� is also a solution�

What is nonlinear dynamics� �

Traditionally� a di�erential equation is �solved by �nding a func�tion that satis�es the di�erential equation� A trajectory is then de�termined by starting the solution with a particular initial condition�For example� if we want to predict the position of a comet ten yearsfrom now we need to measure its current position and velocity� writedown the di�erential equation for its motion� and then integrate thedi�erential equation starting from the measured initial condition� Thetraditional view of a solution thus centers on �nding an individual orbitor trajectory� That is� given the initial condition and the rule� we areasked to predict the future position of the comet� Before Poincar�e�swork it was thought that a nonlinear system would always have a so�lution� we just needed to be clever enough to �nd it�

Poincar�e�s discovery of chaotic behavior in the three�body problemshowed that such a view is wrong� No matter how clever we are we won�tbe able to write down the equations that solve many nonlinear systems�This is not wholly unexpected� After all� in a �bounded� closed formsolution we might expect that any small change in initial conditionsshould produce a proportional change in the predicted trajectories� Buta chaotic system can produce large di�erences in the long�term trajec�tories even when two initial conditions are close� Poincar�e realized thefull implications of this simple discovery� and he immediately rede�nedthe notion of a �solution to a di�erential equation�

Poincar�e was less interested in an individual orbit than in all pos�sible orbits� He shifted the emphasis from a local solution�knowingthe exact motion of an individual trajectory�to a global solution�knowing the qualitative behavior of all possible trajectories for a givenclass of systems� In our comet example� a qualitative solution for thedi�erential equation governing the comet�s trajectory might appear eas�ier to achieve since it would not require us to integrate the equations ofmotion to �nd the exact future position of the comet� The qualitativesolution is often di�cult to completely specify� though� because it re�quires a global view of the dynamics� that is� the possible examinationof a large number of related systems�

Finding individual solutions is the traditional approach to solv�ing a di�erential equation� In contrast� recurrence is a key theme inPoincar�e�s quest for the qualitative solution of a di�erential equation�To understand the recurrence properties of a dynamical system� we

� Introduction

need to know what regions of space are visited and how often the orbitreturns to those regions� We can seek to statistically characterize howoften a region of space is visited� this leads to the so�called ergodic�

theory of dynamical systems� Additionally� we can try to understandthe geometric transformations undergone by a group of trajectories�this leads to the so�called topological theory of dynamical systems em�phasized in this book�

There are many di�erent levels of recurrence� For instance� thecomet could crash into a planet� After that nothing much happens�unless you�re a dinosaur�� Another possibility is that the comet couldgo into an orbit about a star and from then on follow a periodic motion�In this case the comet will always return to the same points along theorbit� The recurrence is strictly periodic and easily predicted� But thereare other possibilities� In particular� the comet could follow a chaoticpath exhibiting a complex recurrence pattern� visiting and revisitingdi�erent regions of space in an erratic manner�

To summarize� Poincar�e advocated the qualitative study of di�er�ential equations� We may lose sight of some speci�c details about anyindividual trajectory� but we want to sketch out the patterns formedby a large collection of di�erent trajectories from related systems� Thisglobal view is motivated by the fact that it is nonsensical to studythe orbit of a single trajectory in a chaotic dynamical system� To un�derstand the motions that surround us� which are largely governed bynonlinear laws and interactions� requires the development of new qual�itative techniques for analyzing the motions in nonlinear dynamicalsystems�

What is in this book�

This book introduces qualitative �bifurcation theory� symbolic dynam�ics� etc�� and quantitative �perturbation theory� numerical methods�etc�� methods that can be used in analyzing a nonlinear dynamicalsystem� Further� it provides a basic set of experimental techniques re�quired to set up and observe nonlinear phenomena in the laboratory�

�V� I� Arnold and A� Avez� Ergodic problems of classical mechanics �W� A�Benjamin� New York� ��

What is in this book� �

Some of these methods go back to Poincar�e�s original work in the lastcentury� while many others� such as computer simulations� are morerecent� A wide assortment of seemingly disparate techniques is usedin the analysis of the humblest nonlinear dynamical system� Whereaslinear theory resembles an edi�ce built upon the principle of superposi�tion� nonlinear theory more closely resembles a toolbox� in which manyof the essential tools have been borrowed from the laboratories of manydi�erent friends�

To paraphrase Tolstoy� all linear systems resemble one another� buteach nonlinear system is nonlinear in its own way� Therefore� on our�rst encounter with a new nonlinear system we need to search our tool�box for the proper diagnostic tools �power spectra� fractal dimensions�periodic orbit extraction� etc�� so that we can identify and characterizethe nonlinear and chaotic structures� And next� we need to analyze andunfold these structures with the help of additional tools and methods�computer simulations� simpli�ed geometric models� universality the�ory� etc�� to �nd those properties that are common to a large class ofnonlinear systems�

The tools in our toolbox are collected from scientists in a widerange of disciplines� mathematics� physics� computing� engineering�economics� and so on� Each discipline has developed a di�erent di�alect� and sometimes even a new language� in which to discuss nonlinearproblems� And so one challenge facing a new researcher in nonlineardynamics is to develop some �uency in these di�erent dialects�

It is typical in many �elds� from cabinet making to mathematics� tointroduce the tyro �rst to the tedious elements and next� when thesebasic elements are mastered� to introduce her to the joys of the celestialwhole� The cabinet maker �rst learns to sweep and sand and measureand hold� Likewise� the aspiring mathematician learns how to expresslimits� take derivatives� calculate integrals� and make substitutions�

All too often the consequence of an introduction through tedium isthe destruction of the inquisitive� eager spirit of inquiry� We hope todiminish this tedium by tying the eagerness of the student to projectsand experiments that illustrate nonlinear concepts� In a few words�we want the student to get her hands dirty� Then� with maturity andinsight born from �rsthand experience� she will be ready to �ll in thebig picture with rigorous de�nitions and more comprehensive study�

� Introduction

The study of nonlinear dynamics is eclectic� selecting what appearsto be most useful among various and diverse theories and methods� Thisposes an additional challenge since the skills required for research innonlinear dynamics can range from a knowledge of some sophisticatedmathematics �hyperbolicity theory� to a detailed understanding of thenuts and bolts of computer hardware �binary arithmetic� digitizers��Nonlinear dynamics is not a tidy subject� but it is vital�

The common thread throughout all nonlinear dynamics is the needand desire to solve nonlinear problems� by hook or by crook� In�deed� many tools in the nonlinear dynamicist�s toolbox originally werecrafted as the solution to a speci�c experimental problem or applica�tion� Only after solving many individual nonlinear problems did thecommon threads and structures slowly emerge�

The �rst half of this book� Chapters �� and �� uses a similar exper�imental approach to nonlinear dynamics and is suitable for an advancedundergraduate course� Our approach seeks to develop and motivate thestudy of nonlinear dynamics through the detailed analysis of a few spe�ci�c systems that can be realized by desktop experiments� the perioddoubling route to chaos in a bouncing ball� the symbolic analysis ofthe quadratic map� and the quasiperiodic and chaotic vibrations of astring� The detailed analysis of these examples develops intuition for�and motivates the study of�nonlinear systems� In addition� analysisand simulation of these elementary examples provide ample practicewith the tools in our nonlinear dynamics toolbox� The second half ofthe book� Chapters and �� provides a more formal treatment of thetheory illustrated in the desktop experiments� thereby setting the stagefor an advanced or graduate level course in nonlinear dynamics� Inaddition� Chapters and � provide the more advanced student or re�searcher with a concise introduction to the mathematical foundationsof nonlinear dynamics� as well as introducing the topological approachtoward the analysis of chaos�

The pedagogical approach also di�ers between the �rst and secondhalf of the book� The �rst half tends to introduce new concepts andvocabulary through usage� example� and repeated exposure� We believethis method is pedagogically sound for a �rst course and is reminiscentof teaching methods found in an intensive foreign language course� Inthe second half of the book examples tend to follow formal de�nitions�

Some Terminology �

as is more common in a traditional mathematics course�Although a linear reading of the �rst three chapters of this text is

the most useful� it is also possible to pick and choose material from dif�ferent sections to suit speci�c course needs� A mixture of mathematical�theoretical� experimental� and computational methods are employed inthe �rst three chapters� The following table provides a rough road mapto the type of material found in each section�

Mathematical Theoretical Experimental Computational��

�� Mathematica usage

�� Appendix B Appendices

A�C�D�E�F�G

The mathematical sections present the core material for a mathe�matical dynamical systems course� The experimental sections presentthe core experimental techniques used for laboratory work with a low�dimensional chaotic system� For instance� those interested in experi�mental techniques could turn directly to the experimental sections forthe information they seek�

Some Terminology�

Maps� Flows� and Fractals

In this section we heuristically introduce some of the basic terminologyused in nonlinear dynamics� This material should be read quickly�as background to the rest of the book� It might also be helpful toread Appendix H� Historical Comments� before delving into the moretechnical material� For precise de�nitions of the mathematical notionsintroduced in this section we highly recommend V� I� Arnold�s masterfulintroduction to the theory of ordinary di�erential equations�� The goalin this section is to begin using the vocabulary of nonlinear dynamicseven before this vocabulary is precisely de�ned�

�V� I� Arnold� Ordinary di�erential equations �MIT Press� Cambridge� MA�� Also see D� K� Arrowsmith and C� M� Place� Ordinary di�erential equations�Chapman and Hall� New York� ��

�� Introduction

Figure �� a� The surfaces of the three�dimensional objects are two�dimensional manifolds� �b� Examples of objects that are not manifolds�

Flows and Maps

A geometric formulation of the theory of di�erential equations says thata di�erential equation is a vector �eld on a manifold� To understandthis de�nition we present an informal description of a manifold and avector �eld�

A manifold is any smooth geometric space �line� surface� solid�� Thesmoothness condition ensures that the manifold cannot have any sharpedges� An example of a one�dimensional manifold is an in�nite straightline� A di�erent one�dimensional manifold is a circle� Examples oftwo�dimensional manifolds are the surface of an in�nite cylinder� thesurface of a sphere� the surface of a torus� and the unbounded real plane�Fig� �� Three�dimensional manifolds are harder to visualize� Thesimplest example of a three�dimensional manifold is unbounded three�space� R�� The surface of a cone is an example of a two�dimensionalsurface that is not a manifold� At the apex of the cone is a sharp point�which violates the smoothness condition for a manifold� Manifolds areuseful geometric objects because the smoothness condition ensures thata local coordinate system can be erected at each and every point onthe manifold�

A vector �eld is a rule that smoothly assigns a vector �a directed

Some Terminology ��

line segment� to each point of a manifold� This rule is often writtenas a system of �rst�order di�erential equations� To see how this works�consider again the linear di�erential equation

d�x

dt�� x�

Let us rewrite this second�order di�erential equation as a system oftwo �rst�order di�erential equations by introducing the new variable v�velocity� de�ned by dx�dt � v� so that

dx

dt� v�

dv

dt� �x�

The manifold in this example is the real plane� R�� which consists ofthe ordered pair of variables �x� v�� Each point in this plane repre�sents an individual state� or possible initial condition� of the system�And the collection of all possible states is called the phase space of thesystem� A process is said to be deterministic if both its future andpast states are uniquely determined by its present state� A process iscalled semideterministic when only the future state� but not the past�is uniquely determined by the present state� Not all physical systemsare deterministic� as the bouncing ball system �which is only semide�terministic� of Chapter � demonstrates� Nevertheless� full determinismis commonly assumed in the classical scienti�c world view�

A system of �rst�order di�erential equations assigns to each pointof the manifold a vector� thereby forming a vector �eld on the manifold�Fig� �� In our example each point of the phase plane �x� v� getsassigned a vector �v��x�� which forms rings of arrows about the origin�Fig� �� A solution to a di�erential equation is called a trajectory

or an integral curve� since it results from �integrating the di�erentialequations of motion� An individual vector in the vector �eld deter�mines how the solution behaves locally� It tells the trajectory to �gothataway� The collection of all solutions� or integral curves� is calledthe �ow �Fig� ��

When analyzing a system of di�erential equations it is importantto present both the equations and the manifold on which the equations

� Introduction

Figure �� Examples of vector �elds on di�erent manifolds�

Figure �� Vector �eld and �ow for a linear di�erential equation�

are speci�ed� It is often possible to simplify our analysis by transferringthe vector �eld to a di�erent manifold� thereby changing the topologyof the phase space �see section �� Topology is a kind of geometrywhich studies those properties of a space that are unchanged undera reversible continuous transformation� It is sometimes called rubbersheet geometry� A basketball and a football are identical to a topologist�They are both �topological spheres� However� a torus and a sphereare di�erent topological spaces as you cannot push or pull a sphere intoa torus without �rst cutting up the sphere� Topology is also de�nedas the study of closeness within neighborhoods� Topological spacescan be analyzed by studying which points are �close to or �in theneighborhood of other points� Consider the line segment between �


Figure �� Typical motions in a planar vector �eld� �a� source� �b�sink� �c� saddle� and �d� limit cycle�

and �� The endpoints � and � are far away� they aren�t neighbors� Butif we glue the ends together to form a circle� then the endpoints becomeidentical� and the points around � and � have a new set of neighbors�

In its grandest form� Poincar�e�s program to study the qualitative be�havior of ordinary di�erential equations would require us to analyze thegeneric dynamics of all vector �elds on all manifolds� We are nowherenear achieving this goal yet� Poincar�e was inspired to carry out thisprogram by his success with the Swedish mathematician Ivar Bendixsonin analyzing all typical behavior for di�erential equations in the plane�As illustrated in Figure �� the Poincar�e�Bendixson Theorem says thattypically no more than four kinds of motion are found in a planar vector�eld� those of a source� sink� saddle� and limit cycle� In particular� nochaotic motion is possible in time�independent planar vector �elds� Toget chaotic motion in a system of di�erential equations one needs threedimensions� that is� a vector �eld on a three�dimensional manifold�

� Introduction

The asymptotic motions �t�� limit sets� of a �ow are character�ized by four general types of behavior� In order of increasing complexitythese are equilibrium points� periodic solutions� quasiperiodic solutions�and chaos�

An equilibrium point of a �ow is a constant� time�independent so�lution� The equilibrium solutions are located where the vector �eldvanishes� The source in Figure ��a� is an example of an unstableequilibrium solution� Trajectories near to the source move away fromthe source as time goes by� The sink in Figure ��b� is an example ofa stable equilibrium solution� Trajectories near the sink tend toward itas time goes by�

A periodic solution of a �ow is a time�dependent trajectory thatprecisely returns to itself in a time T � called the period� A periodictrajectory is a closed curve� Like an equilibrium point� a periodic tra�jectory can be stable or unstable� depending on whether nearby trajec�tories tend toward or away from the periodic cycle� One illustration ofa stable periodic trajectory is the limit cycle shown in Figure ��d��A quasiperiodic solution is one formed from the sum of periodic solu�tions with incommensurate periods� Two periods are incommensurateif their ratio is irrational� The ability to create and control periodicand quasiperiodic cycles is essential to modern society� clocks� elec�tronic oscillators� pacemakers� and so on�

An asymptotic motion that is not an equilibrium point� periodic�or quasiperiodic is often called chaotic� This catchall use of the termchaos is not very speci�c� but it is practical� Additionally� we requirethat a chaotic motion is a bounded asymptotic solution that possessessensitive dependence on initial conditions� two trajectories that beginarbitrarily close to one another on the chaotic limit set start to divergeso quickly that they become� for all practical purposes� uncorrelated�Simply put� a chaotic system is a deterministic system that exhibitsrandom �uncorrelated� behavior� This apparent random behavior ina deterministic system is illustrated in the bouncing ball system �seesection �� A more rigorous de�nition of chaos is presented in section��

All of the stable asymptotic motions �or limit sets� just described�e�g�� sinks� stable limit cycles�� are examples of attractors� The unsta�ble limit sets �e�g�� sources� are examples of repellers� The term strange


attractor �strange repeller� is used to describe attracting �repelling�limit sets that are chaotic� We will get our �rst look at a strange at�tractor in a physical system when we study the bouncing ball systemin Chapter ��

Maps are the discrete time analogs of �ows� While �ows are speci�edby di�erential equations� maps are speci�ed by di�erence equations�A point on a trajectory of a �ow is indicated by a real parameter t�which we think of as the time� Similarly� a point in the orbit of amap is indexed by an integer subscript n� which we think of as thediscrete analog of time� Maps and �ows will be the two primary typesof dynamical systems studied in this book�

Maps �di�erence equations� are easier to solve numerically than�ows �di�erential equations�� Therefore� many of the earliest numericalstudies of chaos began by studying maps� A famous map exhibitingchaos studied by the French astronomer Michel H�enon �� nowknown as the H�enon map� is

xn�� x�n � �yn�

yn�� xn�

where n is an integer index for this pair of nonlinear coupled di�er�ence equations� with � � �� and � � �� being the parameter valuesmost commonly studied� The H�enon map carries a point in the plane��x�� y�� to some new point� �x�� y�� An orbit of a map is the sequenceof points generated by some initial condition of a map� For instance� ifwe start the H�enon map at the point �x�� y�� we �nd thatthe orbit for this pair of initial conditions is

x� � ��

y� � ��

x� � ��

y� � ��

and so on to generate �x�� y�� x�� y�� etc� Unlike planar di�eren�tial equations� this two�dimensional di�erence equation can generatechaotic orbits� In fact� in Chapter we will study a one�dimensional

�� Introduction

Figure �� A Poincar�e map for a three�dimensional �ow with a two�dimensional cross section�

di�erence equation called the quadratic map� which can also generatechaotic orbits�

The H�enon map is an example of a di�eomorphism of a manifold �inthis case the manifold is the plane R�� A map is a homeomorphism ifit is bijective �one�to�one and onto�� continuous� and has a continuousinverse� A di�eomorphism is a di�erentiable homeomorphism� A mapwith an inverse is called invertible� A map without an inverse is callednoninvertible�

Maps exhibit similar types of asymptotic behavior as �ows� equi�librium points� periodic orbits� quasiperiodic orbits� and chaotic orbits�There are many similarities and a few important di�erences betweenthe theory and language describing the dynamics of maps and �ows�For a detailed comparison of these two theories see Arrowsmith andPlace� An introduction to dynamical systems�

The dynamics of �ows and maps are closely related� The study ofa �ow can often be replaced by the study of a map� One prescriptionfor doing this is the so�called Poincar�e map of a �ow� As illustratedin Figure �� a cross section of the �ow is obtained by choosing somesurface transverse to the �ow� A cross section for a three�dimensional


�ow is shown in the illustration and is obtained by choosing the x�yplane� �x� y� z � �� The �ow de�nes a map of this cross section to itself�and this map is an example of a Poincar�e map �also called a �rst returnmap�� A trajectory of the �ow carries a point �x�t�� y�t�� into a newpoint �x�t�� y�t�� And this in turn goes to the point �x�t�� y�t��In this way the �ow generates a map of a portion of the plane� and anorbit of this map consists of the sequence of points

�x�� y�� x�t�� y�t�� z � �� z � ��

�x�� y�� x�t�� y�t�� z � �� z � ��

�x�� y�� x�t�� y�t�� z � �� z � ��

and so on�

There is another reason for studying maps� To quote Steve Smaleon the �di�eomorphism problem� �

�T�here is a second and more important reason for study�ing the di�eomorphism problem �besides its great naturalbeauty�� That is� the same phenomena and problems ofthe qualitative theory of ordinary di�erential equations arepresent in their simplest form in the di�eomorphism prob�lem� Having �rst found theorems in the di�eomorphismcase� it is usually a secondary task to translate the resultsback into the di�erential equations framework�

The �rst dynamical system we will study� the bouncing ball system�illustrates more fully the close connection between maps and �ows�

Binary Arithmetic

Before turning to nonlinear dynamics proper� we need some familiaritywith the binary number system� Consider the problem of converting afraction between � and � �x� � �� written in decimal �base �� to abinary number �base �� The formal expansion for a binary fraction in

�S� Smale� Di�erential dynamical systems� Bull� Am� Math� Soc� ��

�� Introduction

powers of is

x� ��

��

��

��

��

� � � � �decimal�

��

��

��

��

��

� � � �� binary�

where �i � f�� g� The goal is to �nd the �i�s for a given decimalfraction� For example� if x� � �� then

x� ��

��

�

�

�

�

��

�

��

�

��

� �� binary��

The general procedure for converting a decimal fraction less thanone to binary is based on repeated doublings in which the ones or�carry digit is used for the �i�s� This is illustrated in the followingcalculation for x� � ��

� ��

� ��

� ��

� ��

� ��

� ��

sox� � �� binary��

Fractals

Nature abounds with intricate fragmented shapes and structures� in�cluding coastlines� clouds� lightning bolts� and snow�akes� In ��Benoit Mandelbrot coined the term fractal to describe such irregularshapes� The essential feature of a fractal is the existence of a similar


Figure �� Construction of Cantor�s middle thirds set�

structure at all length scales� That is� a fractal object has the propertythat a small part resembles a larger part� which in turn resembles thewhole object� Technically� this property is called self�similarity and istheoretically described in terms of a scaling relation�

Chaotic dynamical systems almost inevitably give rise to fractals�And fractal analysis is often useful in describing the geometric structureof a chaotic dynamical system� In particular� fractal objects can beassigned one or more fractal dimensions� which are often fractional�that is� they are not integer dimensions�

To see how this works� consider a Cantor set� which is de�ned re�cursively as follows �Fig� �� At the zeroth level the construction ofthe Cantor set begins with the unit interval� that is� all points on theline between � and �� The �rst level is obtained from the zeroth levelby deleting all points that lie in the �middle third� that is� all pointsbetween �� and �� The second level is obtained from the �rst levelby deleting the middle third of each interval at the �rst level� that is�all points from �� to �� and �� to �� In general� the next levelis obtained from the previous level by deleting the middle third of allintervals at the previous level� This process continues forever� and theresult is a collection of points that are tenuously cut out from the unitinterval� At the nth level the set consists of n segments� each of which

� Introduction

has length ln � ��n� so that the length of the Cantor set is

limn��

n�

�

�

�n� ��

In the ��s the mathematician Hausdor� developed another wayto �measure the size of a set� He suggested that we should examinethe number of small intervals� N�� needed to �cover the set at a scale�� The measure of the set is calculated from

lim��

N��

�

�

�df

�

An example of a fractal dimension is obtained by inverting this equa�tion�

df � lim��

�� lnN��

ln��

��A �

Returning to the Cantor set� we see that at the nth level the length of

the covering intervals are � ��

�n� and the number of intervals needed

to cover all segments at the nth level is N�� n� Taking the limitsn�� we �nd

df � lim��

�� lnN��

ln��

��A � lim

n��

ln n

ln �n�

ln

ln � ��

The middle�thirds Cantor set has a simple scaling relation� becausethe factor �� is all that goes into determining the successive levels� Afurther elementary discussion of the middle�thirds Cantor set is foundin Devaney�s Chaos� fractals� and dynamics� In general� fractals arisingin a chaotic dynamical system have a far more complex scaling relation�usually involving a range of scales that can depend on their locationwithin the set� Such fractals are called multifractals�

References and Notes �

References and Notes

Some popular accounts of the history and folklore of nonlinear dynam�ics and chaos include�

A� Fisher� Chaos� The ultimate asymmetry� MOSAIC �� pp� �� Jan�

uary�February ��

J� P� Crutch�eld� J� D� Farmer� N� H� Packard� and R� S� Shaw� Chaos� Sci� Am�

�� pp� ��

J� Gleick� Chaos� Making a new science �Viking� New York� ��

I� Stewart� Does god play dice� The mathematics of chaos �Basil Blackwell� Cam�

bridge� MA� ��

The following books are helpful references for some of the material cov�ered in this book�

R� Abraham and C� Shaw� Dynamics�The geometry of behavior� Vol� �� Aerial

Press� Santa Cruz� CA� ��

D� K� Arrowsmith and C� M� Place� An introduction to dynamical systems �Cam�

bridge University Press� New York� ��

G� L� Baker and J� P� Gollub� Chaotic dynamics �Cambridge University Press� New

York� ��

P� Berg�e� Y� Pomeau� and C� Vidal� Order within chaos �John Wiley� New York�

��

R� L� Devaney� An introduction to chaotic dynamical systems� second ed�

�Addison�Wesley� New York� ��

E� A� Jackson� Perspectives of nonlinear dynamics� Vol� �� Cambridge University

Press� New York� ��

F� Moon� Chaotic vibrations �John Wiley� New York� ��

J� Thompson and H� Stewart� Nonlinear dynamics and chaos �John Wiley� New

York� ��

S� Rasband� Chaotic dynamics of nonlinear systems �John Wiley� New York� ��

Introduction

S�Wiggins� Introduction to applied nonlinear dynamical systems and chaos �Springer�

Verlag� New York� ��

These review articles provide a quick introduction to the current re�search problems and methods of nonlinear dynamics�

J��P� Eckmann� Roads to turbulence in dissipative dynamical systems� Rev� Mod�

Phys� �� pp� ��

J��P� Eckmann and D� Ruelle� Ergodic theory of chaos and strange attractors� Rev�

Mod� Phys� �� pp� ��

C� Grebogi� E� Ott� and J� Yorke� Chaos� strange attractors� and fractal basin

boundaries in nonlinear dynamics� Science �� pp� ��

E� Ott� Strange attractors and chaotic motions of dynamical systems� Rev� Mod�

Phys� �� pp� � ��

T� Parker and L� Chua� Chaos� A tutorial for engineers� Proc� IEEE �� pp�

��

R� Shaw� Strange attractors� chaotic behavior� and information �ow� Z� Naturforsch�

��a� pp� ��

Advanced theoretical results are described in the following books�

V� I� Arnold� Geometrical methods in the theory of ordinary di�erential equations�

second ed� �Springer�Verlag� New York� ��

J� Guckenheimer and P� Holmes� Nonlinear oscillations� dynamical systems� and

bifurcations of vector �elds� second printing �Springer�Verlag� New York� ��

D� Ruelle� Elements of di�erentiable dynamics and bifurcation theory �Academic

Press� New York� ��

S� Wiggins� Global bifurcations and chaos �Springer�Verlag� New York� ��

Chapter �

Bouncing Ball

�� Introduction

Consider the motion of a ball bouncing on a periodically vibrating ta�ble� The bouncing ball system is illustrated in Figure �� and arisesquite naturally as a model problem in several engineering applications�Examples include the generation and control of noise in machinery suchas jackhammers� the transportation and separation of granular solidssuch as rice� and the transportation of components in automatic assem�bly devices� which commonly employ oscillating tracks� These vibratingtracks are used to transport parts much like a conveyor belt ��

Assume that the ball�s motion is con�ned to the vertical directionand that� between impacts� the ball�s height is determined by Newton�slaws for the motion of a particle in a constant gravitational �eld� Anonlinear force is applied to the ball when it hits the table� At impact�the ball�s velocity suddenly reverses from the downward to the upwarddirection �Fig� ��

The bouncing ball system is easy to study experimentally �� Oneexperimental realization of the system consists of little more than a ballbearing and a periodically driven loudspeaker with a concave opticallens attached to its surface� The ball bearing will rattle on top ofthis lens when the speaker�s vibration amplitude is large enough� Thecurvature of the lens is chosen so as to help focus the ball�s motion inthe vertical direction�

�

CHAPTER �� BOUNCING BALL

Figure �� Ball bouncing on an oscillating table�

Impacts between the ball and lens can be detected by listening tothe rhythmic clicking patterns produced when the ball hits the lens� Apiezoelectric �lm� which generates a small current every time a stressis applied� is fastened to the lens and acts as an impact detector� Thepiezoelectric �lm generates a voltage spike at each impact� This spikeis monitored on an oscilloscope� thus providing a visual representationof the ball�s motion� A schematic of the bouncing ball machine isshown in Figure �� More details about its construction are providedin reference ��

The ball�s motion can be described in several equivalent ways� Thesimplest representation is to plot the ball�s height and the table�s height�measured from the ground� as a function of time� Between impacts� thegraph of the ball�s vertical displacement follows a parabolic trajectoryas illustrated in Figure ��a�� The table�s vertical displacement variessinusoidally� If the ball�s height is recorded at discrete time steps�

fx�t�� x�t�� x�ti�� x�tn�g� ��

then we have a time series of the ball�s height where x�ti� is the heightof the ball at time ti�

Another view of the ball�s motion is obtained by plotting the ball�sheight on the vertical axis� and the ball�s velocity on the horizontal axis�The plot shown in Figure ��b� is essentially a phase space represen�tation of the ball�s motion� Since the ball�s height is bounded� so isthe ball�s velocity� Thus the phase space picture gives us a description

�� INTRODUCTION �

Figure �� Schematic for a bouncing ball machine�

of the ball�s motion that is more compact than that given by a plot ofthe time series� Additionally� the sudden reversal in the ball�s velocityat impact �from positive to negative� is easy to see at the bottom ofFigure ��b�� Between impacts� the graph again follows a parabolictrajectory�

Yet another representation of the ball�s motion is a plot of the ball�svelocity and the table�s forcing phase at each impact� This is the so�called impact map and is shown in Figure ��c�� The impact map goesto a single point for the simple periodic trajectory shown in Figure ��The vertical coordinate of this point is the ball�s velocity at impact andthe horizontal coordinate is the table�s forcing phase� This phase� ��is de�ned as the product of the table�s angular frequency� �� and thetime� t�

� � �t� � � �T� ��

where T is the forcing period� Since the table�s motion is �periodicin the phase variable �� we usually consider the phase mod � whichmeans we divide � by and take the remainder�

� mod � remainder��

� CHAPTER �� BOUNCING BALL

Figure �� Simple periodic orbit of a bouncing ball� �a� height vs� time��b� phase space �height vs� velocity�� c� impact map �velocity andforcing phase at impact�� Generated by the Bouncing Ball program��

A time series� phase space� and impact map plot are presented to�gether in Figure �� for a complex motion in the bouncing ball system�This particular motion is an example of a nonperiodic orbit known asa strange attractor� The impact map� Figure ��c�� is a compact andabstract representation of the motion� In this particular example wesee that the ball never settles down to a periodic motion� in which itwould impact at only a few points� but rather explores a wide rangeof phases and velocities� We will say much more about these strangetrajectories throughout this book� but right now we turn to the detailsof modeling the dynamics of a bouncing ball�

�� MODEL �

Figure �� Strange orbit of a bouncing ball� �a� height vs� time��b� phase space� �c� impact map� �Generated by the Bouncing Ballprogram��

�� Model

To model the bouncing ball system we assume that the table�s massis much greater than the ball�s mass and the impact between the balland the table is instantaneous� These assumptions are realistic for theexperimental system described in the previous section and simply meanthat the table�s motion is not a�ected by the collisions� The collisionsare usually inelastic� that is� a little energy is lost at each impact� If noenergy is lost then the collisions are called elastic� We will examine bothcases in this book� the case in which energy is dissipated �dissipative�and the case in which energy is conserved �conservative��


�� Stationary Table

First� though� we must �gure out how the ball�s velocity changes ateach impact� Consider two di�erent reference frames� the ball�s motionas seen from the ground �the ground�s reference frame� and the ball�smotion as seen from the table �the table�s reference frame�� Begin byconsidering the simple case where the table is stationary and the tworeference frames are identical� As we will show shortly� understandingthe stationary case will solve the nonstationary case�

Let v�k be the ball�s velocity right before the kth impact� and letvk be the ball�s velocity right after the kth impact� The prime nota�tion indicates a velocity immediately before an impact� If the tableis stationary and the collisions are elastic� then vk � �v�k� the ballreverses direction but does not change speed since there is no energyloss� If the collisions are inelastic and the table is stationary� then theball�s speed will be reduced after the collision because energy is lost�vk � ��v�k �� where � is the coe�cient of restitution� Theconstant � is a measure of the energy loss at each impact� If � � ��the system is conservative and the collisions are elastic� The coe�cientof restitution is strictly less than one for inelastic collisions��

�� Impact Relation for the Oscillating Table

When the table is in motion� the ball�s velocity immediately after animpact will have an additional term due to the kick from the table� Tocalculate the change in the ball�s velocity� imagine the motion of theball from the table�s perspective� The key observation is that in thetable�s reference frame the table is always stationary� The ball� however�appears to have an additional velocity which is equal to the oppositeof the table�s velocity in the ground�s reference frame� Therefore� tocalculate the ball�s change in velocity we can calculate the change invelocity in the table�s reference frame and then add the table�s velocityto get the ball�s velocity in the ground�s reference frame� In Figure ��we show the motion of the ball and the table in both the ground�s andthe table�s reference frames�

�The coe�cient of restitution � is called the damping coe�cient in the BouncingBall program�

�� MODEL �

Figure �� Motion of the ball in the reference frame of the ground �a�and the table �b��

Let uk be the table�s velocity in the ground�s reference frame� Fur�ther� let !v�k and !vk be the velocity in the table�s reference frame imme�diately before and after the kth impact� respectively� The bar denotesmeasurements in the table�s reference frame� the unbarred coordinatesare measurements in the ground�s reference frame� Then� in the table�sreference frame�

!vk � ��!v�k� ��

since the table is always stationary� To �nd the ball�s velocity in theground�s reference frame we must add the table�s velocity to the ball�sapparent velocity�

vk � !vk � uk� v�k � !v�k � uk�

or equivalently�!vk � vk � uk� !v�k � v�k � uk� ��

Therefore� in the ground�s reference frame� equation �� becomes

vk � uk � ��v�k � uk��

when it is rewritten using equation �� Rewriting equation ��gives the velocity vk after the kth impact as

vk � �� uk � �v�k� ��

This last equation is known as the impact relation� It says the kickfrom the table contributes �� uk to the ball�s velocity�

�� CHAPTER �� BOUNCING BALL

�� The Equations of Motion� � Phase

and Velocity Maps

To determine the motion of the ball we must calculate the times� hencephases �from eq� �� when the ball and the table collide� An impactoccurs when the di�erence between the ball position and the table posi�tion is zero� Between impacts� the ball goes up and down according toNewton�s law for the motion of a projectile in a constant gravitational�eld of strength g� Since the motion between impacts is simple� we willpresent the motion of the ball in terms of an impact map� that is� somerule that takes as input the current values of the impact phase andimpact velocity and then generates the next impact phase and impactvelocity�

Let�

x�t� � xk � vk�t� tk��

g�t� tk�

� ��

be the ball�s position at time t after the kth impact� where xk is theposition at the kth impact and tk is the time of the kth impact� and let

s�t� � A�sin��t � ��

be the table�s position with an amplitude A� angular frequency �� andphase �� at t � �� We add one to the sine function to ensure that thetable�s amplitude is always positive� The di�erence in position betweenthe ball and table is

d�t� � x�t�� s�t��

which should always be a non�negative function since the ball is neverbelow the table� The �rst value at which d�t� � �� t tk� implicitlyde�nes the time of the next impact� Substituting equations �� and�� into equation �� and setting d�t� to zero yields

� � xk � vk�tk�� tk��

g�tk�� tk�

��A�sin��tk��

�For a discussion of the motion of a particle in a constant gravitational �eld seeany introductory physics text such as R� Weidner and R� Sells� Elementary Physics�Vol� � �Allyn and Bacon� Boston� �� pp� ��

�� MODEL ��

Equation �� can be rewritten in terms of the phase when the iden�ti�cation � � �t� �� is made between the phase variable and the timevariable� This leads to the implicit phase map of the form�

� � A�sin��k� � �� vk

��

��k�� k�

�

��

g�

�

��k�� k�

��A�sin��k��

where �k�� is the next � for which d�� In deriving equation�� we used the fact that the table position and the ball position areidentical at an impact� that is� xk � A�sin��k� � ��

An explicit velocity map is derived directly from the impact relation�equation �� as

vk�� A cos��tk�� vk � g�tk�� tk��

or� in the phase variable�

vk�� A cos��k�� vk � g

��

��k�� k�

��

noting that the table�s velocity is just the time derivative of the table�sposition� u�t� � s�t� � ds�dt � A� cos��t � �� and that� betweenimpacts� the ball is subject to the acceleration of gravity� so its velocityis given by vk � g�t � tk�� The overdot is Newton�s original notationdenoting di�erentiation with respect to time�

The implicit phase map �eq� �� and the explicit velocity map�eq� �� constitute the exact model for the bouncing ball system�The dynamics of the bouncing ball are easy to simulate on a computerusing these two equations� Unfortunately� the phase map is an implicitalgebraic equation for the variable �k�� that is� �k�� cannot be iso�lated from the other variables� To solve the phase function for �k�� anumerical algorithm is needed to locate the zeros of the phase func�tion �see Appendix A�� Still� this presents little problem for numericalsimulations� or even� as we shall see� for a good deal of analytical work�

�� Parameters

The parameters for the bouncing ball system should be determinedbefore we continue our analysis� The relevant parameters� with typical


Parameter Symbol Experimental values

Coe�cient of restitution � ��Table�s amplitude A �� cmTable�s period T �� sGravitational acceleration g �� cm�s�

Frequency f f � ��TAngular frequency � � � fNormalized acceleration � � � �� A�g

Table �� Reference values for the Bouncing Ball System�

experimental values� are listed in Table �� In an experimental system�the table�s frequency or the table�s amplitude of oscillation is easy toadjust with the function generator� The coe�cient of restitution canalso be varied by using balls composed of di�erent materials� Steelballs� for instance� are relatively hard and have a high coe�cient ofrestitution� Balls made from brass� glass� plastic� or wood are softerand tend to dissipate more energy at impact�

As we will show in the next section� the physical parameters listed inTable �� are related� By rescaling the variables� it is possible to showthat there are only two fundamental parameters in this model� Forour purposes we will take these to be �� the coe�cient of restitution�and a new parameter �� which is essentially proportional to A�� Theparameter � is� in essence� a normalized acceleration and it measuresthe violence with which the table oscillates up and down�

�� High Bounce Approximation

In the high bounce approximation we imagine that the table�s displace�ment amplitude is always small compared to the ball�s maximum height�This approximation is depicted in Figure �� where the ball�s trajectoryis perfectly symmetric about its midpoint� and therefore

v�k�� vk� ��

�� HIGH BOUNCE APPROXIMATION ��

Figure �� Symmetric orbit in the high bounce approximation�

The velocity of the ball between the kth and k � �st impacts is givenby

v�t� � vk � g�t� tk��

At the k � �st impact� the velocity is v�k�� and the time is tk�� so

v�k�� vk � g�tk�� tk��

Using equation �� and simplifying� we get

tk�� tk �

gvk� ��

which is the time map in the high bounce approximation�To �nd the velocity map in this approximation we begin with the

impact relation �eq� ��

vk�� uk�� v�k�� uk�� vk� ��

where the last equality follows from the high bounce approximation�equation �� The table�s velocity at the k � �st impact can bewritten as

uk�� A cos��tk��

� �A cos��tk � vk�g� � ��

when the time map� equation �� is used� Equations �� and�� give the velocity map in the high bounce approximation�

vk�� vk � �� A cos��tk � vk�g� � ��


The impact equations can be simpli�ed somewhat by changing tothe dimensionless quantities

� � �t� ��

� � �v�g� and ��

� � �� A�g� ��

which recasts the time map �eq� �� and the velocity map �eq�� into the explicit mapping form

f � f��

��k�� k � �k��k�� k � � cos��k � �k��

��

In the special case where � � �� this system of equations is known as thestandard map �� The subscripts of f�� explicitly show the dependenceof the map on the parameters � and �� The mapping equation ��is easy to solve on a computer� Given an initial condition �� themap explicitly generates the next impact phase and impact velocity asf�� and this in turn generates f�� f�� f �f�� etc�� where� in this notation� the superscript n infn indicates functional composition �see section �� Unlike the exactmodel� both the phase map and the velocity map are explicit equationsin the high bounce approximation�

The high bounce approximation shares many of the same qualita�tive properties of the exact model for the bouncing ball system� and itwill serve as the starting point for several analytic calculations� How�ever� for comparisons with experimental data� it is worthwhile to putthe extra e�ort into numerically solving the exact equations becausethe high bounce model fails in at least two major ways to model theactual physical system �� First� the high bounce model can generatesolutions that cannot possibly occur in the real system� These unphys�ical solutions occur for very small bounces at negative table velocities�where it is possible for the ball to be projected downward beneath thetable� That is� the ball can pass through the table in this approxi�mation� Second� this approximation cannot reproduce a large class ofreal solutions� called �sticking solutions� which are discussed in sec�tion �� Fundamentally� this is because the map in the high bounce

�� QUALITATIVE DESCRIPTION OF MOTIONS ��

approximation is invertible� whereas the exact model is not invertible�In the exact model there exist some solutions�in particular the stick�ing solutions�for which two or more orbits are mapped to the sameidentical point� Thus the map at this point does not have a uniqueinverse�

�� Qualitative Description of Motions

In specifying an individual solution to the bouncing ball system� weneed to know both the initial condition� that is� the initial impactphase and impact velocity of the ball �� v�� and the relevant systemparameters� �� A� and T � Then� to �nd an individual trajectory� all weneed to do is iterate the mapping for the appropriate model� However��nding the solution for a single trajectory gives little insight into theglobal dynamics of the system� As stressed in the Introduction� weare not interested so much in solving an individual orbit� but ratherin understanding the behavior of a large collection of orbits and� whenpossible� the system as a whole�

An individual solution can be represented by a curve in phase space�In considering a collection of solutions� we will need to understand thebehavior not of a single curve in phase space� but rather of a bundle ofadjacent curves� a region in phase space� Similarly� in the impact mapwe want to consider a collection of initial conditions� a region in theimpact map� In general� the future of an orbit is well de�ned by a �owor mapping� The fate of a region in the phase space or the impact mapis de�ned by the collective futures of each of the individual curves orpoints� respectively� in the region as is illustrated in Figure ��

A number of questions can� and will� be asked about the evolutionof a region in phase space �or in the impact map�� Do regions in thephase space expand or contract as the system evolves" In the bounc�ing ball system� a bundle of initial conditions will generally contract inarea whenever the system is dissipative�a little energy is lost at eachimpact� and this results in a shrinkage of our initial region� or patch� inphase space �see section �� for details�� Since this region is shrink�ing� this raises many questions that will be addressed throughout thisbook� such as where do all the orbits go� how much of the initial area


Figure �� a� Evolution of a region in phase space� �b� Recurrentregions in the phase space of a nonlinear system�

remains� and what do the orbits do on this remaining set once they getto wherever they�re going" This turns out to be a subtle collection ofquestions� For instance� even the question of what we mean by �area gets tricky because there is more than one useful notion of the area�or measure� of a set� Another related question is� do these regions in�tersect with themselves as they evolve �see Figure ��" The answeris generally yes� they do intersect� and this observation will lead us tostudy the rich collection of recurrence structures of a nonlinear system�

A simple question we can answer is� does there exist a closed�simply�connected subset� or region� of the whole phase space �or impactmap� such that all the orbits outside this subset eventually enter into it�and� once inside� they never get out again" If such a subset exists� it iscalled a trapping region� Establishing the existence of a trapping regioncan simplify our general problem somewhat� because instead of consid�ering all possible initial conditions� we need only consider those initialconditions inside the trapping region� since all other initial conditionswill eventually end up there�

�� Trapping Region

To �nd a trapping region for the bouncing ball system we will �rst �ndan upper bound for the next outgoing velocity� vk�� by looking at the


previous value� vk� We will then �nd a lower bound for vk�� Thesebounds give us the boundaries for a trapping region in the bouncingball�s impact map ��i� vi�� which imply a trapping region in phase space�

To bound the outgoing velocity� we begin with equation �� inthe form

vk�� vk � �� A cos��tk�� g�tk�� tk��

The �rst term on the right�hand side is easy to bound� To bound thesecond term� we �rst look at the average ball velocity between impacts�which is given by

!vk � vk � �

g�tk�� tk��

Rearranging this expression gives

tk�� tk �

g�vk � !vk��

Equation �� now becomes

vk�� vk � �� A� cos��tk�� !vk � �vk� ��

Noting that the average table velocity between impacts is the same asthe average ball velocity between impacts �see Prob� �� we �nd that

vk�� vk �� A��

If we de�ne

vmax ��

�� A��

and let vk vmax� then vk�� vk � �� vk� or

vk�� vk�

In this case it is essential that the system be dissipative �� for atrapping region to exist� In the conservative limit no trapping regionexists�it is possible for the ball to reach in�nite heights and velocitieswhen no energy is lost at impact�


To �nd a lower bound for vk�� we simply realize that the velocityafter impact must always be at least that of the table�

vk�� A� � vmin� ��

For the bouncing ball system the compact trapping region� D� givenby

D � f�� v� j vmin v vmaxg ��

is simply a strip bounded by vmin and vmax� To prove that D is atrapping region� we also need to show that v cannot approach vmax

asymptotically� and that once inside D� the orbit cannot leave D �thesecalculations are left to the reader�see Prob� �� The previous cal�culations show that all orbits of the dissipative bouncing ball systemwill eventually enter the region D and be �trapped there�

�� Equilibrium Solutions

Once the orbits enter the trapping region� where do they go next" Toanswer this question we �rst solve for the motion of a ball bouncing ona stationary table� Then we will imagine slowly turning up the tableamplitude�

If the table is stationary� then the high bounce approximation isno longer approximate� but exact� Setting A � � in the velocity map�equation �� immediately gives

vk�� vk� ��

Using the time map� tk�� tk � ��g�vk� the coe�cient of restitution iseasy to measure �� by recording three consecutive impact times� since

� �tk�� tk��tk�� tk

� ��

To �nd how long it takes the ball to stop bouncing� consider the sumof the di�erences of consecutive impact times�

# ��Xn�

�n � �� k � tk�� tk� ��


Since �k�� k� # is the summation of a geometric series�

# � �� Xn�

��n �

��

� ��

which can be summed for � � �� After an in�nite number of bounces�the ball will come to a halt in a �nite time�

For these equilibrium solutions� all the orbits in the trapping regioncome to rest on the table� When the table�s acceleration is small� thepicture does not change much� The ball comes to rest on the oscillatingtable and then moves in unison with the table from then on�

�� Sticking Solutions

Now that the ball is moving with the table� what happens as we slowlyturn up the table�s amplitude while keeping the forcing frequency �xed"Initially� the ball will remain stuck to the table until the table�s maxi�mum acceleration is greater than the earth�s gravitational acceleration�g� The table�s acceleration is given by

$s � �A�� sin��t � ��

The maximum acceleration is thus A�� When A�� is greater thang� the ball becomes unstuck and will �y free from the table until itsnext impact� The phase at which the ball becomes initially unstuckoccurs when

�g � �A�� sin��unstuck� �� unstuck � arcsin�

g

A��

��

Even in a system in which the table�s maximum acceleration ismuch greater than g� the ball can become stuck� An in�nite number ofimpacts can occur in a �nite stopping time� #� The sum of the timesbetween impacts converges in a �nite time much less than the table�speriod� T � The ball gets stuck again at the end of this sequence ofimpacts and moves with the table until it reaches the phase �unstuck�This type of sticking solution is an eventually periodic orbit� After its�rst time of getting stuck� it will exactly repeat this pattern of gettingstuck� and then released� forever�


Figure �� Sticking solutions in the bouncing ball system�

However� these sticking solutions are a bit exotic in several respects�Sticking solutions are not invertible� that is� an in�nite number of initialconditions can eventually arrive at the same identical sticking solution�It is impossible to run a sticking solution backward in time to �nd theexact initial condition from which the orbit came� This is because ofthe geometric convergence of sticking solutions in �nite time�

Also� there are an in�nite number of di�erent sticking solutions�Three such solutions are illustrated in Figure �� To see how some ofthese solutions are formed� let�s turn the table amplitude up a little sothat the stopping time� #� is lengthened� Now� it happens that the balldoes not get stuck in the �rst table period� T � but keeps bouncing oninto the second or third period� However� as it enters each new period�the bounces get progressively lower so that the ball does eventually getstuck after several periods� Once stuck� it again gets released whenthe table�s acceleration is greater than g� and this new pattern repeatsitself forever�

�� QUALITATIVE DESCRIPTION OF MOTIONS �

Figure �� Convergence to a period one orbit�

Figure �� Period two orbit of a bouncing ball�

�� Period One Orbits and Period Doubling

As we increase the table�s amplitude we often see that the orbit jumpsfrom a sticking solution to a simple periodic motion� Figure �� showsthe convergence of a trajectory of the bouncing ball system to a periodone orbit� The ball�s motion converges toward a periodic orbit with aperiod exactly equal to that of the table� hence the term period one

orbit �see Prob� ��What happens to the period one solution as the forcing amplitude

of the table increases further" We discover that the period one orbitbifurcates �literally� splits in two� to the period two orbit illustratedin Figure �� Now the ball�s motion is still periodic� but it bounceshigh� then low� then high again� requiring twice the table�s period tocomplete a full cycle� If we gradually increase the table�s amplitude stillfurther we next discover a period four orbit� and then a period eightorbit� and so on� In this period doubling cascade we only see orbits ofperiod

P � n � ��


Figure �� Chaotic orbit of a bouncing ball�

and not� for instance� period three� �ve� or six�The amplitude ranges for which each of these period n orbits is

observable� however� gets smaller and smaller� Eventually it convergesto a critical table amplitude� beyond which the bouncing ball systemexhibits the nonperiodic behavior illustrated in Figure �� This lasttype of motion found at the end of the period doubling cascade neversettles down to a periodic orbit and is� in fact� our �rst physical exampleof a chaotic trajectory known as a strange attractor� This motion isan attractor because it is the asymptotic solution arising from manydi�erent initial conditions� di�erent motions of the system are attractedto this particular motion� At this point� the term strange is used todistinguish this motion from other motions such as periodic orbits orequilibrium points� A more precise de�nition of the term strange isgiven in section ��

At still higher table amplitudes many other types of strange andperiodic motions are possible� a few of which are illustrated in Fig�ure �� The type of motion depends on the system parameters andthe speci�c initial conditions� It is common in a nonlinear system formany solutions to coexist� That is� it is possible to see several di�erentperiodic and chaotic motions for the same parameter values� Thesecoexisting orbits are only distinguished by their initial conditions�

The �period doubling route to chaos we saw above is common toa wide variety of nonlinear systems and will be discussed in depth insection ��


Figure �� A zoo of periodic and chaotic motions seen in the bouncingball system� �Generated by the Bouncing Ball program��

�� Chaotic Motions

Figure �� shows the impact map of the strange attractor discoveredat the end of the period doubling route to chaos� This strange attractorlooks almost like a simple curve �segment of an upside�down parabola�with gaps� Parts of this curve look chopped out or eaten away� However�on magni�cation� this curve appears not so simple after all� Rather� itseems to resemble an intricate web of points spread out on a narrowcurved strip� Since this chaotic solution is not periodic �and hence�never exactly repeats itself� it must consist of an in�nite collection ofdiscrete points in the impact �velocity vs� phase� space�


Figure �� Strange attractor in the bouncing ball system arising atthe end of a period doubling cascade� �Generated by the Bouncing Ballprogram��

This strange set is generated by an orbit of the bouncing ball system�and it is chaotic in that orbits in this set exhibit sensitive dependenceon initial conditions� This sensitive dependence on initial conditions iseasy to see in the bouncing ball system when we solve for the impactphases and velocities for the exact model with the numerical proceduredescribed in Appendix A� First consider two slightly di�erent trajecto�ries that converge to the same period one orbit� As shown in Table ��these orbits initially di�er in phase by �� This phase di�erenceincreases a little over the next few impacts� but by the eleventh impactthe orbits are indistinguishable from each other� and by the eighteenthimpact they are indistinguishable from the period one orbit� Thus thedi�erence between the two orbits decreases as the system evolves�

An attracting periodic orbit has both long�term and short�term pre�dictability� As the last example indicates� we can predict� from an ini�


Hit Phase Phase

� ��

Table �� Convergence of two di�erent initial conditions to a period oneorbit� The digits in bold are where the orbits di�er� At the zeroth hitthe orbits di�er in phase by �� Note that the di�erence betweenthe orbits decreases so that after �� impacts both orbits are indistin�guishable from the period one orbit� The operating parameters are�A � �� cm� frequency � �� Hz� � � �� and the initial ball velocityis �� cm%s� The impact phase is presented as � �� mod ��so that it is normalized to be between zero and one�


Hit Phase Phase

� ��

Table �� Divergence of initial conditions on a strange attractor il�lustrating sensitive dependence on initial conditions� The parametervalues are the same as in Table �� except for A � �� cm� The bolddigits show where the impact phases di�er� The orbits di�er in phaseby �� at the zeroth hit� but by the twenty�third impact they di�erat every digit�

�� ATTRACTORS �

tial condition of limited resolution� where the ball will be after a fewbounces �short�term� and after many bounces �long�term��

The situation is dramatically di�erent for motion on a strange at�tractor� Chaotic motions may still possess short�term predictability�but they lack long�term predictability� In Table �� we show two dif�ferent trajectories that again di�er in phase by �� at the zerothimpact� However� in the chaotic case the di�erence increases at a re�markable rate with the evolution of the system� That is� given a smalldi�erence in the initial conditions� the orbits diverge rapidly� By thetwelfth impact the error is greater than �� by the twentieth impact�� and by the twenty�fourth impact the orbits show no resemblance�Chaotic motion thus exhibits sensitive dependence on initial conditions�Even if we increase our precision� we still cannot predict the orbit�s fu�ture position exactly�

As a practical matter we have no exact long�term predictive powerfor chaotic motions� It does not really help to double the resolution ofour initial measurement as this will just postpone the problem� Thebouncing ball system is both deterministic and unpredictable� Chaoticmotions of the bouncing ball system are unpredictable in the practicalsense that initial measurements are always of limited accuracy� andany initial measurement error grows rapidly with the evolution of thesystem�

Strange attractors are common to a wide variety of nonlinear sys�tems� We will develop a way to name and dissect these critters inChapters and ��

�� Attractors

An attracting set A in a trapping region D is de�ned as a nonemptyclosed set formed from some open neighborhood�

A ��n��

fn�D��

We mentioned before that for a dissipative bouncing ball system thetrapping region is contracting� so the open neighborhood typically con�sists of a collection of smaller and smaller regions as it approaches theattracting set�


Figure �� A periodic attractor and its transient�

An attractor is an attempt to de�ne the asymptotic solution of adynamical system� It is that part of the solution that is left after the�transient part is thrown out� Consider Figure �� which shows theapproach of several phase space trajectories of the bouncing ball systemtoward a period one cycle� The orbits appear to consist of two parts�the transient�the initial part of the orbit that is spiraling toward aclosed curve�and the attractor�the closed periodic orbit itself�

In the previous section we saw examples of several di�erent typesof attractors� For small table amplitudes� the ball comes to rest onthe table� For these equilibrium solutions the attractor consists of asingle point in the phase space of the table�s reference frame� At highertable amplitudes periodic orbits can exist� in which case the attractoris a closed curve in phase space� In a dissipative system this closedcurve representing a periodic motion is also known as a limit cycle� Atstill higher table amplitudes� a more complicated set called a strangeattractor can appear� The phase space plot of a strange attractor is acomplicated curve that never quite closes� After a long time� this curveappears to sketch out a surface� Each type of attractor�a point� closedcurve� or strange attractor �something between a curve and a surface��represents a di�erent type of motion for the system�equilibrium� pe�riodic� or chaotic�

Except for the equilibrium solutions� each of the attractors just

�� BIFURCATION DIAGRAMS �

described in the phase space has its corresponding representation inthe impact map� In general� the representation of the attractor in theimpact map is a geometric object of one dimension less than in phasespace� For instance� a periodic orbit is a closed curve in phase space�and this same period n orbit consists of a collection of n points in theimpact map� The impact map for a chaotic orbit consists of an in�nite

collection of points�For a nonlinear system� many attractors can coexist� This naturally

raises the question as to which orbits and collections of initial conditionsgo to which attractors� For a given attractor� the domain of attraction�or basin of attraction� is the collection of all those initial conditionswhose orbits approach and always remain near that attractor� Thatis� it is the collection of all orbits that are �captured by an attractor�Like the attractors themselves� the basins of attraction can be simpleor complex ��

Figure �� shows a diagram of basins of attraction in the bouncingball system� The phase space is dominated by the black regions� whichindicate initial conditions that eventually become sticking solutions�The white sinusoidal regions at the bottom of Figure �� show un�physical initial conditions�phases and velocities that the ball cannotobtain� The gray regions represent initial conditions that approach aperiod one orbit� �See Plate � for a color diagram of basins of attractionin the bouncing ball system��

�� Bifurcation Diagrams

A bifurcation diagram provides a nice summary for the transition be�tween di�erent types of motion that can occur as one parameter of thesystem is varied� A bifurcation diagram plots a system parameter onthe horizontal axis and a representation of an attractor on the verticalaxis� For instance� for the bouncing ball system� a bifurcation diagramcan show the table�s forcing amplitude on the horizontal axis and theasymptotic value of the ball�s impact phase on the vertical axis� as il�lustrated in Figure �� At a bifurcation point� the attracting orbitundergoes a qualitative change� For instance� the attractor literallysplits in two �in the bifurcation diagram� when the attractor changes


Figure �� Basins of attraction in the bouncing ball system� �Gener�ated by the Bouncing Ball program��

�� BIFURCATION DIAGRAMS ��

Figure �� Bouncing ball bifurcation diagram� �Generated by theBouncing Ball program��

from a period one orbit to a period two orbit�This bouncing ball bifurcation diagram �Fig� �� shows the classic

period doubling route to chaos� For table amplitudes between �� cmand �� cm a stable period one orbit exists� the ball impacts withthe table at a single phase� For amplitudes between �� cm and�� cm� a period two orbit exists� The ball hits the table at twodistinct phases� At higher table amplitudes� the ball impacts at moreand more phases� The ball hits at an in�nity of distinct points �phases�when the motion is chaotic�

� Bouncing Ball


�� For some recent examples of engineering applications in which the bouncing ballproblem arises� see M��O� Hongler� P� Cartier� and P� Flury� Numerical studyof a model of vibro�transporter� Phys� Lett� A �� M��O�Hongler and J� Figour� Periodic versus chaotic dynamics in vibratory feeders�Helv� Phys� Acta �� T� O� Dalrymple� Numerical solutions tovibroimpact via an initial value problem formulation� J� Sound Vib� ��

�� The bouncing ball problem has proved to be a useful system for experimentallyexploring several new nonlinear e�ects� Examples of some of these experimentsinclude S� Celaschi and R� L� Zimmerman� Evolution of a two�parameter chaoticdynamics from universal attractors� Phys� Lett� A �� N� B� Tu�llaro� T� M� Mello� Y� M� Choi� and A� M� Albano� Period dou�bling boundaries of a bouncing ball� J� Phys� �Paris� �� K�Wiesenfeld and N� B� Tu�llaro� Suppression of period doubling in the dynam�ics of the bouncing ball� Physica D �� P� Piera�nski� Jumpingparticle model� Period doubling cascade in an experimental system� J� Phys��Paris� �� P� Piera�nski� Z� Kowalik� and M� Franaszek� Jump�ing particle model� A study of the phase of a nonlinear dynamical system belowits transition to chaos� J� Phys� �Paris� �� M� Franaszek andP� Piera�nski� Jumping particle model� Critical slowing down near the bifurca�tion points� Can� J� Phys� �� P� Piera�nski and R� Bartolino�Jumping particle model� Modulation modes and resonant response to a peri�odic perturbation� J� Phys� �Paris� �� M� Franaszek and Z�J� Kowalik� Measurements of the dimension of the strange attractor for theFermi�Ulam problem� Phys� Rev� A �� P� Piera�nski andJ� Ma�lecki� Noisy precursors and resonant properties of the period doublingmodes in a nonlinear dynamical system� Phys� Rev� A �� P� Piera�nski and J� Ma�lecki� Noise�sensitive hysteresis loops and period dou�bling bifurcation points� Nuovo Cimento D � �� P� Piera�nski�Direct evidence for the suppression of period doubling in the bouncing ballmodel� Phys� Rev� A �� Z� J� Kowalik� M� Franaszek�and P� Piera�nski� Self�reanimating chaos in the bouncing ball system� Phys�Rev� A ��

�� A description of some simple experimental bouncing ball systems� suitable forundergraduate labs� can be found in A� B� Pippard� The physics of vibration�Vol� � �New York� Cambridge University Press� �� p� � �� N� B� Tu�llaroand A� M� Albano� Chaotic dynamics of a bouncing ball� Am� J� Phys� �� R� L� Zimmerman and S� Celaschi� Comment on �Chaoticdynamics of a bouncing ball�� Am� J� Phys� �� T�M� Mello and N� B� Tu�llaro� Strange attractors of a bouncing ball� Am� J�Phys� �� R� Minnix and D� Carpenter� Piezoelectric �lm

Problems ��

reveals f versus t of ball bounce� The Physics Teacher� �� March �� The KYNAR piezoelectric �lm used for the impact detector is available fromPennwalt Corporation� �� First Avenue� P�O� Box C� King of Prussia� PA�� The experimenter�s kit containing an assortment of piezoelectric�lms costs about ��

�� Some approximate models of the bouncing ball system are presented by G� M�Zaslavsky� The simplest case of a strange attractor� Phys� Lett� A �� P� J� Holmes� The dynamics of repeated impacts with a sinusoidallyvibrating table� J� Sound Vib� �� M� Franaszek� E�ect ofrandom noise on the deterministic chaos in a dissipative system� Phys� Lett�A �� R� M� Everson� Chaotic dynamics of a bouncingball� Physica D �� A� Mehta and J� Luck� Novel temporalbehavior of a nonlinear dynamical system� The completely inelastic bouncingball� Phys� Rev� Lett� ��

�� A detailed comparison between the exact model and the high bounce model isgiven by C� N� Bapat� S� Sankar� and N� Popplewell� Repeated impacts on asinusoidally vibrating table reappraised� J� Sound Vib� ��

�� A� D� Bernstein� Listening to the coe�cient of restitution� Am� J� Phys� ��

�� H� M� Isom!aki� Fractal properties of the bouncing ball dynamics� in Nonlineardynamics in engineering systems� edited by W� Schiehlen �Springer�Verlag�New York� �� pp� ��

Problems

Problems for section ��

�� For a period one orbit in the exact model show that

�a� vk � �v�k �

�b� vk � gT��

�c� the impact phase is exactly given by

cos��P�� gT �

��A

��

� � �

��

� Bouncing Ball

�� Assuming only that vk � �v�k��a� Show that the solution in Problem �� can be generalized to an nth�order symmetric periodic ��equispaced�� orbit satisfying vk � ngT�� forn � ��

�b� Show that the impact phase is exactly given by

cos��Pn� �ngT �

��A

��

� � �

��

�c� Draw pictures of a few of these orbits� Why are they called �equispaced��

�d� Find parameter values at which a period one and period two �n � �� equispaced orbit can coexist�

�� If T � �� s and � � �� what is the smallest table amplitude for which aperiod one orbit can exist �use Prob� ��c�� For these parameters� estimatethe maximumheight the ball bounces and express your answer in units of theaverage thickness of a human hair� Describe how you arrived at this thickness�

�� Describe a numerical method for solving the exact bouncing ball map� Howdo you determine when the ball gets stuck� How do you propose to �nd thezeros of the phase map� equation ��

�� Derive equation �� from equation ��

Section ��

�� Calculate ��n� �n� in equation �� for n � �� when � � �� and ��

�� Con�rm that the variables �� and � given by equations �� aredimensionless�

�� Verify the derivation of equation �� the standard map�

�� Calculate the inverse of the standard map �eq� ��

�� Write a computer program to iterate the model of the bouncing ball systemgiven by equation �� the high bounce approximation�

Section ��

�� a� Calculate the stopping time �eq� �� for a �rst impact time � � ��and a damping coe�cient � � �� Also calculate �� and ��

�b� Calculate � and the stopping time for an initial velocity of �� m�s ��cm�s� and a damping coe�cient of ��

Problems ��

�� For the high bounce approximation �eq� �� show that when � ��

�a� j�j��j � �j�jj� ��

�b� A trapping region is given by a strip bounded by �vmax� where vmax �� Note� The reader may assume� as the book does at the end ofsection �� that the vi cannot approach vmax asymptotically� and that onceinside the strip� the orbit cannot leave��

�� Calculate �unstuck �eq� �� for A � �� cm and T � �� s� What is thespeed and acceleration of the table at this phase� Is the table on its way upor down� Are there table parameters for which the ball can become unstuckwhen the table is moving up� Are there table parameters for which the ballcan become unstuck when the table is moving down�

�� Show that the average table velocity between impacts equals the average ballvelocity between impacts�

�� These problems relate to the trapping region discussion in section ��

�a� Prove that vi cannot approach the vmax given by equation �� asymp�totically� �Hint� It is acceptable to increase vmax by some small � � ��

�b� Prove that� for the trapping region D given by equation �� once theorbit enters D� it can never escape D�

�c� Use the trapping region D in the impact map to �nd a trapping region inphase space� Hint� Use the maximum outgoing velocity �vmax� to calculate aminimum incoming velocity and a maximum height�

�d� The trapping region found in the text is not unique� in fact� it is fairly�loose�� Try to obtain a smaller� tighter trapping region�

�� Derive equation ��

Section ��

�� How many period one orbits can exist according to Problem ��c�� and howmany of these period one orbits are attractors�

Section ��

�� Write a computer program to generate a bifurcation diagram for the bouncingball system�

�� Bouncing Ball

Chapter �

Quadratic Map

�� Introduction

A ball bouncing on an oscillating table gives rise to complicated phe�nomena which appear to defy our comprehension and analysis� Themotions in the bouncing ball system are truly complex� However� partof the problem is that we do not� as yet� have the right language withwhich to discuss nonlinear phenomena� We thus need to develop avocabulary for nonlinear dynamics�

A good �rst step in developing any scienti�c vocabulary is the de�tailed analysis of some simple examples� In this chapter we will beginby exploring the quadratic map� In linear dynamics� the correspondingexample used for building a scienti�c vocabulary is the simple harmonicoscillator �see Figure �� As its name implies� the harmonic oscillatoris a simple model which illustrates many key notions useful in the studyof linear systems� The image of a mass on a spring is usually not farfrom one�s mind even when dealing with the most abstract problems inlinear physics�

The similarities among linear systems are easy to identify becauseof the extensive development of linear theory over the past century�Casual inspection of nonlinear systems suggests little similarity� Care�ful inspection� though� reveals many common features� Our originalintuition is misleading because it is steeped in linear theory� Nonlin�ear systems possess as many similarities as di�erences� However� the

��

�� CHAPTER �� QUADRATIC MAP

Figure �� Simple harmonic oscillator�

vocabulary of linear dynamics is inadequate to name these commonstructures� Thus� our task is to discover the common elements of non�linear systems and to analyze their structure�

A simple model of a nonlinear system is given by the di�erenceequation known as a quadratic map�

xn�� xn � �x�n�

� �xn�� xn��

For instance� if we set the value � � and initial condition x� � ��in the quadratic map we �nd that

x� � ��

x� � ��

x� � ��

etc��

and in this case the value xn appears to be approaching ��Phenomena illustrated in the quadratic map arise in a wide variety

of nonlinear systems� The quadratic map is also known as the logistic

�� INTRODUCTION ��

map� and it was studied as early as �� by P� F� Verhulst as a modelfor population growth� Verhulst was led to this di�erence equation bythe following reasoning� Suppose in any given year� indexed by thesubscript n� that the �normalized� population is xn� Then to �nd thepopulation in the next year �xn�� it seems reasonable to assume thatthe number of new births will be proportional to the current population�xn� and the remaining inhabitable space� �� xn� The product of thesetwo factors and � gives the quadratic map� where � is some parameterthat depends on the fertility rate� the initial living area� the averagedisease rate� and so on�

Given the quadratic map as our model for population dynamics� itwould now seem like an easy problem to predict the future population�Will it grow� decline� or vary in a cyclic pattern" As we will see� theanswer to this question is easy to discover for some values of �� but notfor others� The dynamics are di�cult to predict because� in additionto exhibiting cyclic behavior� it is also possible for the population tovary in a chaotic manner�

In the context of physical systems� the study of the quadratic mapwas �rst advocated by E� N� Lorenz in �� At the time� Lorenz waslooking at the convection of air in certain models of weather prediction�Lorenz was led to the quadratic map by the following reasoning� whichalso applies to the bouncing ball system as well as to Lorenz�s originalmodel �or� for that matter� to any highly dissipative system�� Considera time series that comes from the measurement of a variable in somephysical system�

fx�� x�� x�� x�� xi� � � � � xn�� xn � � �g� ��

For instance� in the bouncing ball system this time series could consistof the sequence of impact phases� so that x� � �� x� � �� x� � ��and so on� We require that this time series arise from motion on anattractor� To meet this requirement� we throw out any measurementsthat are part of the initial transient motion� In addition� we assumeno foreknowledge of how to model the process giving rise to this timeseries� Given our ignorance� it then seems natural to try to predict then��st element of the time series from the previous nth value� Formally�


we are seeking a function� f � such that

xn�� f�xn��

In the bouncing ball example this idea suggests that the next impactphase would be a function of the previous impact phase� that is� �n�� f��n��

If such a simple relation exists� then it should be easy to see byplotting yn � xn�� on the vertical axis and xn on the horizontal axis�Formally� we are taking our original time series� equation �� and cre�ating an embedded time series consisting of the ordered pairs� �x� y� ��xn� xn��

f�x�� x�� x�� x�� x�� x�� xn�� xn�� xn� xn�� g� ��

The idea of embedding a time series will be central to the experimentalstudy of nonlinear systems discussed in section �� In Figure � weshow an embedded time series of the impact phases for chaotic motionsin the bouncing ball system� The points for this embedded time seriesappear to lie close to a region that resembles an upside�down parabola�The exact details of the curve depend� of course� on the speci�c param�eter values� but as a �rst approximation the quadratic map providesa reasonable �t to this curve �see Figure �� Note that the curve�smaximum amplitude �located at the point x � �� rises as the param�eter � increases� We think of the parameter � in the quadratic map asrepresenting some parameter in our process� � could be analogous tothe table�s forcing amplitude in the bouncing ball system� Such single�humped maps often arise when studying highly dissipative nonlinearsystems� Of course� more complicated many�humped maps can and dooccur� however� the single�humped map is the simplest� and is thereforea good place to start in developing our new vocabulary�

�� Iteration and Dierentiation

In the previous section we introduced the equation

f�x� � �x�� x��

�� ITERATION AND DIFFERENTIATION ��

Figure �� Embedded time series of chaotic motion in the bouncingball system� �Generated by the Bouncing Ball program��

� CHAPTER �� QUADRATIC MAP

Figure �� The quadratic function� �Generated by the Quadratic Mapprogram��

�� ITERATION AND DIFFERENTIATION ��

known as the quadratic map� We write f��x� when we want to makethe dependence of f on the parameter � explicit�

In this section we review two mathematical tools we will need forthe rest of the chapter� iteration and di�erentiation� We think of amap f � xn �� xn�� as generating a sequence of points� With the seedx�� de�ne xn � fn�x�� and consider the sequence x�� x�� x�� x�� asan orbit of the map� That is� the orbit is the sequence of points

x�� x� � f�x�� x� � f��x�� x� � f��x��

where the nth iterate of x� is found by functional composition n times�

f� � f � f�f� � f � f � f�

fn �

nz � �f � f � � � � f � f �

When determining the stability of an orbit we will need to calcu�late the derivative of these composite functions �see section �� Thederivative of a composite function evaluated at a point x � x� is writtenas

�fn��x��

�d

dxfn�x�

�jxx� � ��

The left�hand side of equation �� is a shorthand form for the right�hand side that tells us to do the following when calculating the deriva�tive� First� construct the nth composite function of f � call it fn� Sec�ond� compute the derivative of fn� And third� as the bar notation�jxx�� tells us� evaluate this derivative at x � x�� For instance� iff�x� � x�� n � � and x� � �� then

�f��

�d

dxf��x�

�jx��

�d

dx�f � f��x� �

d

dx�f�x��

d

dx�x��

� x�jx� � ��

Notice that we suppressed the bar notation during the intermediatesteps� This is common practice when the meaning is clear from context�


You may sometimes see the even shorter notation for evaluating thederivative at a point x� as

fn�

�x��

�d

dxfn�x�

�jxx� � ��

which is su�ciently terse to be legitimately confusing�An examination of the dynamics of the quadratic map provides an

excellent introduction to the rich behavior that can exist in nonlinearsystems� To �nd the itinerary of an individual orbit all we need is apocket calculator or a computer program something like the followingC program��

�� quadratic�c� calculate an orbit for the quadratic map

input� l x�

output� � x�

� x�

� x�

etc�

��include stdio�h�

main

fint n�

float lambda� x zero� x n�

printf Enter� lambda x zeronn � scanf �f �f � �lambda� �x zero�

x n � x zero�

forn � �� n � �� n fx n � lambda � x n � � � x n� �� the quadratic map ��printf �d �fnn � n� x n�

gg

�A nice� brief introduction to the C programming language su�cient for most ofthe programs in this book is Chapter �� A tutorial introduction� of B� W� Kernighanand D� M� Ritchie� The C Programming Language �Prentice�Hall� Englewood Cli�s�NJ� �� pp� ��

�� GRAPHICAL METHOD ��

Figure �� Graphical method for iterating the quadratic map� �Gen�erated by the Quadratic Map program��

�� Graphical Method

In addition to doing a calculation� there is a graphical procedure for�nding the itinerary of an orbit� This graphical method is illustratedin Figure � for the same parameter value and initial condition usedin the previous example and is based on the following observation� To�nd xn�� from xn we note that xn�� f�xn�� graphically� to get f�xn�we start at xn on the horizontal axis and move vertically until we hitthe graph y � f�x�� Now this current value of y must be transferredfrom the vertical axis back to the horizontal axis so that it can be usedas the next seed for the quadratic map� The simplest way to transferthe y axis to the x axis is by folding the x�y plane through the diagonalline y � x since points on the vertical axis are identical to points onthe horizontal axis on this line� This insight suggests the followinggraphical recipe for �nding the orbit for some initial condition x��


Figure �� Graphical iteration of the quadratic map for a chaotic orbit��Generated by the Quadratic Map program��

�� Start at x� on the horizontal axis�

� Move vertically up until you hit the graph f�x��

�� Move horizontally until you hit the diagonal line y � x�

� Move vertically�up or down�until you hit the graph f�x��

�� Repeat steps � and to generate new points�

For the example in Figure � it is clear that the orbit is convergingto the point �� This same graphical technique is also illustrated inFigure �� for the more complicated orbit that arises when � � ��

We can also ask again about the fate of a whole collection of initialconditions� instead of just a single orbit� In particular we can considerthe transformation of all initial conditions on the unit interval�

I � fxjx � �� g � ��

�� GRAPHICAL METHOD ��

Figure �� Stretching and folding in the quadratic map ��

subject to the quadratic map� equation �� As shown in Figure ��the quadratic map for � � can be viewed as transforming the unitinterval in two steps� The �rst step is a stretch� which takes the intervalto twice its length� fstretch � �� The second step is a fold �which takes the lower half of the interval to the whole unit interval�and the upper half of the interval also to the whole unit interval withits direction reversed� ffold � �� and �� Thesetwo operations of stretching and folding are the key geometric construc�tions leading to the complex behavior found in nonlinear systems� Thestretching operation tends to quickly separate nearby points� while thefolding operation ensures that all points will remain bounded in someregion of phase space�

Another way to visualize this stretching and folding process is pre�sented in Figure �� Imagine taking a rubber sheet and dividing it intotwo sections by slicing it down the middle� The sheet separates intotwo branches at the gap �the upper part of the sheet where the slicebegins�� the left branch is �at� while the right branch has a half�twistin it� These two branches are rejoined� or glued together again� at thebranch line seen at the bottom of the diagram� Notice that the leftbranch passes behind the right branch� Next� imagine that there is asimple rule� indicated by the wide arrows in the diagram� that smoothlycarries points at the top of the sheet to points at the bottom� In partic�ular� the unit interval at the top of the sheet gets stretched and foldedso that it ends up as the bent line segment indicated at the bottom


Figure �� Rubber sheet model of stretching and folding in thequadratic map�

of the sheet� At this point Figure �� is just a visual aid illustratinghow stretching and folding can occur in a dynamical system� However�observe that the resulting folded line segment resembles a horseshoe�These horseshoes were �rst identi�ed and analyzed as recurring ele�ments in nonlinear systems by the mathematician Steve Smale� Wewill say much more about these horseshoes� and make more extensiveuse of such diagrams� in section �� and Chapter � when we try tounravel the topological organization of strange sets�

�� Fixed Points

A simple linear map f � R �� R of the real line R to itself is givenby f�x� � mx� Unlike the quadratic map� this linear map can havestretching� but no folding� The graphical analysis shown in Figure ��quickly convinces us that for x� � only three possible asymptoticstates exist� namely�

limn�� xn � �� if m ��limn�� xn � �� if m � �� andxn�� xn� for m � ��

�� FIXED POINTS ��

Figure �� Graphical iteration of the linear map�

The period one points of a map �points that map to themselves afterone iteration� are also called �xed points� If m � �� then the originis an attracting �xed point or sink since nearby points tend to � �seeFigure ��a�� If m �� then the origin is still a �xed point� However�because points near the origin always tend away from it� the origin iscalled a repelling �xed point or source �see Figure ��b�� Lastly� ifm � �� then all initial conditions lead immediately to a period oneorbit de�ned by y � x� All the periodic orbits that lie on this line haveneutral stability�

The story for the more complicated function f�x� � x� is not muchdi�erent� For this parabolic map a simple graphical analysis shows thatas n��

fn�x� �� if jxj ��fn�x� � �� if jxj � ��fn�� for all n�fn�� if n ��

In this case all initial conditions tend to either � or �� except for thepoint x � �� which is a repelling �xed point since all nearby orbitsmove away from �� The special initial condition x� � �� is said to beeventually �xed because� although it is not a �xed point itself� it goesexactly to a �xed point in a �nite number of iterations� The stickingsolutions of the bouncing ball system are examples of orbits that couldbe called eventually periodic since they arrive at a periodic orbit in a�nite time�


Figure �� The local stability of a �xed point is determined by theslope of f at x��

Graphical analysis also allows us to see why certain �xed points arelocally attracting and others repelling� As Figure �� illustrates� thelocal stability of a �xed point is determined by the slope of the curvepassing through the �xed point� If the absolute value of the slope is lessthan one�or equivalently� if the absolute value of the derivative at the�xed point is less than one�then the �xed point is locally attracting�Alternatively� if the absolute value of the derivative at the �xed pointis greater then one� then the �xed point is repelling�

An orbit of a map is periodic if it repeats itself after a �nite numberof iterations� For instance� a point on a period two orbit has the prop�erty that f��x�� x�� and a period three point satis�es f��x�� x��that is� it repeats itself after three iterations� In general a period npoint repeats itself after n iterations and is a solution to the equation

fn�x�� x��

�� PERIODIC ORBITS ��

In other words� a period n point is a �xed point of the nth composite

function of f� Accordingly� the stability of this �xed point and of thecorresponding period n orbit is determined by the derivative of fn�x��

Our discussion about the �xed points of a map is summarized inthe following two de�nitions concerning �xed points� periodic points�and their stability �� A more rigorous account of periodic orbits andtheir stability in presented in section ��

De�nition� Let f � R �� R� The point x� is a �xed point for fif f�x�� x�� The point x� is a periodic point of period n for f iffn�x�� x� but f i�x�� x� for � � i � n� The point x� is eventuallyperiodic if fm�x�� fm�n�x�� but x� is not itself periodic�

De�nition� A periodic point x� of period n is attracting if j�fn��x��j �� The prime denotes di�erentiation with respect to x� The peri�odic point x� is repelling if j�fn��x��j �� The point x� is neutral ifj�fn��x��j � ��

We have just shown that the dynamics of the linear map and theparabolic map are easy to understand� By combining these two mapswe arrive at the quadratic map� which exhibits complex dynamics� Thequadratic map is then� in a way� the simplest map exhibiting nontrivialnonlinear behavior�

�� Periodic Orbits

From our de�nition of a period n point� namely�

fn�x�� x��

we see that �nding a period n orbit for the quadratic map requires�nding the zeros for a polynomial of order n� For instance� the periodone orbits are given by the roots of

f�x� � �x�� x� � x� ��

which is a polynomial of order � The period two orbits are found byevaluating

f��x� � f�f�x��


� f ��x�� x��

� ��x�� x�� x�� x�� x� ��

which is a polynomial of order � Similarly� the period three orbits aregiven by solving a polynomial of order �� and so on� Unfortunately�except for small n� solving such high�order polynomials is beyond themeans of both mortals and machines�

Furthermore� our de�nition for the stability of an orbit says thatonce we �nd a point of a period n orbit� call it x�� we next need toevaluate the derivative of our polynomial at that point� For instance�the stability of a period one orbit is determined by evaluating

f ��x�� d

dx�x�� x�jxx� � �� x��

Similarly� the stability of a period two orbit is determined from theequation

�f��x�� d

dx��x�� x�� x � �x��jxx�

� �� x�� x� � �x��

Again� these stability polynomials quickly become too cumbersome toanalyze as n increases�

Any periodic orbit of period n will have n points in its orbit� We willgenerally label this collection of points by the subscript i � �� n� �� so that

x� � fx�� x�� x�� xi� � � � � x�n�� x�n��g� ��

where i labels an individual point of the orbit� The boldface notationindicates that x� is an n�tuple of real numbers� Another complicationwill arise� in some cases it is useful to write our indexing subscript insome base other than ten� For instance� it is useful to work in base twowhen studying one�humped maps� In general� it is convenient to workin base n � � where n is the number of critical points of the map� Itwill be advantageous to label the orbits in the quadratic map accordingto some binary scheme�


Lastly� the question arises� which element of the periodic orbit dowe use in evaluating the stability of an orbit" In Problem �� weshow that all periodic points in a periodic orbit give the same value

for the stability function� �fn�� So we can use any point in theperiodic sequence� This fact is good to keep in mind when evaluatingthe stability of an orbit�

�� Graphical Method

Although the algebra is hopeless� the geometric interpretation for thelocation of periodic orbits is straightforward� As we see in Figure��a�� the location of the period one orbits is given by the intersectionof the graphs y � f�x� and y � x� The latter equation is simply astraight line passing through the origin with slope �� In the case ofthe quadratic map� f� is an inverted parabola also passing through theorigin� These two graphs can intersect at two distinct points� givingrise to two distinct period one orbits� One of these orbits is always atthe origin and the other�s exact location depends on the height of thequadratic map� that is� the speci�c value of � in the quadratic map�

To �nd the location of the period two orbit we need to plot y � xand f��x�� The graph shown in Figure ��b� shows three points of in�tersection in addition to the origin� The middle point �the open circle�is the period one orbit found above� The two remaining intersectionpoints are the two points belonging to a single period two orbit� Adashed line indicates where these period two points sit on the originalquadratic map �the two dark circles�� and the simple graphical con�struction of section �� should convince the reader that this is� in fact�a period two orbit�

The story for higher�order orbits is the same �see Fig� ��a� and�b�� The graph of the third iterate� y � f��x�� shows eight pointsof intersection with the straight line� Not all eight intersection pointsare elements of a period three orbit� Two of these points are just thepair of period one orbits� The remaining six points consist of a pair ofperiod three orbits� The graph for the period four orbits shows sixteenpoints of intersection� Again� not all the intersection points are part ofa period four orbit� Two intersection points are from the pair of periodone orbits� and two are from the period two orbit� That leaves twelve


Figure �� First and second iterates of the quadratic map ��


Figure �� Third and fourth iterates of the quadratic map ��


remaining points of intersection� each of which is part of some periodfour orbit� Since there are twelve remaining points� there must be three�� points % points per orbit� distinct period four orbits�

The number of intersection points of fn� depends on �� If � � � � �and n � there are only two intersection points� the two distinct pe�riod one orbits� In dramatic contrast� if � � then it is easy to showthat there will be n intersection points� and counting arguments likethose just illustrated allow us to determine how many of these intersec�tion points are new periodic points of period n �� One fundamentalquestion is� how can a system as simple as the quadratic map changefrom having only two to having an in�nite number of periodic orbits"Like many aspects of the quadratic map� the answers are surprising�Before we tackle this problem� let�s resume our analysis of the periodone and period two orbits�

�� Period One Orbits

Solving equation �� for x we �nd two period one solutions��

x��

and

x��

��

The �rst period one orbit� labeled x�� always remains at the origin�while the location of the second period one orbit� x�� depends on ��From equation �� the stability of each of these orbits is determinedfrom

f ��x��

�The subscript n to x�n is labeling two distinct periodic orbits� This is potentiallyconfusing notation since we previously reserved this subscript to label di�erentpoints in the same periodic orbit� In practice this notation will not be ambiguoussince this label will be a binary index� the length of which determines the periodof the orbit� Di�erent cyclic permutations of this binary index will correspond todi�erent points on the same orbit� A noncyclic permutation must then be a pointon a distinct period n orbit� The rules for this binary labeling scheme are spelledout in section ��


andf ��x��

Clearly� if � � � � � then jf ��x��j � � and jf ��x��j �� so the periodone orbit x�� is stable and x�� is unstable� At � � � these two orbitscollide and exchange stability so that for � � � � �� x�� is unstable andx�� is stable� For � �� both orbits are unstable�

�� Period Two Orbit

The location of the period two orbit is found from equation ��

x��

�

��

p��

��

and

x��

�

�� p��

��

These two points belong to the period two orbit� We label the left pointx�� and the right point x�� Note that the location of the period twoorbit produces complex numbers for � � �� This indicates that theperiod two orbit exists only for � �� which is obvious geometricallysince y � f��x� begins a new intersection with the straight line y � xat � � ��

The stability of this period two orbit is determined by rewritingequation �� as

�f��x�� x�� x��

where we used equations �� and �� for x�� and x�� A plot of thestability for the period two orbit is presented in Figure �� A closeexamination of this �gure shows that� for � � � � �� the absolutevalue of the stability function is less than one� that is� the period twoorbit is stable� For � �� the period two orbit is unstable�

The range in � for which the period two orbit is stable can actuallybe obtained analytically� The period two orbit is stable as long as

�� f��x��

The period two orbit �rst becomes stable when �f��x�� whichoccurs at � � �� and it loses stability at �f��x�� which thereader can verify takes place at � � � �

p� �� see Prob� ��


Figure �� Stability of period two orbit�

�� Stability Diagram

The location and stability of the two period one orbits and the singleperiod two orbit are summarized in the orbit stability diagram shownin Figure �� The vertical axis shows the location of the periodicorbit x�n as a function of the parameter �� Stable orbits are denoted bysolid lines� unstable orbits by dashed lines� Two �bifurcation points are evident in the diagram� The �rst occurs when the two period oneorbits collide and exchange stability at � � �� The second occurs withthe birth of a stable period two orbit from a stable period one orbit at� � ��

�� Bifurcation Diagram

To explore the dynamics of the quadratic map further� we can choosean initial condition x� and a parameter value �� and then iterate themap using the program in section � to see where the orbit goes� Wewould notice a few general results if we play this game long enough�

First� if � � and x� �� then the graphical analysis of section�� shows us that all points not in the unit interval will run o� to

�� BIFURCATION DIAGRAM ��

Figure �� Orbit stability diagram�

in�nity� Further� if � � � � �� then the same type of graphical analysisshows that the dynamics of the quadratic map are simple� there is onlyone attracting �xed point and one repelling �xed point� These �xedpoints are the period one orbits calculated in the previous section�

Second� the initial condition we pick is usually not important indetermining the attractor� although the value of � is very important�We seem to end up with the same attractor no matter what x� � �� we pick�� The quadratic map usually has one� and only one� attractor�whereas most nonlinear systems can have more than one attractor ��The bouncing ball system� for example� can have two or more coexistingattractors�

Third� as we will show in section �� almost all initial conditionsrun o� to in�nity for all � � There are no attractors in this case�

Therefore� when studying the quadratic map� it will usually su�ce

�Some initial conditions do not converge to the attractor� For instance� any xbelonging to an unstable periodic orbit will not converge to the attractor� Unstableorbits are� by de�nition� not attractors� so that almost any orbit near an unstableperiodic orbit will diverge from it and head toward some attractor�


to pick a single initial condition from the unit interval� If fn�x�� everleaves the unit interval� then it will run o� to in�nity and never return�provided � �� Further� when studying attractors we can limit ourattention to values of � � �� If � � � � � then the only attractor isa stable �xed point at zero� and if � there are no attractors��

For all these reasons� a bifurcation diagram is a particularly powerfulmethod for studying the attractors in the quadratic map� Recall that abifurcation diagram is a plot of an asymptotic solution on the verticalaxis and a control parameter on the horizontal axis� To construct abifurcation diagram for the quadratic map only requires some simplemodi�cations of our previous program for iterating the quadratic map�As seen below� the new algorithm consists of the following steps�

�� Set � � �� and x� � �� almost any x� will do��

� Iterate the quadratic map �� times to remove the transientsolution� and then print � and xn for the next �� points�which are presumably part of the attractor�

�� Increment � by a small amount� and set x� to the last valueof xn�

� Repeat steps and � until � � �

A C program implementing this algorithm is as follows�

�� bifquad�c� calculate bifurcation diagram for the quadratic map�

input� none

output� l� x��

l� x��

etc��

l� x��

l� x��

etc�

��include stdio�h�

main

fint n�

�Technically� the phase space of the quadratic map� R� can be compacti�edthereby making the point at in�nity a valid attractor�

�� BIFURCATION DIAGRAM ��

Figure �� Bifurcation diagram for the quadratic map�

float lambda� x n�

x n � ��

forlambda � �� lambda � �� lambda �� fforn � �� n � �� n f

x n � lambda � x n � � � x n�

ifn � ��

printf �f �fnn � lambda� x n�

gg

g

When plotted in Figure �� for �� the output of oursimple program produces a bifurcation diagram of stunning complex�ity� Above the diagram we provide comments on the type of attractorobserved� and on the horizontal axis signi�cant parameter values areindicated� This bifurcation diagram shows many qualitative similari�ties to bifurcation diagrams from the bouncing ball system �compareto Figure �� Both exhibit the period doubling route to chaos� Forthe quadratic map an in�nite number of period doublings occur for


� � � � �� Both also show periodic windows �white bands� withinthe chaotic regions� For the quadratic map a period three windowbegins at � �� and a period �ve window begins at � �� Look�ing closely at the periodic windows we see that each branch of theseperiodic windows also undergoes a period doubling cascade�

Bifurcation diagrams showing only attracting solutions can be some�what misleading� Much of the structure in the bifurcation diagram canonly be understood by keeping track of both the stable attracting solu�tions and the unstable repelling solutions� as we did in constructing theorbit stability diagram �Figure �� Just as there are stable periodicorbits and chaotic attractors� there are also unstable periodic orbitsand chaotic repellers� In Figure �� for instance� we indicate the ex�istence of an unstable period one orbit by the dashed line beginningat the �rst period doubling bifurcation� As we show in section ��this unstable period one orbit is simply the continuation of the stableperiod one orbit that exists before this period doubling bifurcation �seeFigures �� and ��

�� Local Bifurcation Theory

Poincar�e used the term bifurcation to describe the �splitting of asymp�totic states of a dynamical system� Figure �� shows a bifurcationdiagram for the quadratic map �see Plate for a color version of aquadratic map bifurcation diagram�� As we examine Figure �� wesee that several di�erent types of changes can occur� We would liketo analyze and classify these bifurcations� At a bifurcation value� thequalitative nature of the solution changes� It can change to� or from� anequilibrium� periodic� or chaotic state� It can change from one type ofperiodic state to another� or from one type of chaotic state to another�

For instance� in the bouncing ball system we are initially in anequilibrium state� with the ball moving in unison with the table� Aswe turn up the table amplitude we �rst �nd sticking solutions� Aswe increase the amplitude further� we �nd a critical parameter valueat which the ball switches from the sticking behavior to bouncing in aperiod one orbit� Such a change from an equilibrium state to a periodicstate is an example of a saddle�node bifurcation� As we further turn

�� LOCAL BIFURCATION THEORY ��

Figure �� Saddle�node bifurcation diagram�

up the table amplitude� we �nd that there is a second critical tableamplitude at which the ball switches from a period one to a periodtwo orbit� The analogous period doubling bifurcation in the quadraticmap occurs at � � �� Both the saddle�node and the period doublingbifurcations are examples of local bifurcations� At their birth �or death�all the orbits participating are localized in phase space� that is� they allstart out close together� Global bifurcations can also occur� althoughtypically these are more di�cult to analyze since they can give birth toan in�nite number of periodic orbits� In this section we analyze threesimple types of local bifurcations that commonly occur in nonlinearsystems� These are the saddle�node� period doubling� and transcriticalbifurcations ��

�� Saddle�node

In a saddle�node bifurcation a pair of periodic orbits are created �outof nothing� One of the periodic orbits is always unstable �the saddle��while the other periodic orbit is always stable �the node�� The basicbifurcation diagram for a saddle�node bifurcation looks like that shownin Figure �� The saddle�node bifurcation is fundamental to the studyof nonlinear systems since it is one of the most basic processes by whichperiodic orbits are created�


Figure �� Tangency mechanism for a saddle�node bifurcation�

A saddle�node bifurcation is also referred to as a tangent bifurcation

because of the mechanism by which the orbits are born� Consider thenth composite of some mapping function f� which is near to a tangencywith the line y � x� Let �sn be the value at which a saddle�nodebifurcation occurs� Notice in Figure �� that fn� is tangent to the liney � x at �sn� For � � �sn� no period n orbits exist in this neighborhood�but for � �sn two orbits are born� The local stability of a point ofa map is determined by �fn� �� Since fn� �x� is tangent to y � x at abifurcation� it follows that at �sn��

�fn�sn��x��

Tangent bifurcations abound in the quadratic map� For instance� apair of period three orbits are created by a tangent bifurcation in thequadratic map when � � � �

p� �� As illustrated in Figure ��

for � �� there are eight points of intersection� Two of the inter�section points belong to the period one orbits� while the remaining sixmake up a pair of period three orbits� Near to tangency� the absolutevalue of the slope at three of these points is greater than one�this is theunstable period three orbit� The remaining three points form the sta�ble periodic orbit� The birth of this stable period three orbit is clearlyvisible as the period three window in our numerically constructed bi�furcation diagram of the quadratic map� Figure �� In fact� all the

�See reference " # for more details�


Figure �� A pair of period three orbits created by a tangent bi�furcation in the quadratic map �shown with the unstable period oneorbits��

odd�period orbits of the quadratic map are created by some sort oftangent bifurcation�

�� Period Doubling

Period doubling bifurcations are evident when we consider an even num�ber of compositions of the quadratic map� In Figure �� we show thesecond iteration of the quadratic map near a tangency� Below the pe�riod doubling bifurcation� a single stable period one orbit exists� As �is increased� the period one orbit becomes unstable� and a stable periodtwo orbit is born� This information is summarized in the bifurcationdiagram presented in Figure �� Let �pd be the parameter value atwhich the period doubling bifurcation occurs� At this parameter valuethe period one and the nascent period two orbit coincide� As illustratedin Figure �� f��pd�x

�� is tangent to y � x so that �f��pd��x��

However� �fn�pd��x�� that is� at a period doubling bifurcation the

function determining the local stability of the periodic orbit is always�� Figure �� shows f �� just after period doubling� In general� for a


Figure �� Second iterate of the quadratic map near a tangency�

Figure �� Period doubling ��ip� bifurcation diagram�


Figure �� Tangency mechanism near a period doubling ��ip� bifur�cation�

period n to period n bifurcation� �f�n�pd��x�� and

�fn�pd��x��

A period doubling bifurcation is also known as a �ip bifurcation� Inthe period one to period two bifurcation� the period two orbit �ips fromside to side about its period one parent orbit� This is because f ��pd�x

��

�� see Prob� �� The �rst �ip bifurcation in the quadratic mapoccurs at � � � and was analyzed in sections �� where weconsidered the location and stability of the period one and period twoorbits in the quadratic map�

�� Transcritical

The last bifurcation we illustrate with the quadratic map is a trans�

critical bifurcation� in which an unstable and stable periodic orbit col�lide and exchange stability� A transcritical bifurcation occurs in thequadratic map when � � �tc � �� As in a saddle�node bifurcation�f ��tc � �� at a transcritical bifurcation� However� a transcritical bi�furcation also has an additional constraint not found in a saddle�nodebifurcation� namely�

f�tc�x��


For the quadratic map this �xed point is just the period one orbit atthe origin� x�� found from equation ��

A summary of these three types of bifurcations is presented in Figure�� Other types of local bifurcations are possible� a more completetheory for both maps and �ows is given in reference ��

�� Period Doubling Ad In nitum

A view of the bifurcation diagram for the quadratic map for � between�� and �� is presented in Figure �� This diagram reveals notone� but rather an in�nite number of period doubling bifurcations� As� is increased a period two orbit becomes a period four orbit� andthis in turn becomes a period eight orbit� and so on� This sequence ofperiod doubling bifurcations is known as a period doubling cascade� Thisprocess appears to converge at a �nite value of � around �� beyondwhich a nonperiodic motion appears to exist� This period doublingcascade often occurs in nonlinear systems� For instance� a similar perioddoubling cascade occurs in the bouncing ball system �Figure �� Theperiod doubling route is one common way� but certainly not the onlyway� by which a nonlinear system can progress from a simple behavior�one or a few periodic orbits� to a complex behavior �chaotic motionand the existence of an in�nity of unstable periodic orbits��

In �� Feigenbaum began to wonder about this period doublingcascade� He started playing some numerical games with the quadraticmap using his HP�� hand�held calculator� His wondering soon led to aremarkable discovery� At the time� Feigenbaum knew that this perioddoubling cascade occurred in one�dimensional maps of the unit inter�val� He also had some evidence that it occurred in simple systems ofnonlinear di�erential equations that model� for instance� the motion ofa forced pendulum� In addition to looking at the qualitative similaritiesbetween these systems� he began to ask if there might be some quan�

titative similarity�that is� some numbers that might be the same inall these di�erent systems exhibiting period doubling� If these numberscould be found� they would be �universal in the sense that they wouldnot depend on the speci�c details of the system�

Feigenbaum was inspired in his search� in part� by a very success�

�� PERIOD DOUBLING AD INFINITUM ��

Figure �� Summary of bifurcations�


Figure �� Bifurcation diagram showing period doubling in thequadratic map�

ful theory of universal numbers for second�order phase transitions inphysics�� A phase transition takes place in a system when a change ofstate occurs� During the ��s it was discovered that there were quan�titative measurements characterizing phase transitions that did not de�pend on the details of the substance used� Moreover� these universalnumbers in the theory of phase transitions were successfully measuredin countless experiments throughout the world� Feigenbaum wonderedif there might be some similar universality theory for dissipative non�linear systems ��

�Feigenbaum introduced the renormalization group approach of critical phenom�ena to the study of nonlinear dynamical systems� Additional early contributionsto these ideas came from Cvitanovi�c� and also Collet� Coullet� Eckmann� Lanford�and Tresser� The geometric convergence of the quadratic map was noted as earlyas �� by Myrberg �see C� Mira� Chaotic dynamics �World Scienti�c� New Jersey�� and also by Grossmann and Thomae in ��

�� PERIOD DOUBLING AD INFINITUM ��

��

Table �� Period doubling bifurcation values for the quadratic map�

By de�nition� such universal numbers are dimensionless� the speci�cmechanical details of the system must be scaled out of the problem�Feigenbaum began his search for universal numbers by examining theperiod doubling cascade in the quadratic map� He recorded� with thehelp of his calculator� the values of � at which the �rst few period dou�bling bifurcations occur� We have listed the �rst eight values �orbits upto period � in Table �� While staring at this sequence of bifurcationpoints� Feigenbaum was immediately struck by the rapid convergenceof this series� Indeed� he recognized that the convergence appears tofollow that of a geometric series� similar to the one we saw in equation�� when we studied the sticking solutions of the bouncing ball�

Let �n be the value of the nth period doubling bifurcation� andde�ne �� as limn�� n� Based on his inspiration� Feigenbaum guessedthat this sequence obeys a geometric convergence�� that is�

�� n � c� n �n��

where c is a constant� and is a constant greater than one� Usingequation �� and a little algebra it follows that if we de�ne by

� limn��

�n � �n��n�� n

� ��

then is a dimensionless number characterizing the rate of convergenceof the period doubling cascade�

The three constants in this discussion have been calculated as

�� and c � ��

�For a review of geometric series see any introductory calculus text� such as C�Edwards and D� Penny� Calculus and analytic geometry �Prentice�Hall� EnglewoodCli�s� NJ� �� p� ��


The constant is now called �Feigenbaum�s delta� because Feigenbaumwent on to show that this number is universal in that it arises in awide class of dissipative nonlinear systems that are close to the single�humped map� This number has been measured in experiments withchicken hearts� electronic circuits� lasers� chemical reactions� and liquidsin their approach to a turbulent state� as well as the bouncing ballsystem ��

To experimentally estimate Feigenbaum�s delta all one needs to dois measure the parameter values of the �rst few period doublings� andthen substitute these numbers into equation �� The geometricconvergence of is a mixed blessing for the experimentalist� In practiceit means that n converges very rapidly to �� so that only the �rstfew �n�s are needed to get a good estimate of Feigenbaum�s delta� Italso means that only the �rst few n�s can be experimentally measuredwith any accuracy� since the higher n�s bunch up too quickly to ��To continue with more technical details of this story� see Rasband�saccount of renormalization theory for the quadratic map ��

Feigenbaum�s result is remarkable in two respects� Mathematically�he discovered a simple universal property occurring in a wide class ofdynamical systems� Feigenbaum�s discovery is so simple and fundamen�tal that it could have been made in �� or in �� for that matter�Still� he had some help from his calculator� It took a lot of numericalwork to develop the intuition that led Feigenbaum to his discovery� andit seems unlikely that the computational work needed would have oc�curred without help from some sort of computational device such as acalculator or computer� Physically� Feigenbaum�s result is remarkablebecause it points the way toward a theory of nonlinear systems in whichcomplicated di�erential equations� which even the fastest computerscannot solve� are replaced by simple models�such as the quadraticmap�which capture the essence of a nonlinear problem� including itssolution� The latter part of this story is still ongoing� and there aresurely other gems to be discovered with some inspiration� perspiration�and maybe even a workstation�

�� SARKOVSKII�S THEOREM ��

�� Sarkovskii�s Theorem

In the previous section we saw that for � � � � �� an in�nite numberof periodic orbits with period n are born in the quadratic map� In aperiod doubling cascade we know the sequence in which these periodicorbits are born� A period one orbit is born �rst� followed by a periodtwo orbit� a period four orbit� a period eight orbit� and so on� Forhigher values of �� additional periodic orbits come into existence� Forinstance� a period three orbit is born when � � � �

p� �� as we

showed in section �� In this section� we will explicitly show that allpossible periodic orbits exist for � � One of the goals of bifurcationtheory is to understand the di�erent mechanisms for the birth and deathof these periodic orbits� Pinning down all the details of an individualproblem is usually very di�cult� often impossible� However� there isone qualitative result due to Sarkovskii of great beauty that applies toany continuous mapping of the real line to itself�

The positive integers are usually listed in increasing order�� However� let us consider an alternative enumeration thatre�ects the order in which a sequence of period n orbits is created� Forinstance� we might list the sequence of integers of the form n as

n � � � � � � � � � � � � � ��

where the symbol � means �implies� In the quadratic map systemthis ordering says that the existence of a period n orbit implies theexistence of all periodic orbits of period i for i � n� We saw thisordering in the period doubling cascade� A period eight orbit thusimplies the existence of both period four and period two orbits� Thisordering diagram says nothing about the stability of any of these orbits�nor does it tell us how many periodic orbits there are of any givenperiod�

Consider the ordering of all the integers given by

� � � � � � � � � � � � � � � � � � � � � � � � � � �

� n � � � n � � � n � � � n � � � � � �

� n � � � � � ��

with n � �� Sarkovskii�s theorem says that the ordering found inequation �� holds� in the sense of the n ordering above� for any


continuous map of the real line R to itself�the existence of a periodi orbit implies the existence of all periodic orbits of period j where jfollows i in the ordering� Sarkovskii�s theorem is remarkable for its lackof hypotheses �it assumes only that f is continuous�� It is of great helpin understanding the structure of one�dimensional maps�

In particular� this ordering holds for the quadratic map� For in�stance� the existence of a period seven orbit implies the existence ofall periodic orbits except a period �ve and a period three orbit� Andthe existence of a single period three orbit implies the existence ofperiodic orbits of all possible periods for the one�dimensional map�Sarkovskii�s theorem forces the existence of period doubling cascadesin one�dimensional maps� It is also the basis of the famous statementof Li and Yorke that �period three implies chaos� where chaos looselymeans the existence of all possible periodic orbits� An elementaryproof of Sarkovskii�s theorem� as well as a fuller mathematical treat�ment of maps as dynamical systems� is given by Devaney in his bookAn Introduction to Chaotic Dynamical Systems ��

Sarkovskii�s theorem holds only for mappings of the real line toitself� It does not hold in the bouncing ball system because it is a mapin two dimensions� It does not hold for mappings of the circle� S�� toitself� Still� Sarkovskii�s theorem is a lovely result� and it does point theway to what might be called �qualitative universality� that is� generalstatements� usually topological in nature� that are expected to hold fora large class of dynamical systems�

�� Sensitive Dependence

In section �� we saw how a measurement of �nite precision in thebouncing ball system has little predictive value in the long term� Suchbehavior is typical of motion on a chaotic attractor� We called such

In section �� we show there exists a close connection between the existenceof an in�nity of periodic orbits and the existence of a chaotic invariant set� notnecessarily an attractor� The term �chaos� in nonlinear dynamics is due to Li andYorke� although the current usage di�ers somewhat from their original de�nition�see T� Y� Li and J� A� Yorke� Period three implies chaos� Am� Math� Monthly ��

�� SENSITIVE DEPENDENCE ��

behavior sensitive dependence on initial conditions� For the specialvalue � � in the quadratic map we can analyze this behavior in somedetail�

Consider the transformation xn � sin��n� applied to the quadraticmap when � � � Making use of the identity

sin � � sin� cos��

we �nd

xn�� xn�� xn� ��sin��n�� sin��n�� sin��n��

� sin��n� cos��n�

� � sin��n� cos��n��

� �sin��n�� n�� n mod ��

This last linear di�erence equation has the explicit solution

�n � n�� mod ��

Sensitive dependence on initial conditions is easy to see in this ex�ample when we express the initial condition as a binary number�

�� b�

�b�

�b��

� � � � ��Xi�

bii��

� bi � f�� g� ��

Now the action of equation �� on an initial condition �� is a shiftmap� At each iteration we multiply the previous iterate by two �� inbinary�� which is a left shift� and then apply the mod function� whicherases the integer part� For example� if �� in binary�then

��

�� mod �

� �� shift left and drop the integer part�

��

��

��


and we see the precision of our initial measurement evaporating beforeour eyes�

We can even quantify the amount of sensitive dependence the systemexhibits� that is� the rate at which an initial error grows� Assuming thatour initial condition has some small error �� the growth rate of the erroris

fn�� fn�� n�� n�� n� � �en�ln��

If we think of n as time� then the previous equation is of the form eat

with a � ln � In this example the error grows at a constant exponentialrate of ln � The exponential growth rate of an initial error is thede�ning characteristic of motion on a chaotic attractor� This rate ofgrowth is called the Lyapunov exponent� A strictly positive Lyapunovexponent� such as we just found� is an indicator of chaotic motion�The Lyapunov exponent is never strictly positive for a stable periodicmotion�

�� Fully Developed Chaos

The global dynamics of the quadratic map are well understood for� � � � �� namely� almost all orbits beginning on the unit intervalare asymptotic to a period one �xed point� We will next show thatthe orbit structure is also well understood for � � This is known asthe hyperbolic regime� This parameter regime is �fully developed inthe sense that all of the possible periodic orbits exist and they are allunstable�� No chaotic attractor exists in this parameter regime� butrather a chaotic repeller� Almost all initial conditions eventually leave�or are repelled from� the unit interval� However� a small set remains�This remaining invariant set is an example of a fractal�

The analysis found in this book is based substantially on sections ��to �� of Devaney�s An Introduction to Chaotic Dynamical Systems ��

For a well�illustrated exploration of the Lyapunov exponent in the quadraticmap system see A� K� Dewdney� Leaping into Lyapunov space� Sci� Am� �� pp� ��

��Technically� the system is �structurally stable�� See section �� of Devaney�sbook for more details� Hyperbolicity and structural stability usually go hand�in�hand�

�� FULLY DEVELOPED CHAOS ��

This section is more advanced mathematically than previous sections�The reader should consult Devaney�s book for a complete treatment�Section �� contains a more pragmatic description of the symbolicdynamics of the quadratic map and can be read independently of thecurrent section�

�� Hyperbolic Invariant Sets

We begin with some de�nitions�

De�nition� A set or region # is said to be invariant under the map fif for any x� � # we have fn�x�� # for all n�

The simplest example of an invariant set is the collection of pointsforming a periodic orbit� But� as we will see shortly� there are morecomplex examples� such as strange invariant sets� which are candidatesfor chaotic attractors or repellers�

De�nition� For mappings of R �� R� a set # � R is a repelling�resp�� attracting� hyperbolic set for f if # is closed� bounded� andinvariant under f and there exists an N � such that j�fn��x�j ��resp�� for all n N and all x � # ��

This de�nition says that none of the derivatives of points in theinvariant set are exactly equal to one� A simple example of a hyperbolicinvariant set is a periodic orbit that is either repelling or attracting� butnot neutral� In higher dimensions a similar de�nition of hyperbolicityholds� namely� all the points in the invariant set are saddles�

The existence of both a simple periodic regime and a complicatedfully developed chaotic �yet well understood� hyperbolic regime turnsout to be quite common in low�dimensional nonlinear systems� In Chap�ter � we will show how information about the hyperbolic regime� whichwe can often analyze in detail using symbolic dynamics� can be ex�ploited to determine useful physical information about a nonlinear sys�tem�

In examining the dynamics of the quadratic map for � weproceed in two steps� �rst� we examine the invariant set� and second�we describe how orbits meander on this invariant set� The set itself is


Figure �� Quadratic map for � � �Generated by the QuadraticMap program��

a fractal Cantor set �� and to describe the dynamics on this fractalset we employ the method of symbolic dynamics�

Since f�� for � there exists an open interval centered at�� with points that leave the unit interval after one iteration� never toreturn� Call this open set A� �see Figure �� These are the points inA� whose image under f is greater than one� On the second iteration�more points leave the unit interval� In fact� these are the points thatget mapped to A� after the �rst iteration� A� � fx � Ijf�x� � A�g�Inductively� de�ne An � fx � Ijf i�x� � I for i n but fn��x� �� Ig�that is� An consists of all points that escape from I at the n � �stiteration� Clearly� the invariant limit set� call it &� consists of all theremaining points

& � I ��

��n�

An

��

What does & look like" First� note that An consists of n disjoint


open intervals� so &n � I � �A�S��

SAn� is n�� disjoint closed in�

tervals� Second� fn�� monotonically maps each of these intervals ontoI� The graph of fn�� is a polynomial with n humps� The maximalsections of the humps are the collection of intervals An that get mappedout of I� but more importantly this polynomial intersects the y � xline n�� times� Thus� &n has n�� periodic points in I�

The set & is a Cantor set if & is a closed� totally disconnected�perfect subset� A set is disconnected if it contains no intervals� a set isperfect if every point is a limit point� It is not too hard to show thatthe invariant set de�ned by equation �� is a Cantor set �� Thus�we see that the invariant limit set arising from the quadratic map for� is a fractal Cantor set with a countable in�nity of periodic orbits�

�� Symbolic Dynamics

Our next goal is to unravel the dynamics on &� In beginning this taskit is useful to think how the unit interval gets stretched and folded witheach iteration� The transformation of the unit interval under the �rstthree iterations for � � is illustrated in Figure �� This diagramshows that the essential ingredients that go into making a chaotic limitset are stretching and folding� The technique of symbolic dynamics isa bookkeeping procedure that allows us to systematically follow thisstretching and folding process� For one�dimensional maps the completesymbolic theory is also known as kneading theory ��

We begin by de�ning a symbol space for symbolic dynamics� Let'� � fs � �s�s�s��jsj � � or �g� '� is known as the sequence space

on the symbols � and �� We sometimes use the symbols L �Left� andR �Right� to denote the symbols � and � �see Figure �� If we de�nethe distance between two sequences s and t by

d�s� t� ��Xi�

jsi � tiji

� ��

then '� is a metric space� The metric d�s� t� induces a topology on'� so we have a notion of open and closed sets in '�� For instance�if s � �� and t � �� then the metric d�s� t� ��


Figure �� Keeping track of the stretching and folding of the quadraticmap with symbolic dynamics�


A dynamic on the space '� is given by the shift map � � '� �� '�

de�ned by ��s�s�s�� s�s�s�� That is� the shift map drops the�rst entry and moves all the other symbols one place to the left� Theshift map is continuous� Brie�y� for any � �� pick n such that ��n �� and let � ��n�� Then the usual �� proof goes through when weuse the metric given by equation ��

What do the orbits in '� look like" Periodic points are identi�edwith exactly repeating sequences� s � �s� � � � sn�� s� � � � sn�� Forinstance� there are two distinct period one orbits� given by �� and �� The period two orbit takes the form �� and �� and one of the period three orbits looks like�� and �� and so on� Evi�dently� there are n periodic points of period n� although some of thesepoints are of a lower period� But there is more� The periodic pointsare dense in '�� that is� any nonperiodic point can be represented asthe limit of some periodic sequence� Moreover� the nonperiodic pointsgreatly outnumber the periodic points�

What does this have to do with the quadratic map� or more exactlythe map f� restricted to the invariant set &" We now show that it is the�same map� and thus to understand the orbit structure and dynamicsof f� on & we need only understand the shift map� �� on the space oftwo symbols� '�� We can get a rough idea of the behavior of an orbitby keeping track of whether it falls to the left �L or �� or right �R or�� at the nth iteration� See Figure �� for a picture of this partition�That is� the symbols � and � tell us the fold which the orbit lies on atthe nth iteration�

Accordingly� de�ne the itinerary of x as the sequence S�x� � s�s�s��where sj � � if f j��x� � �� and sj � � if f j��x� �� Thus� theitinerary of x is an in�nite sequence of �s and �s� it �lives in '�� Fur�ther� we think of S as a map from & to '�� If � � then it can beshown that S � & �� '� is a homeomorphism �a map is a homeomor�phism if it is a bijection and both f and f�� are continuous�� Thislast result says that the two sets & and '� are the same� To show theequivalence between the dynamics of f� on & and � on '�� we need thefollowing theorem� which is quoted from Devaney�

Theorem� If � �p

�� then S � & �� '� is a homeomorphism and


S � f� � � � S �

Proof� See section �� in Devaney�s book �� This theorem holds forall � � but the proof is more subtle�

As we show in the next section� the essential idea in this proof isto keep track of the preimages of points not mapped out of the unitinterval� The symbolic dynamics of the invariant set gives us a way touniquely name the orbits in the quadratic map that do not run o� toin�nity� In particular� the itinerary of an orbit allows us to name� andto �nd the relative location of� all the periodic points in the quadraticmap� Symbolic dynamics is powerful because it is easy to keep track ofthe orbits in the symbol space� It is next to impossible to do this usingonly the quadratic map since it would involve solving polynomials ofarbitrarily high order�

�� Topological Conjugacy

This last example suggests the following notion of equivalence of dy�namical systems� which was originally put forth by Smale �� and isfundamental to dynamical systems theory�

De�nition� Let f � A �� A and g � B �� B be two maps� Thefunctions f and g are said to be topologically conjugate if there existsa homeomorphism h � A �� B such that h � f � g � h�

The homeomorphism is called a topological conjugacy� and is morecommonly de�ned by simply stating that the following diagram com�mutes�

f

A �� A

h � � hg

B �� B

Using the theorem of the previous section� we know that if � �p

�then f� �the quadratic map� is topologically conjugate to � �the shift


map�� Topologically conjugate systems are the same system insofar asthere is a one�to�one correspondence between the orbits of each system�Sometimes this is too restrictive and we only require that the mappingbetween orbits be many�to�one� In this latter case we say the twodynamical systems are semiconjugate�

In nonlinear dynamics� it is often advantageous to establish a conju�gacy or a semiconjugacy between the dynamical system in question andthe dynamics on some symbol space� The properties of the dynamicalsystem are usually easy to see in the symbol space and� by the conju�gacy or semiconjugacy� these properties must also exist in the originaldynamical system� For instance� the following properties are easy toshow in '� and must also hold in &� namely�

�� The cardinality of the set of periodic points �often writtenas Pern�f�� is n�

� Per�f�� is dense in &�

�� f� has a dense orbit in & ��

Although there is no universally accepted de�nition of chaos� mostde�nitions incorporate some notion of sensitive dependence on initialconditions� Our notions of topological conjugacy and symbolic dynam�ics give us a promising way to analyze chaotic behavior in a speci�cdynamical system�

In the context of one�dimensional maps� we say that a map f � I ��I possesses sensitive dependence on initial conditions if there exists a � such that� for any x � I and any neighborhood N of x� there existy � N and n � such that jfn�x� � fn�y�j � This says that smallerrors due either to measurement or round�o� errors become magni�edupon iteration�they cannot be ignored�

Let S� denote the unit circle� Here we will think of the members ofS� as being normalized to the range �� A simple example of a mapthat is chaotic in the above sense is given by g � S� �� S� de�ned byg�� As we saw in section �� when � is written in base two�g�� is simply a shift map on the unit circle� In ergodic theory the aboveshift map is known as a Bernoulli process� If we think of each symbol� as a Tail �T�� and each symbol � as a Head �H�� then the above shift


map is topologically conjugate to a coin toss� our intuitive model ofa random process� Each shift represents a toss of the coin� We nowshow that the shift map is essentially the same as the quadratic mapfor � � � that is� the quadratic map �a fully deterministic process� canbe as random as a coin toss�

If f��x� � x�� x�� then the limit set is the whole unit inter�val I � �� since the maximum f�� that is� the map isstrictly onto �the map is measure�preserving and is roughly analogousto a Hamiltonian system that conserves energy�� To continue withthe analysis� de�ne h� � S� �� by h� � cos�� Also de�neq�x� � x� � �� Then

h� � g�� cos��

� cos��

� q � h��

so h� conjugates g and q� Note� however� that h� is two�to�one at mostpoints so that we only have a semiconjugacy� To go further� if we de�neh� � �� by h��t� � �

�� t�� then f� � h� � h� � q� Thenh� � h� � h� is a topological semiconjugacy between g and f�� we haveestablished the semiconjugacy between the chaotic linear circle mapand the quadratic map when � � � The reader is invited to workthrough a few examples to see how the orbits of the quadratic map� thelinear circle map� and a coin toss can all be mapped onto one another�

�� Symbolic Coordinates

In the previous section we showed that when � � the dynamics ofthe quadratic map restricted to the invariant set & are �the same asthose given by the shift map � on the sequence space on two symbols�'�� We established this correspondence by partitioning the unit inter�val into two halves about the maximum point of the quadratic map�x � �� The left half of the unit interval is labeled � while the right halfis labeled �� as illustrated in Figure �� To any orbit of the quadraticmap fn�x�� we assign a sequence of symbols s � �s�s�s� � � ��for ex�ample� ��called the itinerary� or symbolic future� of the orbit�

�� SYMBOLIC COORDINATES ��

Each si represents the half of the unit interval in which the ith iterationof the map falls�

In part� the theorem of section �� says that knowing an orbit�sinitial condition is exactly equivalent to knowing an orbit�s itinerary�Indeed� if we imagine that the itinerary is simply an expression for somebinary number� then perhaps the correspondence is not so surprising�That is� the mapping fn� takes some initial coordinate number x� andtranslates it to a binary number �� s�� s�� constructed from thesymbolic future� which can be thought of as a �symbolic coordinate�

From a practical point of view� the renaming scheme described bysymbolic dynamics is very useful in at least two ways�

�� Symbolic dynamics provides a good way to label all the periodicorbits�

� The symbolic itinerary of an orbit provides the location of theorbit in phase space to any desired degree of resolution�

We will explain these two points further and in the process show thecorrespondence between � and x��

In practical applications we shall be most concerned with keepingtrack of the periodic orbits� Symbolic itineraries of periodic orbits arerepeating �nite strings� which can be written in various forms� such as

�s�s� � � � sn�� s�s� � � � sn�� s�s� � � � sn�� s�s� � � � sn��

To see the usefulness of the symbolic description� let us consider thefollowing problem� for � � �nd the approximate location of all theperiodic orbits in the quadratic map�

�� Whats in a name Location�

As discussed in section �� the exact location of a period n orbit isdetermined by the roots of the �xed point equation� fn�x� � x� Thisnaive method of locating the periodic points is impractical in generalbecause it requires �nding the roots of an arbitrarily high�order poly�nomial� We now show that the problem is easy to solve using symbolicdynamics if we ask not for the exact location� but only for the locationrelative to all the other periodic orbits�


If � � then there exists an interval centered about x � �� forwhich f�x� �� Call this interval A� �see Figure �� Clearly� noperiodic orbit exists in A� since all points in A� leave the unit intervalI at the �rst iteration� and thereafter escape to �� As we arguedin section �� the periodic points must be part of the invariant set�those points that are never mapped into A��

As shown in Figure �� the points in the invariant set can beconstructed by considering the preimages of the unit interval foundfrom the inverse map f�� The �rst iteration of f�� produces twodisjoint intervals�

f�� I��

I� I�

which are labeled I� �the left interval� and I� �the right interval�� Asindicated by the arrows in Figure �� I� preserves orientation� whileI� reverses orientation� The orientation of the interval is simply deter�mined by the slope �derivative� of f��x��

f ��x� � if x � �� preserves orientation�

f ��x� � � if x �� reverses orientation�

We view f��I� as a �rst�level approximation to the invariant set&� In particular� f��I� gives us a very rough idea as to the location ofboth period one orbits� one of which is located somewhere in I�� whilethe other is located somewhere in I��

To further re�ne the location of these periodic orbits� consider theapplication of f�� to both I� and I��

f�� I�� f�� I��

I�� I�� I�� I��

The second iteration gives rise to four disjoint intervals� Two of thesecontain the distinct period one orbits� and the remaining two intervalscontain the period two orbit�

I�� I��f�� I��


Figure �� Symbolic coordinates and the alternating binary tree�


I�� I��f�� I��

I�� I��f�� I��

I�� I��f�� I��

In general we can de�ne n disjoint intervals at the nth level of re�ne�ment by

Is�s��sn�� Is�

�f�� Is��

��

�f��n�� Isn��

��

With each new re�nement� we hone in closer and closer to the periodicorbits�

The one�to�one correspondence between x� and s is easy to see ge�ometrically by observing that� as n��

n��

Is�s��sn

forms an in�nite intersection of nested nonempty closed intervals thatconverges to a unique point in the unit interval�� The invariant limitset is the collection of all such limit points� and the periodic points areall those limit points indexed by periodic symbolic strings�

�� Alternating Binary Tree

We must keep track of two pieces of information to �nd the location ofthe orbits at the nth level� the relative location of the interval Is�s��sn��

and its orientation� A very convenient way to encode both pieces of datais through the construction of a binary tree that keeps track of all theintervals generated by the inverse function� f�� The quadratic mapgives rise to the �alternating binary tree illustrated in Figure ��b��

The nth level of the alternating binary tree has n nodes� which arelabeled from left to right by the sequence

nth level �

�nz � ��

��A partition of phase space that generates a one�to�one correspondence betweenpoints in the limit set and points in the original phase space is known in ergodictheory as a generating partition� Physicists loosely call such a generating partitiona �good partition� of phase space�


This sequence starts at the left with a zero� It is followed by a pair ofones� then a pair of zeros� and so on until n digits are written down�To form the alternating binary tree� we construct the above list of �sand �s from level one to level n and then draw in the pair of lines fromeach i� �st level node to the adjacent nodes at the ith level�

Now� to �nd the symbolic name for the interval at the nth level�Is�s��sn��

� we start at the topmost node� s�� and follow the path downthe alternating binary tree to the nth level� reading o� the appropriatesymbol name at each level along the way� By construction� we seethat the symbolic name read o� at the nth level of the tree mimics thelocation of the interval containing a period n orbit�

More formally� we identify the set of repeating sequences of periodn in '� with the set of �nite strings s�s� � � � sn�� Let ��s�� s�� sn��denote the fraction between � and �n � ��n giving the order� fromleft to right� generated by the alternating binary tree� Further� letN�s�� s�� sn�� denote the integer position between � and n � �and let B denote N in binary form� It is not too di�cult to show thatB�s�� s�� sn�� b�b� � � � bn�� where bi � � or �� and

��s�� s�� sn�� b�

�b�

� � � ��bn��n

��

N�s�� s�� sn�� b� � n�� b� � n�� bn��

bi �iX

j�

sj mod � ��

An application of the ordering relation can be read directly o� of Figure��b� for n � � and is presented in Table �� As expected� the left�most orbit is the string �� which corresponds to the period one orbitat the origin� x�� Less obvious is the position of the other period oneorbit� x�� which occupies the �fth position at the third level�

The itinerary of a periodic orbit is generated by a shift on the re�peating string �s�s� � � � sn��

��s�s� � � � sn�� s�s� � � � sn��s��

In this case� the shift is equivalent to a cyclic permutation of the sym�bolic string� For instance� there are two period three orbits shown in


N B s�s�s�x position binary x position symbolic name

� ��

Table �� Symbolic coordinate for n � � from the alternating binarytree�

Table �� their itineraries and positions are

s�s�s� � �� N � � ��

ands�s�s� � ��

N � ��

So the itinerary of a periodic orbit is generated by cyclic permutationsof the symbolic name�

�� Topological Entropy

To name a periodic orbit� we need only choose one of its cyclic per�mutations� The number of distinct periodic orbits grows rapidly withthe length of the period� The symbolic names for all periodic orbitsup to period eight are presented in Table �� A simple indicator ofthe complexity of a dynamical system is its topological entropy� In theone�dimensional setting� the topological entropy� which we denote byh� is a measure of the growth of the number of periodic cycles as a


� ��

Table �� Symbolic names for all periodic orbits up to period eightoccurring in the quadratic map for � � All names related by a cyclicpermutation are equivalent�

function of the symbol string length �period��

h � limn��

lnNn

n� ��

where Nn is the number of distinct periodic orbits of length n� Forinstance� for the fully developed quadratic map� Nn is of order n� so

h � ln ��

The topological entropy is zero in the quadratic map for any value of� below the accumulation point of the �rst period doubling cascadebecause Nn is of order n in this regime� The topological entropyis a continuous� monotonically increasing function between these twoparameter values� The topological entropy increases as periodic orbitsare born by di�erent bifurcation mechanisms� A strictly positive valuefor the topological entropy is sometimes taken as an indicator for theamount of �topological chaos�

In addition to its theoretical importance� symbolic dynamics willalso be useful experimentally� It will help us to locate and organizethe periodic orbit structure arising in real experiments� In Chapter� we will show how periodic orbits can be extracted and identi�ed


from experimental data� We will further describe how to constructa periodic orbit�s symbolic name directly from experiments and howto compare this with the symbolic name found from a model� such asthe quadratic map� Reference �� describes an additional re�nement ofsymbolic dynamics called kneading theory� which is useful for analyzingnonhyperbolic parameter regions� such as occur in the quadratic mapfor � � � � �

Notice that the ordering relation described by the alternating bi�nary tree between the periodic orbits does not change for any � � Asimple observation� which will nevertheless be very important from anexperimental viewpoint� is the following� this ordering relation� which

is easy to calculate in the hyperbolic regime� is often maintained in the

nonhyperbolic regime� This is the case� for instance� in the quadraticmap for all � �� This observation is useful experimentally because itwill give us a way to name and locate periodic orbits in an experimen�tal system at parameter values where a nonhyperbolic strange attractorexists� That is� we can name and identify periodic orbits in a hyper�bolic regime� where the system can be analyzed analytically� and thencarry over the symbolic name for the periodic orbit from the hyperbolicregime to the nonhyperbolic regime� where the system is more di�cultto study rigorously� Symbolic dynamics and periodic orbits will be our�breach through which we may attempt to penetrate an area hithertodeemed inaccessible ��

The reader might notice that our symbolic description of thequadratic map used very little that was speci�c to this map� Thesame description holds� in fact� for any single�humped �unimodal� mapof the interval� Indeed� the topological techniques we described herein terms of binary trees extend naturally to k symbols on k�ary treeswhen a map with many humps is encountered�

This concludes our introduction to the quadratic map� There arestill many mysteries in this simple map that we have not yet begunto explore� such as the organization of the periodic window structure�but at least we can now continue our journey into nonlinear lands withwords and pictures to describe what we might see ��

Usage of Mathematica ��

Usage of Mathematica

In this section� we illustrate how Mathematica�� can be used for re�search or course work in nonlinear dynamics� Mathematica is a com�plete system for doing symbolic� graphical� and numerical manipula�tions on symbols and data� and is commonly available on a wide rangeof machines from microcomputers ��s and Macs� to mainframes andsupercomputers�

Mathematica is strong in two� and three�dimensional graphics� and�where appropriate� we would encourage its use in a nonlinear dynamicscourse� It can serve at least three important functions� �� a means ofgenerating complex graphical representations of data from experimentsor simulations� �� a method for double�checking complex algebraicmanipulations �rst done by hand� and �� a general system for writ�ing both numerical and symbolic routines for the analysis of nonlinearequations�

What follows is text from a typical Mathematica session� typesetfor legibility� used to produce some of the graphics and to double�checksome of the algebraic results presented in this chapter� Of course� areal Mathematica session would not be so heavily commented�� This is a Mathematica comment statement� Mathematica ignores

everything between star parentheses� �

� To try out this Mathematica session yourself� type everything in

bold that is not enclosed in the comment statements� The following

line is Mathematica�s answer typed in italic� Mathematica�s output

from graphical commands are not printed here� but are left for the

reader to discover� �

� This notebook is written on a Macintosh� �� nbt� �

� In this notebook we will analytically solve for the period one and

period two orbits in the quadratic map and plot their locations as a

function of the control parameter� lambda� �

��Mathematica is a trademark of Wolfram Research Inc� For a brief introductionto Mathematica see S� Wolfram� Mathematica� a system for doing mathematics bycomputer �Addison�Wesley� New York� �� pp� ��

�� Quadratic Map

� First we define the quadratic function with the Mathematica

Function�fx�y� ��g� fx�y� �� command� which takes two

arguments� the first of which is a list of variables� and the second

of which is the function� The basic data type in Mathematica

is a list� and all lists are enclosed between braces�

fx�� x�� x�� g� Note that all arguments and functions in

Mathematica are enclosed in square brackets f�� which differs from

the standard mathematical notation of parentheses f� This is� in

part� because square brackets are easier to reach on the keyboard�

Also� note that in defining a variable one must always put a space

around it� so xy is equal to a single variable named xy � while

x y with a space between is equal to two variables� x and y� �

� To evaluate a Mathematica expression tap the enter key� not the

return key� Now to our first Mathematica command� �

f � Function flambda� xg� lambda x �� x��

Function�flambda� xg� lambda x � x��

� Mathematica should respond by saying Out�� which tells us that

Mathematica successfully processed the first command and has put the

result in the variable Out�� as well as the variable we created� f�

To evaluate the quadratic map� we now can feed f two arguments�

the first of which is lambda� and the second of which is x� �

f ��

� Mathematica should respond with the function Out�� which

contains the quadratic map evaluated at lambda � � and x � �� To

evaluate a list of values for x we could let the x variable be a

list� i�e�� a series of numbers enclosed in braces fx�� x�� x�� g�To try this command� type� �

f �� f� �� g�f�� g

Usage of Mathematica ��

� To plot the quadratic map we use the Mathematica plot command�

Plot�f� fx� xmin� xmaxg�� For instance a plot of the quadratic map

for lambda � � is given by� �

Plot f ��x�� fx� � �g�

� It is as easy as pie to take a composite function in Mathematica�

just type f�f�x�� so to plot ffx for the quadratic map we

simply type� �

Plot f ��f ��x�� fx� � �g�

� Now let�s find the locations of the period one orbits� given by

the roots of fx � x� To find the roots� we use the Mathematica

command Solve�eqns� vars�� Notice that we are going to rename the

parameter lambda to a just to save some space when printing the

answer� The double equals �� in Mathematica is equivalent to the

single equal � of mathematics� �

Solve f a� x�� x�� x��

ffx �� a��g� fx �� gg

� As expected� Mathematica finds two roots� Let�s make a function

out of the first root so that we can plot it later using Plot� To do

this we need the following sequence of somewhat cryptic commands� �

r� � � ��

fx �� a��g

� The roots are saved in a list of two items� To pull out the

first item we used the � command� a Mathematica variable that always

holds the value of the last expression� In this case it holds the

list of two roots� and the double square bracket notation ��

tells Mathematica we want the first item in the list of two items�

Now we must pull out the last part of the expression� � � ��a� with

the replacement command Replace�expr� rules��


x� � Replace x�� r��

� a��

� To plot the location of the period one orbit we just use the Plot

command again� �

Plot x�� fa� �� g�

� To find the location of the period two orbit�

we solve for ffx � x� �

Solve f a�f a� x�� x�� x��

ffx �� g� fx �� a��g�fx �� a�� a�� a�� a�� g�fx �� a�� a�� a�� a�� gg

� We find four roots� as expected� Before proceeding further� it�s

a good idea to try and simplify the algebra for the last two new

roots by applying the Simplify command to the last expression� �

rt � Simplify ��

ffx �� g� fx �� a��g�fx �� a� � �a�� a�� g�fx �� a� � �a�� a�� gg

� And we can now pull out the positive and negative roots of the

period two orbit� �

x�plus � Replace x�� rt ��

x�minus � Replace x�� rt ��

� � � � ��a� � �a�� a��

� As a last step� we can plot the location of the period one orbit

and both branches of the period two orbit by making a list of

functions to be plotted� �

References and Notes ��

Plot fx�� x�plus� x�minusg� fa� �� g�

� This is how we originally plotted the orbit stability diagram

in the text� Mathematica can go on to find higher�order periodic

orbits by numerically finding the roots of the nth composite of f�

if no exact solution exists� �


�� There are several excellent reviews of the quadratic map� Some of the oldestare still the best� and all of the following are quite accessible to undergraduates�E� N� Lorenz� The problem of deducing the climate from the governing equa�tions� Tellus �� E� N� Lorenz� Deterministic nonperiodic �ow� J�Atmos� Sci� �� R� M� May� Simple mathematical models withvery complicated dynamics� Nature �� M� J� Feigenbaum�Universal behavior in nonlinear systems� Los Alamos Science �� These last two articles are reprinted in the book� P� Cvitanovi�c� ed�� Universal�ity in Chaos �Adam Hidgler Ltd� Bristol� �� All four are models of goodexpository writing� A simple circuit providing an analog simulation of thequadratic map suitable for an undergraduate lab is described by T� Mishina�T� Kohmoto� and T� Hashi� Simple electronic circuit for the demonstration ofchaotic phenomena� Am� J� Phys� ��

�� A good review of the dynamics of the quadratic map from a dynamical systemsperspective is given by R� L� Devaney� Dynamics of simple maps� in Chaos andfractals� The mathematics behind the computer graphics� Proc� Symp� AppliedMath� �� edited by R� L� Devaney and L� Keen �AMS� Rhode Island��

�� For an elementary proof� see E� A� Jackson� Perspectives in nonlinear dynamics�Vol� � �Cambridge University Press� New York� �� pp� � ��

�� Counting the number of periodic orbits of period n is a very pretty combina�torial problem� See Hao B��L�� Elementary symbolic dynamics and chaos indissipative systems �World Scienti�c� New Jersey� �� pp� �� An up�dated version for some results in this book can be found in Zheng W��M� andHao B��L�� Applied symbolic dynamics� in Experimental study and characteri�zation of chaos� edited by Hao B��L� �World Scienti�c� New Jersey� �� Alsosee the original analysis published in the article by M� Metropolis� M� L� Stein�and P� R� Stein� On the �nite limit sets for transformations of the unit interval�J� Comb� Theory �� Also reprinted in Cvitanovi�c� reference "�#�


�� For a discussion concerning the existence of a single stable attractor in thequadratic map and its relation to the Schwarzian derivative� see Jackson� ref�erence "�#� pp� �� and Appendix D� pp� ��

�� A more complete mathematical account of local bifurcation theory is presentedby S� N� Rasband� Chaotic dynamics of nonlinear systems �John Wiley $ Sons�New York� �� pp� � �� and pp� �� Chapter � deals with universalitytheory from the quadratic map�

�� G� Iooss and D� D� Joseph� Elementary stability and bifurcation theory �Springer�Verlag� New York� ��

�� For more about this tale see the chapter �Universality� of J� Gleick� Chaos�Making a new science� �Viking� New York� ��

�� See the introduction of P� Cvitanovi�c� ed�� Universality in Chaos �AdamHidglerLtd� Bristol� ��

�� R� L� Devaney� An introduction to chaotic dynamical systems� second edition�Addison�Wesley� New York� �� Section �� covers Sarkovskii�s Theo�rem and section �� covers kneading theory� Another elementary proof ofSarkovskii�s Theorem can be found in H� Kaplan� A cartoon�assisted proof ofSarkovskii�s Theorem� Am� J� Phys� ��

�� K� Falconer� Fractal geometry �John Wiley $ Sons� New York� ��

�� S� Smale� The mathematics of time� Essays on dynamical systems� economicprocesses� and related topics �Springer�Verlag� New York� ��

�� P� Cvitanovi�c� G� H� Gunaratne� and I� Procaccia� Topological and metricproperties of H�enon�type strange attractors� Phys� Rev� A ��

�� Some papers providing a hands�on approach to kneading theory include� P�Grassberger� On symbolic dynamics of one�humped maps of the interval� Z�Naturforsch� ��a� �� J��P� Allouche and M� Cosnard� It�erationsde fonctions unimodales et suites engendr�ees par automates� C� R� Acad� Sc�Paris S�erie I �� and reference "��#� Also see section �� ofDevaney�s book� reference "��#� for a nice mathematical account of kneadingtheory� The classic reference in kneading theory is J� Milnor and W� Thurston�On iterated maps of the interval� Lect� Notes in Math� �� in Dynamical Sys�tems Proceedings� University of Maryland �� edited by J� C� Alexander�Springer�Verlag� Berlin� �� pp� �� The early mathematical con�tributions of Fatou� Julia� and Myrberg to the dynamics of maps� symbolicdynamics� and kneading theory are emphasized by C� Mira� Chaotic dynamics�From the one�dimensional endomorphism to the two�dimensional di�eomor�phism �World Scienti�c� New Jersey� ��

Problems ��

�� H� Poincar�e� Les m�ethodes nouvelles de la m�ecanique c�eleste� Vol� �� Gauthier�Villars� Paris� �� reprinted by Dover� �� English translation� Newmethods of celestial mechanics �NASA Technical Translations� �� See Vol�� section �� for the quote�

�� See sections �� and �� of Hao B��L�� reference "�#� for some results onperiodic windows in the quadratic map�

Problems


�� For the quadratic map �eq� �� show that the interval "�� # is atrapping region for all x in the unit interval and all � �� #�

�� Use the transformation

x � � �� y � ��

to show that the quadratic map �eq� �� can be written as

yn�� y�n� y � "��#� � � �� #� ��

or �using a di�erent x�transformation� as

zn�� z�n� z � "��#� � � �� #� ��

Specify the ranges to which x and are restricted under these transforma�tions�

�� Read the Tellus article by Lorenz mentioned in reference "�#�

Section ��

�� Write a program to calculate the iterates of the quadratic map�

Section ��

�� For f�x� � �x��x� and x� � �� calculate f��x�� by the graphical methoddescribed in section ��


�� Show by graphical analysis that if x� �� "�� # and � � in the quadratic map�then as n � �� fn�x�� Further� show that if � �� then the�xed point at the origin is an attractor�

Section ��

�� Find all the attractors and basins of attraction for the map f�x� � mx� wherem is a constant�

�� The tent map �see sections �� of Rasband� reference "�#� is de�ned by

%� � �� jx� �

�j� � ��

x if � � x � �� x if �� x � ��

�a� Sketch the graph of the tent map for � � �� Why is it called the tentmap�

�b� Show that the �xed points for the tent map are

x�� and� for � � �� x��

�c� Show that x�� is always repelling and that x�� is attracting when � ��

�d� For a one�dimensional map the Lyapunov exponent is de�ned by

�x�� limn��

�

nln

�� ddxfn�x��x�x�

��

Show that for � � �� the Lyapunov exponent for the tent map is � ln ��Hint� For the tent map use the chain rule of di�erentiation to show that

�x�� limn��

�

n

n��Xi��

ln jf ��xi�j � ��

�� Determine the local stability of orbits with � f ��x�� and � f ��x�� using graphical analysis as in Figure ��

Section ��

�� Determine the parameter value�s� for which the quadratic map intersects theline y � x just once�

�� Consider a period two orbit of a one�dimensional map�

f��x�s� � f�f�x�s�� x�s�

Problems ��

�a� Use the chain rule for di�erentiation to show that �f��x�� f��x��

�b� Show that the period two orbit is stable if jf ��x�� f ��x��j �� andunstable if jf ��x�� f ��x��j � ��

�� For � �� use Mathematica to �nd the locations of both period three orbitsin the quadratic map�

�� Consider a p�cycle of a one�dimensional map� If this p�cycle is periodic thenit is a �xed point of fp� Let x�� x�� xp�� represent one orbit of period p�Show that this periodic orbit is stable if

p��Yj��

jf ��xj�j ��

Hint� Show by the chain rule for di�erentiation that for any orbit �not nec�essarily periodic��

dfp

dx

��x�x�

� f ��x��f��x�� f ��xp��

Now assume the orbit is periodic� Then for any two points of a period p orbit�xi and xj � note that �fp��xi� � �fp��xj��

�� Consider a seed near a period two orbit� x� � x� � �� with slope near tof ��x�� Show by graphical analysis that fn�x�� ips� back and forthbetween the two points on the period two orbit� Show by graphical construc�tion that this period two orbit is stable if j�f��x��j �� and unstable ifj�f��x��j � ��

�� For �� show that the quadratic map fn� �x� intersects the y � x line �n

times� but only some of these intersection points belong to new periodic orbitsof period n� For n � � to �� build a table showing the number of di�erentorbits of f� of period n�

�� Prove �see Prob� �� that for prime n� ��n � ��n is an integer �this resultis a special case of the Simple Theorem of Fermat��

�� a� Derive the equation of the graph shown in Figure ��

�b� Verify that the two intersection points shown in the �gure occur at � and� �

p��

�� Equation �� follows directly from equation �� and the chain rule dis�cussion in Problem �� Derive it using only equations �� and��

� Quadratic Map

Section ��

�� Write a program to generate a plot of the bifurcation diagram for the quadraticmap�

Section ��

�� Show that the period two orbit in the quadratic map loses stability at ��

p�� i�e�� f��x�� at this value of �

�� Show that the �equispaced� orbits of the bouncing ball system �see Prob�� are born in a saddle�node bifurcation� Note that equation �� givestwo impact phases for each n� For n � � and for n � � in equation ��which orbit is a saddle near birth� and which orbit is a node� What is theimpact phase of the saddle� What is the impact phase of the node�

�� Use Mathematica �or another computer program� to show that a pair of periodthree orbits are born in the quadratic map by a tangent bifurcation at ��

p�

�� Show that the absolute value of the slope of fn evaluated at all points of aperiod n orbit have the same value at a bifurcation point�

�� In Figure �� the two sets of triangles ��open��white interior and �closed��black interior� represent two di�erent period three orbits� Show that the opentriangles represent an unstable orbit�

Section ��

�� Using Table �� and equations �� and �� estimate �� c� and ��

�� Use the bifurcation diagram of the quadratic map �Fig� �� and a rulerto measure the �rst few period doubling bifurcation values� n� It may behelpful to use a photocopying machine to expand the �gure before doing themeasurements� Based on these measurements� estimate �Feigenbaum�s delta�with equation �� Do the same thing for the bifurcation diagram for thebouncing ball system found in Figure �� How do these two values compare�

Section ��

�� Find a one�dimensional circle map g � S� � S� for which Sarkovskii�s orderingdoes not hold� Hint� Find a map that has no period one solutions by using adiscontinuous map�

�� Consider the periodic orbits of periods �� and �� Orderthese points according to Sarkovskii�s ordering� equation �� Show thatSarkovskii�s ordering uniquely orders all the positive integers�

Problems ��

Section ��

�� For � �� determine the interval A�� as shown in Figure �� as a function of � Verify that points in this interval leave the unit interval and never return�

�� a� Using the metric of equation �� calculate the distance between thetwo period two points� �� and ��

�b� Create a table showing the distances between all six period three points�� and ��

�� Establish an isomorphism between the unit interval and the sequence space&� of section �� Hint� See Devaney "��#� section ��

�� a� De�ne f � R � R by f�x� � mx � b and de�ne g � R � R by g�x� �mx � nb� where m� b� and n � R� Show that f and g are topologicallyconjugate�

�b� De�ne f � S� � S� by f�x� � x � �� De�ne g � "�� # � "�� # byg�x� � �� x� Show that f and g are topologically semiconjugate�

�c� Find a set of functions f � g� and h that satis�es the de�nition of topologicalconjugacy�

Section ��

�� Construct the binary tree up to the fourth level where the nth level is de�nedby the rule

nth level �

�nz � ��

�a� Construct the sixteen symbolic coordinates s�s�s�s� at the fourth levelof this binary tree� and show that the ordering from left to right at the nthlevel is given by

N �s�� s�� sn�� s� �n�� s� �n�� sn��

Why is it called the �binary tree��

�b� Show that the fractional ordering � is given by

��s�� s�� sn�� s��

�s��

� � sn��n

� ��

�c� Give an example of a one�dimensional map �not necessarily continuous�on the unit interval giving rise to the binary tree�

� Quadratic Map

�� Construct the alternating binary tree up to the fourth level and calculate thesymbolic coordinate and position of each of the sixteen points �� Presentthis information in a table�

�� Show that the order on the x�axis of two points x� and y� in the quadraticmap with � � is determined by their itineraries fakg and fbkg as follows�suppose a� � b�� a� � b�� ak � bk� and that ak�� and bk�� i�e��the itineraries of each initial condition are identical up to the kth iteration�and di�er for the �rst time at the k��st iteration�� Then

x� y� ��kXi��

ai mod � � ��

Hint� See Appendix A of reference "��# and theorem �� on page �� ofDevaney� reference "��#�

Chapter �

String

�� Introduction

Like a jump rope� a string tends to swing in an ellipse� a fact well knownto children� When holding both ends of a rope or string� it is di�cult toshake it so that motion is con�ned to a single transverse plane� Insteadof remaining con�ned to planar oscillations� strings appear to preferelliptical or whirling motions like those found when playing jump rope�Borrowing terminology from optics� we would say that a string preferscircular polarization to planar polarization� In addition to whirling�other phenomena are easily observed in forced strings including bi�furcations between planar and nonplanar periodic motions� transitionsto chaotic motions� sudden jumps between di�erent periodic motions�hysteresis� and periodic and aperiodic cycling between large and smallvibrations�

In this chapter we will begin to explore the dynamics of an elasticstring by examining a single�mode model for string vibrations� In theprocess� several new types of nonlinear phenomena will be discovered�including a new type of attractor� the torus� arising from quasiperiodicmotions� and a new route to chaos� via torus doubling� We will alsoshow how power spectra and Poincar�e sections are used in experimentsto identify di�erent types of nonlinear attractors� In this way we willcontinue building the vocabulary used in studying nonlinear phenomena��

��

�� CHAPTER �� STRING

In addition to its intrinsic interest� understanding the dynamics ofa string can also be important for musicians� instrument makers� andacoustical engineers� For instance� nonlinearity leads to the modulationand complex tonal structure of sounds from a cello or guitar� Whirlingmotions account for the rattling heard when a string is strongly plucked�� Linear theory provides the basic outline for the science of theproduction of musical sounds� its real richness� though� comes fromnonlinear elements�

When a string vibrates� the length of the string must �uctuate�These �uctuations can be along the direction of the string� longitu�dinal vibrations� or up and down� vibrations transverse to the string�The longitudinal oscillations occur at about twice the frequency of thetransverse vibrations� The modulation of a string�s length is the essen�tial source of a string�s nonlinearity and its rich dynamical behavior�The coupling between the transverse and longitudinal motions is anexample of a parametric oscillation� An oscillation is said to be para�metric when some parameter of a system is modulated� in this case thestring�s length� Linear theory predicts that a string�s free transverse os�cillation frequency is independent of the string�s vibration amplitude�Experimental measurements� on the other hand� show that the reso�nance frequency depends on the amplitude� Thus the linear theory hasa restricted range of applicability�

Think of a guitar string� A string under a greater tension has ahigher pitch �fundamental frequency�� Whenever a string vibrates itgets stretched a little more� so its pitch increases slightly as its vibrationamplitude increases�

We begin this chapter by describing the experimental apparatuswe�ve used to study the string �section �� In section �� we modelour experiment mathematically� Sections �� to �� examine a specialcase of string behavior� planar motion� which gives rise to the Du�ngequation� Section �� looks at the more general case� nonplanar mo�tion� Finally� in section �� we present experimental techniques usedby nonlinear dynamicists� These experimental methods are illustratedin the string experiment�

�� EXPERIMENTAL APPARATUS ��

Figure �� Schematic of the apparatus used to study the vibrations ofa wire �string��

�� Experimental Apparatus

An experimental apparatus to study the vibrations of a string can beconstructed by mounting a wire between two heavy brass anchors ��As shown in Figure �� a screw is used to adjust the position of theanchors� and hence the tension in the wire �string�� An alternating sinu�soidal current passed through the wire excites vibrations� this current isusually supplied directly from a function generator� An electromagnet�or large permanent magnet� is placed at the wire�s midpoint� The inter�action between this magnetic �eld and the magnetic �eld generated bythe wire�s alternating current causes a periodic force to be applied atthe wire�s midpoint�� If a nonmagnetic wire is used� such as tungsten�then both planar and nonplanar whirling motions are easy to observe�On the other hand� if a magnetic wire is used� such as steel� then themotion always remains restricted to a single plane �� The use of amagnetic wire introduces an asymmetry into the system that causesthe damping rate to depend strongly on the direction of oscillation�

�From the Lorentz force law� a wire carrying a current I� in a magnetic �eld ofstrength B� is acted on by a magnetic force Fmag �

R�I �B�dl� If the current I in

the wire varies sinusoidally� then so does the force on the wire� See D� J� Gri�ths�An introduction to electrodynamics �Prentice�Hall� Englewood Cli�s� NJ� �� pp��


A similar asymmetry is seen in the decay rates of violin� guitar�and piano strings� In these musical instruments the string runs over abridge� which helps to hold the string in place� The bridge damps themotion of the string� however� the damping force applied by the bridgeis di�erent in the horizontal and vertical directions �� The clampsholding the string in our apparatus are designed to be symmetric� andit is easy to check experimentally that the decay rates in di�erent direc�tions �in the absence of a magnetic �eld� show no signi�cant variation�Our experimental apparatus can be thought of as the inverse of thatfound in an electric guitar� There� a magnetic coil is used to detect themotion of a string� In our apparatus� an alternating magnetic �eld isused to excite motions in a wire�

The horizontal and vertical string displacements are monitored witha pair of inexpensive slotted optical sensors consisting of an LED �light�emitting diode� and a phototransistor in a U�shaped plastic housing ��Two optical detectors� one for the horizontal motion and one for thevertical motion� are mounted together in a holder that is fastened toa micropositioner allowing exact placement of the detectors relativeto the string� The detectors are typically positioned near the stringmounts� This is because the detector�s sensitivity is restricted to asmall�amplitude range� and the string displacement is minimal close tothe string mounts� As shown in Figure �� the string is positionedto obstruct the light from the LED and hence casts a shadow on thesurface of the phototransistor� For a small range of the string displace�ments� the size of this shadow is linearly proportional to the position ofthe string� and hence also to the output voltage from the photodetec�tor� This voltage is then monitored on an oscilloscope� digitized witha microcomputer� or further processed electronically to construct anexperimental Poincar�e section as described in section ��

Care must be taken to isolate the rig mechanically and acoustically�In our case� we mounted the apparatus on a �oating optical table� Wealso constructed a plastic cover to provide acoustical isolation� Thestring apparatus is small and easily �ts on a desktop� Typical experi�mental parameters are listed in Table ��

Most of our theoretical analysis will be concerned with single�modeoscillations of a string� If we pluck the string near its center� it tendsto oscillate in a sinusoidal manner with most of its energy at some pri�

�� EXPERIMENTAL APPARATUS ��

Figure �� Module used to detect string displacements� �Adapted fromHanson ��

Parameter Typical experimental value

Length �� mmMass per unit length �� g%mDiameter �� mmPrimary resonance � kHzRange of hysteresis �� HzMagnetic �eld strength �� TCurrent �� AMaximum displacement � mmDamping ��

Table �� Parameters for the string apparatus�


mary frequency called the fundamental� A similar plucking e�ect canbe achieved by exciting wire vibrations with the current and the sta�tionary magnetic �eld� To pluck the string� we switch o� the currentafter we get a large�amplitude string vibration going� The fundamen�tal frequency is recognizable by us as the characteristic pitch we hearwhen the string is plucked� A large�amplitude �resonant� response isexpected when the forcing frequency applied to a string is near to thisfundamental� This is the primary resonance of the string� and it isde�ned by the linear theory as

��

l

�T

�

��

� ��

where � � m�l is the mass per unit length and T is the tension in thestring��

The primary assumption of the linear theory is that the equilibriumlength of the string� l� remains unchanged as the string vibrates� thatis� l�t� � l� where l�t� is the instantaneous length� In other words� thelinear theory assumes that there are no longitudinal oscillations� Indeveloping a simple nonlinear model for the vibrations of a string wemust begin to take into account these longitudinal oscillations and thedependence of the string�s length on the vibration amplitude�

�� Single�Mode Model

A model of a string oscillating in its fundamental mode is presented inFigure �� and consists of a single mass fastened to the central axis by apair of linearly elastic springs �� Although the springs provide a linearrestoring force� the resulting force toward the origin is nonlinear becauseof the geometric con�guration� The ends of the massless springs are�xed a distance l apart where the relaxed length of the spring is l�and the spring constant is k� In the center a mass is attached that isfree to make oscillations in the x�y plane centered at the origin� Themotion in the two transverse directions� x and y� is coupled directly�

�For a review of the linear theory for the vibrations of a stretched string see A�P� French� Vibrations and waves �W� W� Norton� New York� �� pp� ��

�� SINGLE MODE MODEL ��

Figure �� Single�mode model for nonlinear string vibrations� Stringvibrations are assumed to be in the fundamental mode and are mea�sured in the transverse x�y plane by the polar coordinates �r� �� a�Equilibrium length� �b� relaxed length�

and also indirectly� via the longitudinal motion of the spring� Both ofthese coupling mechanisms are nonlinear� The multimode extension ofthis single�mode model would consist of n masses hooked together byn � � springs�

The restoring force on the mass shown in Figure �� is

F � �kr

�� l�p

l� � r�

��

where the position of the mass is given by polar coordinates �r� �� ofthe transverse plane �see Prob� �� Expanding the right�hand side ofequation �� in a Taylor series �r � l�� we �nd that

F � �kr�l � l��r

l�� kl��

r

l��

r

l��

The force can be written as

F � m$r�


so

m$r �k�l� l��r

l

��

l��l � l��

�r

l

��

Note the cubic restoring force� Also note that nonlinearity dominateswhen l l�� That is� the nonlinear e�ects are accentuated when thestring�s tension is low�

De�ne

��

k

m

�l � l��

l��

and

K �l�

l��l � l��

Then from equation �� we get� because of symmetry in the angularcoordinate� the vector equation for r � �x�t�� y�t��

�r� ��r�� Kr��

which is the equation of motion for a two�dimensional conservativecubic oscillator�� The behavior of equation �� depends criticallyupon the ratio �l��l�� If l� � l� the coe�cient of the nonlinear term� K�is positive� the equilibrium point at r � � is stable� and we have a modelfor a string vibrating primarily in its fundamental mode� On the otherhand� if l� l� then K is negative� the origin is an unstable equilibriumpoint� and two stable equilibrium points exist at approximately r � �l�This latter case models the motions of a single�mode elastic beam ��For our purpose we will mostly be concerned with the case l� � l� orK ��

In general� we will want to consider damping and forcing� so equa�tion �� is modi�ed to read

�r� ��r� �� Kr��r � f�t��

where f�t� is a periodic forcing term and � is the damping coe�cient�Usually� the forcing term is just a sinusoidal function applied in oneradial direction� so that it takes the form f�t� � �A cos��t�� For

�The term r� in equation �� is a typical physicist�s notation meaning the dotproduct of the vector� r� � �r r�� x� � y� � r��

�� SINGLE MODE MODEL ��

simplicity� we have assumed that the energy losses are linearly propor�tional to the radial velocity of the string� �r� We also assumed that theends of the string are symmetrically �xed� so that � is a scalar� In gen�eral� the damping rate depends on the radial direction� so the dampingterm is a vector function� This is the case� for instance� when a stringis strung over a bridge that breaks the symmetry of the damping term�

Equation �� was also derived by Gough �� and Elliot �� bothof whom related �� and K to actual string parameters that arise inexperiments� For instance� Gough showed that the natural frequencyis given by

�� c

l��

and the strength of the nonlinearity is

K ��

�l�

��

where � is the longitudinal extension of a string of equilibrium length l�� is the low�amplitude angular frequency of free vibration� and c is thetransverse wave velocity� Again� we see that the nonlinearity parameter�K� increases as the longitudinal extension� �� approaches zero� Thatis� the nonlinearity is enhanced when the longitudinal extension�andhence the tension�is small� Nonlinear e�ects are also ampli�ed whenthe overall string length is shortened� and they are easily observablein common musical instruments� For a viola D�string with a vibrationamplitude of � mm� typical values of the string parameters showingnonlinear e�ects are� l � �� cm� �� Hz� � � �� mm� K �� mm��

Equation �� constitutes our single�mode model for nonlinear stringvibrations and is the central result of this section� For some calcula�tions it will be advantageous to write equation �� in a dimensionlessform� To this end consider the transformation

� � ��t� s �r

l��

which givess�� s� � �� s��s � g��


where the prime denotes di�erentiation with respect to � and

� � �

�� Kl�� g �

f

l��

� and � � �

��

Before we begin a systematic investigation of the single�mode modelit is useful to consider the unforced linear problem� f�t� � �� If thenonlinearity parameter K is zero� then equation �� is simply a two�degree of freedom linear harmonic oscillator with damping that admitssolutions of the form

r � �X� cos��t� Y� sin��t�e��t��

where X� and Y� are the initial amplitudes in the x and y directions�Equation �� is a solution of �� if we discard second�order termsin �� In the conservative limit �� the orbits are ellipses centeredabout the z�axis� As we show in section �� one e�ect of the nonlin�earity is to cause these elliptical orbits to precess� The trajectories ofthese precessing orbits resemble Lissajous �gures� and these precessingorbits will be one of our �rst examples of quasiperiodic motion on atorus attractor�

�� Planar Vibrations� Du�ng Equation

An external magnetic �eld surrounding a magnetic wire restricts theforced vibrations of a wire to a single plane� Alternatively� we couldfasten the ends of the wire in such a way as to constrain the motionto planar oscillations� In either case� the nonlinear equation of motiongoverning the single�mode planar vibrations of a string is the Du�ng

equation�$x � � x � ��

�x�� Kx�� A cos��t��

where equation �� is calculated from equation �� by assumingthat the string�s motion is con�ned to the x�z plane in Figure �� Theforcing term in equation �� is assumed to be a periodic excitation ofthe form

f�t� � A cos��t��

�� PLANAR VIBRATIONS� DUFFING EQUATION ��

where the constant A is the forcing amplitude and � is the forcingfrequency� The literature studying the Du�ng equation is extensive�and it is well known that the solutions to equation �� are alreadycomplicated enough to exhibit multiple periodic solutions� quasiperi�odic orbits� and chaos� A good guide to the nonchaotic properties ofthe Du�ng equation is the book by Nayfeh and Mook �� Highlyrecommended as a pioneering work in nonlinear dynamics is the bookby Hayashi� which deals almost exclusively with the Du�ng equation��

�� Equilibrium States

The �rst step in analyzing any nonlinear system is the identi�cation ofits equilibrium states� The equilibrium states are the stationary pointsof the system� that is� where the system comes to rest� For a system ofdi�erential equations� the equilibrium states are calculated by settingall the time derivatives equal to zero in the unforced system� Setting$x � �� x � �� and A � � in equation �� we immediately �nd thatthe location of the equilibrium solutions is given by

��x�� Kx��

which� in general� has three solutions�

x� � �� and x� � �

s��

K� x� � �

s��

K� ��

Clearly� there is only one real solution if K �� x�� since the other twosolutions� x�� are imaginary in this case� If K � �� then there are threereal solutions�

To understand the stability of the stationary points it is useful torecall the physical model that goes with equation �� If l � l�� thenK � � �see eqs� �� and �� and the Du�ng equation �� is asimple model for a beam under a compressive load� As illustrated inFigure �� the solutions x� correspond to the two asymmetric stablebeam con�gurations� The position x� corresponds to the symmetricunstable beam con�guration�a small tap on the beam would imme�diately send it to one of the x� con�gurations� If l l�� then K �


Figure �� Equilibrium states of a beam under a compressive load�

Figure �� Equilibrium state of a wire under tension�

and the Du�ng equation is a simple model of a string or wire undertension� so there is only one symmetric stable con�guration� x� �Fig��

�� Unforced Phase Plane

Conservative Case

After identifying the equilibrium states� our next step is to understandthe trajectories in phase space in a few limiting cases� In the unforced�conservative limit� a complete account of the orbit structure is given byintegrating the equations of motion by using the chain rule in the form

$x � v �d

dtv�x� �

dv

dx

dx

dt

� vdv

dx� ��

Applying this identity to equation �� with � � � and A � � yields

vdv

dx� ��

�x�� Kx��

which can be integrated to give

�

v� � h� ��

�

�x�

� K

x�

��


where h is the constant of integration�The term on the left�hand side of equation �� is proportional to

the kinetic energy� while the term on the right�hand side�

V �x� � ��

�x�

� K

x�

��

is proportional to the potential energy� Therefore� the constant h isproportional to the total energy of the system� as illustrated in Figure��a��

The phase space is a plot of the position x and the velocity v of allthe orbits in the system� In this case� the phase space is a phase plane�and in the unforced conservative limit we �nd

v � �p�h� V �x��

� �p

�h� ��

�

�x�

� K

x�

��

The last equation allows us to explicitly construct the integral curves �aplot of v�t� vs� x�t�� in the phase plane� Each integral curve is labeledby a value of h� and the qualitative features of the phase plane dependcritically on the signs of ��

� and K�If l l�� then both ��

� and K are positive� A plot of equation�� for several values of h is given in Figure ��c�� If h � h��then the integral curve consists of a single point called a center� Whenh h�� the orbits are closed� bounded� simply connected curves aboutthe center� Each curve corresponds to a distinct periodic motion of thesystem� Going back to the string model again� we see that the centercorresponds to the symmetric equilibrium state of the string� while theintegral curves about the center correspond to �nite�amplitude periodicoscillations about this equilibrium point�

If l � l�� then K is negative� The phase plane has three stationarypoints� This parameter regime models a compressed beam� The left andright stationary points� x�� are centers� but the unstable point at x � ��labeled S in Figure ��b�� is a saddle point because it corresponds toa local maximum of V �x��

Curves that pass through a saddle point are very important andare called separatrices� In Figure ��d� we see that there are two


Figure �� Potential and phase space for a single�mode string �a�c�e�and beam �b�d�f��


integral curves approaching the saddle point S and two integral curvesdeparting from S� These separatrices �separate the phase plane intotwo distinct regions� Each integral curve inside the separatrices goesaround one center� and hence corresponds to an asymmetric periodicoscillation about either the left or the right center� but not both� Theintegral curves outside the separatrices go around all three stationarypoints and correspond to large�amplitude symmetric periodic orbits�Figure ��d�� Thus� the separatrices act like barriers in phase spaceseparating motions that are qualitatively di�erent�

Dissipative Case

If damping is included in the system� then the phase plane changesto that shown in Figure ��e�� For the string� damping destroys allthe periodic orbits� and all the motions are damped oscillations thatconverge to the point attractor at the origin� That is� if we pluck astring� the sound fades away� The string vibrates with a smaller andsmaller amplitude until it comes to rest� Moreover� the basin of attrac�tion for the point attractor is the entire phase plane� This particularpoint attractor is an example of a sink�

The phase plane for the oscillations of a damped beam is a bit moreinvolved� as shown in Figure ��f�� The center points at x� becomepoint attractors� while the stationary point at x� is a saddle� There aretwo separate basins of attraction� one for each point attractor �sink��The shaded region shows all the integral curves that head toward theright sink� Again we see the important role played by separatrices� sincethey separate the basins of attraction of the left and right attractingpoints� In the context of a dissipative system� the separatrix naturallydivides into two parts� the inset consisting of all integral curves thatapproach the saddle point S� and the outset consisting of all pointsdeparting from S� Formally� the outset of S can be de�ned as all pointsthat approach S as time runs backwards� That is� we simply reverse allthe arrows in Figure ��f��

The qualitative analysis of a dynamical system can usually be di�vided into two tasks� �rst� identify all the attractors and repellers ofthe system� and second� analyze their respective insets and outsets�Attractors and repellers are limit sets� Insets� outsets� and limit sets


are all examples of invariant sets �see section �� Thus� much ofdynamical systems theory is concerned not simply with the analysis ofattractors� but rather with the analysis of invariant sets of all kinds� at�tractors� repellers� insets� and outsets� For the unforced� damped beamthe task is relatively easy� There are two attracting points and onesaddle point� The inset and outset of the saddle point spiral aroundthe two attracting points and completely determine the structure of thebasins of attraction �see Figure ��f��

�� Extended Phase Space

To continue with the analysis of planar string vibrations� we now turnour attention to the forced Du�ng equation in dimensionless variables�from eqs� �� and ��

x�� x� � �� x��x � F cos��

where F is the forcing amplitude and � is the normalized forcing fre�quency�

It is often useful to rewrite an nth�order di�erential equation as asystem of �rst�order equations� and to recall the geometric interpreta�tion of a di�erential equation as a vector �eld� To this end� considerthe change of variable v � x�� so that

x� � v�v� � aut�x� v� � g��

��x� v� � R� ��

where aut�x� v� � ��v � �� x��x� is the autonomous� or time�independent� term of v� and g�� F cos�� is the time�dependentterm of v�� The phase space for the forced Du�ng equation is topolog�ically a plane� since each dependent variable is just a copy of R� andthe phase space is formally constructed from the Cartesian product ofthese two sets� R�R � R��

A vector �eld is obtained when to each point on the phase planewe assign a vector whose coordinate values are equal to the di�erentialsystem evaluated at that point� The vector �eld for the unforced� un�damped Du�ng equation is shown in Figure ��a�� This vector �eldis static �time�independent�� In contrast� the forced Du�ng equation


Figure �� Extended phase space for the Du�ng oscillator�

has a time�dependent vector �eld since the value of the vector �eld at�x� v� at � is

�x�� v�� v� aut�x� v� � F cos��

In Figure ��b� we show what the integral curves look like when plottedin the extended phase space� which is obtained by introducing a thirdvariable�

z � ��

With this variable the di�erential system can be rewritten as

x� � v�v� � aut�x� v� � g�z��z� � ��

�� x� v� z� � R� ��

� CHAPTER �� STRING

By increasing the number of dependent variables by one� we can for�mally change the forced �time�dependent� system into an autonomous�time�independent� system�

Moreover� since the vector �eld is a periodic function in z� it issensible to introduce the further transformation

� � �� mod � ��

thereby making the third variable topologically a circle� S�� With thistransformation the forced Du�ng equation becomes

x� � v�v� � ��v � �� x��x� � F cos��

�� x� v� �� R� � S� ��

One last reduction is possible in the topology of the phase space ofthe Du�ng equation� It is usually possible to �nd a trapping regionthat topologically is a disk� a circular subset D � R�� In this lastinstance� the topology of the phase space for the Du�ng equation issimply D � S�� or a solid torus �see Figure ��c��

�� Global Cross Section

The global solution to a system of di�erential equations �the collectionof all integral curves� is also known as a �ow� A �ow is a one�parameterfamily of di�eomorphisms of the phase space to itself �see section ��

To visualize the �ow in the Du�ng equation� imagine the extendedphase space as the solid torus illustrated in Figure �� Each initialcondition in the disk D� at � � �� must return to D when � � �because D is a trapping region and the variable � is �periodic� Thatis� the region D �ows back to itself� An initial point in D labeled�x�� v�� is carried by its integral curve back to some new pointlabeled �x�� v�� also in D� The Du�ng equation satis�es thefundamental uniqueness and existence theorems in the theory of or�dinary di�erential equations �� Hence� each initial point in D getscarried to a unique point back in D and no two integral curves can everintersect in D � S��

As originally observed by Poincar�e� this unique dependence withrespect to initial conditions� along with the existence of some region in


Figure �� Phase space for the Du�ng oscillator as a solid torus�

phase space that is recurrent� allows one to naturally associate a mapto any �ow� The map he described is now called the Poincar�e map� Forthe Du�ng equation this map is constructed from the �ow as follows�De�ne a global cross section '�� of the vector �eld �eq� �� by

'�� f�x� v� �� D � S� j � � �� g� ��

Next� de�ne the Poincar�e map of '�� as

P�� '�� '�� x� �� x�� v� �� v��

where �x�� v�� is the next intersection with '�� of the integral curveemanating from �x�� v�� For the Du�ng equation the Poincar�e map isalso known as a stroboscopic map since it samples� or strobes� the �owat a �xed time interval�

The dynamics of the Poincar�e map are often easier to study thanthe dynamics in the original �ow� By constructing the Poincar�e mapwe reduce the dimension of the problem from three to two� This di�mension reduction is important both for conceptual clarity as well as

� CHAPTER �� STRING

for graphical representations �both numerical and experimental� of thedynamics� For instance� a periodic orbit is a closed curve in the �ow�The corresponding periodic orbit in the map is a collection of pointsin the map� so the �xed point theory for maps is easier to handle thanthe corresponding periodic orbit theory for �ows�

The construction of a map from a �ow via a cross section is gener�ally unique� However� constructing a �ow from a map is generally notunique� Such a construction is called a suspension of the map� Studiesof maps and �ows are intimately related�but they are not identical�For instance� a �xed point of a �ow �an equilibrium point of the di�er�ential system� has no natural analog in the map setting�

A complete account of Poincar�e maps along with a thorough casestudy of the Poincar�e map for the harmonic oscillator is presented byWiggins ��

�� Resonance and Hysteresis

We now turn our attention to resonance in the Du�ng oscillator� Thenotion of a resonance is a physical concept with no exact mathemat�ical de�nition� Physically� a resonance is a large�amplitude response�or output� of a system that is subject to a �xed�amplitude input� Theconcept of a resonance is best described experimentally� and resonancesare easy to see in the string apparatus described in section �� by con�structing a sort of experimental bifurcation diagram for forced stringvibrations�

Imagine that the string apparatus is running with a small excitationamplitude �the amount of current in the wire is small� and a low forcingfrequency �the frequency of the alternating current in the wire is muchless than the natural frequency of free wire vibrations�� To constructa resonance diagram we need to measure the response of the system�by measuring the maximum amplitude of the string vibrations as afunction of the forcing frequency� To do this we slowly increase �scanthrough� the forcing frequency while recording the response of the stringwith the optical detectors� The results of this experiment depend onthe forcing amplitude as well as where the frequency scan begins andends� Decreasing frequency scans can produce di�erent results from

�� RESONANCE AND HYSTERESIS ��

Figure �� Response curve for a harmonic oscillator�

increasing frequency scans�

�� Linear Resonance

For a very small forcing amplitude the string responds with a linear

resonance� such as that illustrated in Figure �� According to lineartheory� the response of the string is maximum when � � �� In other words� it is maximum when the forcing frequency � exactlyequals the natural frequency �� A primary �or main� resonance existswhen the natural frequency and the excitation frequency are close� Theresonance diagram �Fig� �� is called a linear response because it can beobtained by solving the periodically forced� linearly damped harmonicoscillator�

x�� x� � x � F cos��

which has a general solution of the form

x�� x�e�� cos��

F �� cos��

The constants x� and �� are initial conditions� Equation �� is asolution to equation �� if we discard higher�order terms in �� Themaximum amplitude of x� as a function of the driving frequency �� isfound from the asymptotic solution of equation ��

lim��

x�� F cos��

��


Figure �� Schematic of the response curve for a cubic oscillator�

which produces the linear response diagram shown in Figure �� since

a�� xmax�� max�

lim��

x��

�F

��

After the transient solution dies out� the steady�state response has thesame frequency as the forcing term� but it is phase shifted by an amount that depends on �� and F � As with all damped linear systems� thesteady�state response is independent of the initial conditions so that wecan speak of the solution�

In the linear solution� motions of signi�cant amplitude occur whenF is large or when � �� Under these circumstances the nonlinearterm in equation �� cannot be neglected� Thus� even for planarmotion� a nonlinear model of string vibrations may be required when aresonance occurs or where the excitation amplitude is large�

�� Nonlinear Resonance

A nonlinear resonance curve is produced when the frequency is scannedwith a moderate forcing amplitude� F � Figure �� shows the resultsof both a backward and a forward scan� which can be constructedfrom a numerical solution of the Du�ng oscillator� equation �� seeAppendix C on Ode �� for a description of the numerical methods��The two scans are identical except in the region marked by �l � � � �u�Here� the forward scan produces the upper branch of the response curve�This upper branch makes a sudden jump to the lower branch at thefrequency �u� Similarly� the backward �decreasing� scan makes a sudden


Figure �� Nonlinear resonance curve showing secondary resonancesin addition to the main resonance�

jump to the upper branch at �l� In the region �l � � � �u� at leasttwo stable periodic orbits coexist� The sudden jump between these twoorbits is indicated by the upward and downward arrows at �l and �u�This phenomenon is known as hysteresis�

The nonlinear response curve also reveals several other intriguingfeatures� For instance� the maximum response amplitude no longer oc�curs at � � �� but is shifted forward to the value �u� This is expectedin the string because� as the string�s vibration amplitude increases� itslength increases� and this increase in length �and tension� is accompa�nied by a shift in the natural frequency of free oscillations�

Several secondary resonances are evident in Figure �� These sec�ondary resonances are the bumps in the amplitude resonance curve thatoccur away from the main resonance�

The main resonance and the secondary resonances are associatedwith periodic orbits in the system� The main resonance occurs near� � � when the forcing amplitude is small and corresponds to theperiod one orbits in the system� those orbits whose period equals theforcing period� The secondary resonances are located near some ra�tional fraction of the main resonance and are associated with periodicmotions whose period is a rational fraction of �� These periodic orbits�denoted by !x� can often be approximated to �rst order by a sinusoidalfunction of the form

!xm�n�� A cos�m

n� � ��

where A is the amplitude of the periodic orbit� �m�n� is its frequency�


Figure �� Amplitude�modulated� quasiperiodic motions on a torus�

and is the phase shift� These periodic motions are classi�ed by theintegers m and n as follows �m �� n ��

� � �� m� an ultraharmonic�� n� a subharmonic�� m�n� an ultrasubharmonic�

Equation �� is used as the starting point for the method of harmonic

balance� a pragmatic technique that takes a trigonometric series as thebasis for an approximate solution to the periodic orbits of a nonlinearsystem �see Prob� �� It is also possible to have solutions to dif�ferential equations involving frequencies that are not rationally related�Such orbits resemble amplitude�modulated motions and are generallyknown as quasiperiodic motions �see Figure �� A more complete ac�count of nonlinear resonance theory is found in Nayfeh and Mook ��Parlitz and Lauterborn also provide several details about the nonlinearresonance structure of the Du�ng oscillator ��

�� Response Curve

In this section we focus on understanding the hysteresis found at themain resonance of the Du�ng oscillator because hysteresis at the mainresonance and at some secondary resonances is easy to observe experi�mentally� The results in this section can also be derived by the methodof harmonic balance by taking m � n � � in equation �� how�


ever� we will use a more general method that is computationally a littlesimpler�

We generally expect that in a nonlinear system the maximum re�sponse frequency will be detuned from its natural frequency� An esti�mate for this detuning in the undamped� free cubic oscillator�

x�� x��x � ��

is obtained by studying this equation by the method of slowly varying

amplitude �� Write�

x��

�A�� ei� � A�� e�i� � ��

and substitute equation �� into equation �� while assuming A�� varies slowly in the sense that jA��j �� A� Then equation �� isapproximated by

�i��A� � ��

jAj��A � ��

where we ignore all terms not at the driving frequency� Equation ��has a steady�state solution in A� denoted by a � R� In this case�

��

jaj��

since A� � �� and ��!x�� a cos��

To �rst order� the strength of the nonlinearity increases the normalizedfrequency by an amount depending on the amplitude of oscillation andthe nonlinearity parameter� This approximate value for a Du�ng os�cillator is consistent with the results found in Appendix B where exactsolutions for a cubic oscillator are presented�

�If we write A� � � a� � � ib� �� and A�� a� � � ib� �� then it is easy toshow from Euler�s identity that x� � � a� � cos�� b� � sin�� Thus� the useof complex numbers is not essential in this calculation� it is merely a trick thatsimpli�es some of the manipulations with the sinusoidal functions�


Hysteresis is discovered when we apply the slowly varying amplitudeapproximation to the forced� damped Du�ng equation�

x�� x� � �� x��x � F cos��

Substituting equation �� into equation �� and again keepingonly the terms at the �rst harmonic� we arrive at the complex amplitudeequation

�� i��A� �

�� i��

��

jAj�

�A � F� ��

which in steady�state �A� � �� becomes�� i��

��

j !Aj�

�!A � F� ��

To �nd the set of real equations for the steady state� write the complexamplitude in the form

!A � ae�i � ��

where both a and are real constants� Then equation �� separatesinto two real equations�

��a � F sin ��

and

��

a��a � F cos � ��

which collectively determine both the phase and the amplitude of theresponse� Squaring both equations �� and �� and then addingthe results� we obtain a cubic equation in a��

��

a��a� � F ��

illustrated in Figure �� which is known as the response curve� In thisapproximation the steady�state response is given by

!x � a cos��

where a is the maximum amplitude of the harmonic response deter�mined from equation �� and is the phase shift determined fromequations �� and ��


�� Hysteresis

The response curve shown in Figure �� is a plot of a versus � cal�culated from equation �� the nonlinear phase�amplitude relation�This curve shows that the string can exhibit hysteresis near a primaryresonance� a slow scan of the variable � �the so�called quasistatic ap�proximation� results in a sudden jump between the two stable solutionsindicated by the solid lines in Figure �� The jump from the upperbranch to the lower branch takes place at �u� The jump from the lowerbranch to the upper branch takes place at �l�

In the parameter regime �l � � � �u� the response curve revealsthe coexistence of three periodic orbits at the same frequency� but withdi�erent amplitudes� All these orbits are possible solutions to equation�� for the values of a indicated in the diagram� All three orbits areharmonic responses �or period one orbits� since their frequency equalsthe forcing frequency� The middle solution� indicated by the dashedline in Figure �� is an unstable periodic orbit�

�� Basins of Attraction

In a linear system with damping� the attracting periodic orbit is inde�pendent of the initial conditions� In contrast� the existence of two ormore stable periodic orbits for the same parameter values in a nonlin�ear system indicates that the initial conditions play a critical role indetermining the system�s overall response� These attracting periodicorbits are called limit cycles� and their global stability is determined byconstructing their basins of attraction� A very nice three�dimensionalpicture of the basins of attraction for the two stable periodic orbitsfound in the Du�ng oscillator is presented by Abraham and Shaw�� However� this picture of the basins of attraction within the three�dimensional �ow is very intricate� An equivalent picture of the basinsof attraction constructed with a two�dimensional cross section and aPoincar�e map is easier to understand�

A schematic for the basins of attraction in a Du�ng oscillator withthree coexisting orbits is portrayed in Figure �� The cross sectionshows two stable orbits� P� and P�� and one unstable orbit� P�� Inthe region surrounding the inset of P�� a small change in the initial


Figure �� Schematic of the basins of attraction in the Du�ng oscil�lator� �Adapted from Hayashi ��

conditions can produce a large change in the response of the systemsince initial conditions in this region can go to either attracting periodicorbit� The unstable periodic orbit indicated by P� is a saddle �xedpoint in the Poincar�e map� and the stable periodic orbits are sinks inthe Poincar�e map�

The inset of the saddle is the collection of all points that approachP�� This inset divides the Poincar�e map into two distinct regions� theinitial conditions that approach P� and the initial conditions that ap�proach P�� That is� the inset to the saddle determines the boundaryseparating the two basins of attraction� Again� we see the importanceof keeping track of the unstable solutions� as well as the stable solu�tions� when analyzing a nonlinear system� Figure ��a� should becompared to�but not confused with�Figure ��e�� the phase planefor the unforced� damped Du�ng oscillator� In the Poincar�e map each�xed point represents an entire periodic orbit� not just an equilibriumpoint of the �ow as in Figure �� More importantly� in the Poincar�emap� the inset to the saddle point at P� is not a trajectory of the �ow�Rather� it is the collection of all initial conditions that converge toP�� The approach of a single orbit toward P� is a sequence of discrete

�� HOMOCLINIC TANGLES ��

Figure �� Schematic of a homoclinic tangle in the Du�ng oscillator�

points� indicated by the crosses �� in Figure �� In general� the insetand the outset of the saddle represent an in�nite continuum of distinctorbits� all of which share a common property� namely� they arrive ator depart from a periodic point of the map�

�� Homoclinic Tangles

Figure �� shows the Poincar�e map for the �ow arising in the Du�ngoscillator in a parameter region where hysteresis exists� We see thatthe inset to P� consists of two di�erent curves� Similarly� the outsetof P� also consists of two di�erent curves� One branch of the outsetapproaches P�� while the other branch approaches P�� The inset andthe outset of the saddle are not trajectories in the �ow� so they canintersect without violating a fundamental theorem of ordinary di�eren�tial equations� the unique dependence of an orbit with respect to initialconditions�

It is possible to �nd parameter values so that the inset and theoutset of the saddle point at P� do indeed cross� A self�intersectionof the inset of a saddle with its outset is illustrated in Figure ��b�and� as originally observed by Poincar�e� it always gives rise to wildoscillations about the saddle �see section ��

Such a self�intersection of the inset of a saddle with its outset is


called a homoclinic intersection� and it is a fundamental mechanism bywhich chaos is created in a nonlinear dynamical system� The reason isroughly the following� Consider a point at a crossing of the inset andthe outset indicated by the point I in Figure ��b�� By de�nition�this point is part of an orbit that approaches the saddle by both itsinset and its outset� that is� it is doubly asymptotic� Consider thenext intersection point of I with the cross section� P ��I�� This pointmust lie on the inset at a point closer to the saddle� Next� the seconditerate of I under the Poincar�e map� P ��I�� must be even closer to thesaddle� Similarly� the preimage of I approaches the saddle along theoutset of the saddle� The outset and inset get bunched up near thesaddle� creating an image known as a homoclinic tangle� Homoclinictangles beat at the heart of chaos because� in the region of a homoclinictangle� initial conditions are subject to a violent stretching and foldingprocess� the two essential ingredients for chaos� A marvelous pictorialdescription of homoclinic tangles along with an explanation as to theirimportance in dynamical systems is presented by Abraham and Shaw��

Homoclinic tangles are often associated with the existence of strangesets in a system� Indeed� it is thought that in many instances a strangeattractor is nothing but the closure� of the outset of some saddle whenthis outset is bunched up in a homoclinic tangle� Figure ��a� showsthe cross section for a strange attractor of the Du�ng oscillator� Figure��b� shows the cross section of the outset of the period one saddle inthis strange set for the exact same parameter values� The resemblancebetween these two structures is striking� Indeed� developing methods todissect homoclinic tangles will be central to the study of chaos in low�dimensional nonlinear systems� In fact� one could call it the problemof low�dimensional chaos�

�� Nonplanar Motions

Additional dynamical possibilities arise when we consider nonplanarstring vibrations� These vibrations are also easy to excite with thestring apparatus described in section �� When a nonmagnetic wire

�The closure of a set X is the smallest closed set that is a superset of X�

�� NONPLANAR MOTIONS ��

Figure �� Comparison of a strange attractor and the outset of aperiod one saddle in the Du�ng oscillator�


Figure �� Planar periodic� elliptical �nonplanar� periodic� and pre�cessing �quasiperiodic� motions of a string�

is used� out�of�plane motions are observed which are sometimes calledballooning or whirling motions� These nonplanar vibrations arise evenwhen the excitation is only planar�

Indeed� ballooning motions are hard to avoid� Imagine scanning theforcing frequency of the string apparatus through a resonance� The re�sponse of the string increases as the resonance frequency is approached�and the following behavior is typically observed� Well below the reso�nance frequency� the string responds with a planar� periodic oscillation�Fig� ��a�� As the forcing frequency is increased� the amplitude ofthe response also grows until the string �pops out of the plane and be�gins to move in a nonplanar� elliptical� periodic pattern �Fig� ��b��That is� the string undergoes a bifurcation from a planar to a nonpla�nar oscillation� At a still higher frequency the elliptical periodic orbitbecomes unstable and begins to precess� as illustrated in Figure ��c��

We will present a more complete qualitative account of these whirlingmotions in section �� which is based on the recent work of Johnsonand Bajaj �� Miles �� and O�Reilly �� Now� though� we turn ourattention to the whirling motion that occurs when no forcing is present�

�� Free Whirling

If we pluck a string hard and look closely� we typically see the stringwhirling around in an elliptical pattern with a diminishing amplitude�Some understanding of these motions is obtained by considering the free


planar oscillations of a string modeled by the two�dimensional equation

�r� ��r� �� Kr��r � ��

which is equation �� with no forcing term� We noted in section ��that the linear approximation to equation �� results in elliptical mo�tion �eq� �� We shall use this observation to calculate an approxi�mate solution to equation �� using a procedure put forth by Gough�� similar results were obtained by Elliot ��

Transform the problem of nonlinear free vibrations to a referenceframe rotating with an angular frequency (� with ( to be determined�In this rotating frame� equation �� becomes

�u� � �u � � � �u � �� u �(�u � �� Ku��u � ��

where u is the new radial displacement vector� subjected to the additionof Coriolis and centrifugal accelerations� Let us now look for a solutionof the form

u�t� � �x�t�� y�t��

e��t��X� cos )�t � X� cos �)�t� Y� sin )�t� Y� sin �)�t��

where X� and Y� are small compared to X� and Y�� Looking at thex coordinate only� when we substitute equation �� into equation�� and discard appropriate higher�order terms� we get

�� (� � )��X� cos )�t� ��

� � (� � �)��X� cos �)�t�()�Y� cos )�t � ��

�K�X�� cos� )�t � Y �

� sin� )�t�e��tX� cos )�t � ��

a similar relation holds for the y coordinate� On equating sinusoidalterms of the same frequency we�after considerable algebra�discover

)��

��

�K

�X�

� � Y �� e��t

��(��

)�(

��

��K

X�Y�e

��t� ��

andX�

X��

Y�Y�

��K

��X

�� Y �

� �e��t

��)�� (��

� ��


If there is no damping �� then this approximate solution isperiodic in the rotating reference frame and is slightly distorted froman elliptical orbit� The angular frequency )� is detuned from �� byan amount proportional to the mean�square radius X�

� � Y �� In the

original stationary reference frame� equation �� shows us that theorbit precesses at a rate ( proportional to the orbital area X�Y�� Theangular frequency )� in equations �� to �� is measured in therotating reference frame� It is related to the angular frequency in thestationary reference frame � by

�� )� � (��

� �

K

��X�

� � Y �� X�Y��e

��t� ��

Thus� in the stationary reference frame� the undamped motion isquasiperiodic unless )� and ( are accidentally commensurate� in whichcase the orbit is periodic� The damped oscillations are also elliptical incharacter and precess at a rate (� In both cases� the detuning givenby equation �� is due to two sources� the nonlinear planar motiondetuning plus a detuning resulting from the precessional frequency�

�� Response Curve

The case of whirling motions subject to a planar excitation is describedby the equation

�r� ��r � �� Kr��r � �A cos�t� ��

where the phase space is four�dimensional� �x� vx� y� vy� � R�� Theextended phase space� when we add the forcing variable� is �ve�dimen�sional� �x� vx� y� vy� �� R� � S��

Equation �� can be analyzed for periodic motions by a combi�nation of averaging and algebraic techniques not unlike the harmonicbalance method� Because our text is an experimental introduction tononlinear dynamics� we present here a qualitative description of theresults of this analysis� For further details see references �� and��

As mentioned in the introduction to this section� an experimentalfrequency scan that passes through a main resonance can result in thefollowing sequence of motions�


Figure �� Response curve for planar and nonplanar motion��Adapted from Johnson and Bajaj ��

planar periodic �� nonplanar periodic ��

planar periodic�quasiperiodic�precessing elliptical orbit��chaotic

��

�� jump to small�amplitude planar motions�

The basic features of these experimental observations agree with thosepredicted by equation �� and are summarized in the response curveshown in Figure ��

In the parameter range �l � � � �u� the response curve indicatesthe coexistence of three planar periodic motions and one nonplanarperiodic orbit� In this parameter regime� the planar periodic orbitbecomes unstable� the string �pops out of the plane and begins toexecute a whirling motion� At some parameter value �q� the nonplanarperiodic motion itself becomes unstable and the system may do anynumber of things depending on the exact system parameters and ini�tial conditions� For instance� it may hop to the small�amplitude planarperiodic orbit� Or the ballooning orbit itself may become unstable andbegin to precess �quasiperiodic motion�� In addition� chaotic motionscan sometimes be observed in this parameter range� These various dy�namical possibilities are illustrated schematically in Figure �� We


repeat� the motion observed depends on the exact system parametersand the initial conditions� because there can be many coexisting attrac�tors with complicated basins of attraction in this region� In particular�the chaotic motions are di�cult to isolate �and observe experimentally�without a thorough understanding of the system�

�� Torus Attractor

To construct a cross section for nonplanar periodic motion we couldimagine a plot of the position of the center of the string� and the forcingphase� �x� y� �� R� � S� �Fig� �� A map can be associated to anorbit in R��S� by recording the position of the orbit once each forcingperiod� Although this map is not a true Poincar�e map� it is easy toobtain experimentally and will be useful in explaining the notion of atorus attractor �see section ��

An elliptical periodic orbit in the �ow is represented in this crosssection by a discrete set of points that lie on a closed curve� This curveis topologically a circle� S� �Fig� ��b�� Similarly� a precessing ellipse�quasiperiodic motion� can generate an in�nite number of points� thesepoints �ll out this circle �Fig� ��c��

In the extended space �x� y� �� this quasiperiodic motion representsa dense winding of a torus as shown in Figure ��d�� Topologically� atorus is a space constructed from the Cartesian product of two circles�T � � S� � S�� In general� an n torus is constructed from n copies of acircle�

T n �

nz � �S� � S� � � � � S��

and a torus attractor naturally arises whenever quasiperiodic motionis encountered in a dissipative dynamical system�� The torus is an

�A proper cross section would be a manifold transverse to the �ow in R� � S��i�e�� a four manifold such as & � f�x� vx� y� vy� �� j � � �g� The torus attractor arisesfrom a Hopf bifurcation�a bifurcation from a �xed point to an invariant curve� Ina cross section� the limit cycle is represented by a �xed point� At the transition toquasiperiodic motion� this �xed point loses stability and gives birth to an invariantcircle� which is a cross section of the torus in the �ow�

�The attracting torus for nonplanar string vibrations is actually a four torus�T ��


Figure �� Experimental cross section for the nonplanar string vibra�tions�

attractor because it is an invariant set and an attracting limit set� Thisis illustrated in Figure �� which shows how orbits are attracted toa torus� A graph of a quasiperiodic orbit on a torus attractor is anamplitude�modulated time series �Fig� ��

�� Circle Map

We found that the single�humped map of the interval� f� � I �� I� wasa good model for some aspects of the dynamics of the bouncing ballsystem� Similarly� insight about motion near a torus attractor can begained by studying a circle map�

g � S� �� S��


Figure �� Torus attractor�

where the mapping g most often studied is a two�parameter map of theform

�n�� g��n� � �n � (� K

sin��n�� n � ��

This map has a linear term �n� a constant bias term (� and a nonlinearterm whose strength is determined by the constant K�

The frequency of the circle map is monitored by the winding number

W � limn��

gn��K��

n� ��

If the nonlinear term K equals zero� then W � (� The winding numbermeasures the average increase in the angle � per unit time �Fig� ��An orbit of the circle map is periodic if� after q iterations� �n�q ��n � p� for integers p and q� The winding number for a periodic orbit isW � p�q� A quasiperiodic orbit has an irrational winding number ��Circle maps have a devilish dynamical structure� which is explored inreference ��


Figure �� The winding number measures the average increase in theangle of the circle map�

�� Torus Doubling

A nonlinear system can make a transition from quasiperiodic motiondirectly to chaos� This is known as the quasiperiodic route to chaos�It is of great practical and historical importance since it was one ofthe �rst proposed mechanisms leading to the formation of a strangeattractor ��

There are� in fact� many routes to chaos even from a humble T � torusattractor� For instance� when the T � attractor loses stability� a stablehigher�dimensional torus attractor sometimes forms� Another possibil�ity in the string system is the formation of a doubled torus� illustratedschematically in Figure �� In the torus doubling route to chaos� ouroriginal torus �which is a closed curve in cross section� appears to splitinto two circles at the torus doubling bifurcation point �� The torusdoubling route to chaos is reminiscent of the period doubling route tochaos� However� it di�ers in at least two signi�cant ways� First� in mostexperimental systems� there are only a �nite number of torus doublingsbefore the onset of chaotic motion� In fact� no more than two torusdoublings have ever been observed in the string experiment� Second�the torus doubling route to chaos is a higher�dimensional phenomenon�


Figure �� Schematic of a torus doubling bifurcation�

requiring at least a four�dimensional �ow� or a three�dimensional map�It is not observed in one�dimensional maps� unlike the period doublingroute to chaos�

Now that we have reviewed some of the more salient dynamicalfeatures of a string�s motions� let�s turn our attention to assemblingthe tools required to view these motions in a real string experiment�

�� Experimental Techniques

The dynamics of a forced string raises experimental challenges commonto a variety of nonlinear systems� In this section we describe a fewof the experimental diagnostics that help with the visualization andidenti�cation of di�erent attractors� such as�

equilibrium points

limit cycles �periodic orbits�

invariant tori �quasiperiodic orbits�

strange attractors �chaotic orbits�

�� EXPERIMENTAL TECHNIQUES ��

The main tools used in the real�time identi�cation of an attractor areFourier power spectra and real�time Poincar�e maps� In addition� thecorrelation dimension calculated from an experimental time series helpsto con�rm the existence of a strange attractor� as well as providing ameasure of its fractal structure�

The terms �strange attractor and �chaotic attractor are not al�ways interchangeable� Speci�cally� a strange attractor is an attractorthat is a fractal� That is� the term strange refers to a static geomet�ric property of the attractor� The term chaotic attractor refers to anattractor whose motions exhibit sensitive dependence on initial condi�tions� That is� the term chaotic refers to the dynamics on the attractor�We mention this distinction because it is possible for an attractor tobe strange �a fractal�� but not chaotic �exhibit sensitive dependence oninitial conditions� �� Experimental methods are available for quanti�fying both the geometric structure of an attractor �fractal dimensions�and the dynamic properties of orbits on an attractor �Lyapunov expo�nents��

�� Experimental Cross Section

Variables measured directly in the string apparatus include the forcingphase� ��t�� and the displacement amplitudes of the string� x�t� andy�t�� The phase is measured directly from the function generator thatprovides the sinusoidal current to the wire� The string displacementis measured from the optical detectors that record the horizontal andvertical displacement at a �xed point along the wire� In addition� thewire�s velocity can be measured by sending the amplitude displacementsignal through a di�erentiator� a circuit that takes the analog derivativeof an input signal�

Many approaches are possible for constructing an experimentalPoincar�e section� The particular approach taken depends on both thetype of system and the equipment at hand� Here� we assume that thelab is stocked with a dual�trace storage oscilloscope and some basicelectronic components� A di�erent approach might be taken� for in�stance� if we have access to a digital oscilloscope either in the form ofa commercial instrument or a plug�in board to a microcomputer�


Figure �� Schematic for the construction of an experimental Poincar�emap for the string apparatus�

Planar Cross Section

We will record the Poincar�e section on the storage oscilloscope� Our�rst step is to adjust the optical detectors so that the axis for a purelyplanar vibration is well aligned with one of the optical detectors� Next�the signal from this optical detector is sent to one of the input channelsof the oscilloscope� The other channel is used to record the velocity�via the di�erentiator� of this same signal� Lastly� the time�base on theoscilloscope must be set to X�Y mode� thereby allowing both the hori�zontal and vertical oscilloscope sweeps to be controlled by the externalsignals�

The result� as shown schematically in the oscilloscope in Figure ��is an experimental rendering of the phase space trajectory for a string�To construct a Poincar�e map� we need to sample this trajectory onceeach forcing period� That is� instead of recording the entire trajectory�we only want to record a sequence of points on this trajectory� Thiscan be accomplished by turning the oscilloscope�s beam intensity on for


a brief moment once each forcing period� This is easy to do because onthe back of most oscilloscopes is an analog input line labeled �z thatcontrols the oscilloscope�s beam intensity� Finally� we need a triggering

circuit that takes as input the sinusoidal forcing signal and generates asoutput a clock pulse� which is used to brie�y turn on the oscilloscope�sbeam once each forcing period�

Triggering Circuit

A simple triggering circuit can be constructed from a monostable vi�brator and a Schmitt trigger� Here� though� we describe a slightlymore sophisticated approach based on a phase�locked loop �PLL�� Thephase�locked loop circuit has the advantage that it allows us to triggermore than once each forcing period� This feature will be useful whenwe come to digitizing a signal because the phase�locked loop circuit canbe used to trigger a digitizer an integer number k times each forcingperiod� thereby giving us k samples of the trajectory each cycle�

A good account of all things electronic� including phase�locked loops�is presented in the book by Horowitz and Hill� The art of electronics�For our purposes� a phase�locked loop chip contains a phase detector�an ampli�er� and a voltage�controlled oscillator �VCO�� in one pack�age� A PLL� when used in conjunction with a stage counter� gener�ates a clock signal �or triggering pulse� that is ideal for constructing aPoincar�e section� A schematic of the triggering circuit used with thestring apparatus is presented in Figure �� and is constructed fromtwo o��the�shelf chips� a CMOS �� PLL and a �� stage counter�This circuit� with small adjustments� is useful for generating a trigger�ing signal in any forced system� The counter is set to one for a Poincar�esection� that is� it generates one pulse each period� It can be readjustedto produce k pulses per period when digitizing�

Nonplanar Cross Section

To generate an experimental cross section for nonplanar motions wereplace the vx�t� input to the oscilloscope with the output y�t� from thesecond displacement detector� The resulting plot on the oscilloscope isproportional to the actual horizontal and vertical displacement of the


Figure �� Triggering circuit used to construct a Poincar�e map in thestring apparatus� �Courtesy of K� Adams and T� C� A� Molteno��

string� thus providing us with a magni�ed view of the string�s whirling�

�� Embedding

In the string system there is no di�culty in specifying and measur�ing the major system variables� Usually� though� we are not so lucky�Imagine an experimental dynamical system as a black box that gen�erates a time series� x�t�� In practice� we may know little about theprocess inside the black box� Therefore� the experimental constructionof a phase space trajectory� or a Poincar�e map� seems very problematic�

One approach to this problem�the experimental reconstruction of aphase space trajectory�is as follows� We start out by assuming that thetime series is produced by a deterministic dynamical system that canbe modeled by some nth�order ordinary di�erential equation� For thisparticular example we assume that the system is modeled by a third�order di�erential system� as is the case for planar string vibrations�


Then a given trajectory of the system is uniquely speci�ed by the valueof the time series and its �rst and second derivatives at time t� � ��

x�t�� x�t�� $x�t��

This suggests that in reconstructing the phase space we can begin withour measured time series� and then use x�t� to calculate two new phasespace variables y�x�t�� and z�x�t�� de�ned by

y�t� �d

dtx�t��

z�t� �d

dty�t��

In estimating the pointwise derivatives of x�t� in an experiment wecan proceed in at least two ways� �rst� we can process the originalsignal through a di�erentiator� and then record �digitize� the originalsignal along with the di�erentiated signals� or second� we could digitizethe signal� and then compute the derivatives numerically� While bothtechniques are feasible� each is fraught with experimental di�cultiesbecause di�erentiation is an inherently noisy process� This is becauseapproximating a derivative often involves taking the di�erence of twonumbers that are close in value� To see this� consider the numericalderivative of a digitized time series fx�ti�g de�ned by

yi �xi � xi��ti � ti��

� ��

For instance� let xi�� xi � �� and ti�ti�� Thenyi � �� that is� the value of the �rst derivative is already buriedin the noise� and the problem just gets worse when taking higher�orderderivatives�

However� let�s look at equation �� again� If the sampling timeis evenly spaced� then ti � ti�� is constant� so

yi � xi � xi��

That is� almost all the information about the derivative is contained ina variable constructed by taking the di�erence of two points in the orig�inal time series fx�ti�g� This idea can be generalized as follows� Instead


of de�ning the new variables for the reconstructed phase space in termsof the derivatives� we can recover almost all of the same informationabout an orbit from the embedded variables de�ned by ��

yi � xi�r� ��

zi � xi�s� s �� r� ��

where r and s are integers� Each new embedded variable is de�ned bytaking a time delay of the original time series� Clearly� any number ofembedded variables can be created in this way� and this method can beused to reconstruct a phase space of dimension far greater than three�

There are many technical issues associated with the construction ofan embedded phase space� An in�depth discussion of these issues canbe found in reference �� The �rst concern is determining a good choicefor the delay times� r and s� One rule of thumb is to take r to be small�say � or � In fact� we could de�ne the second variable as

vi � xi � xi�r� ��

and think of it as a velocity variable� The second embedding timeshould be much larger than r� but not too large� To be more speci�c�consider the planar oscillations of a string again� In this example anatural cycle time is given by the period of the forcing term� A goodchoice for s is some sizable fraction of the cycle time� For instance� let�ssay our digitizer is set to sample the signal � times each period� Thena sensible choice for r might be � and for s might be �� or one�quarterof the forcing period� Figure �� shows a plot of a chaotic trajectoryin the Du�ng oscillator in both the original phase space and the phasespace reconstructed from the embedding variables� The similarity ofthe two representations lends support to the claim that a trajectoryin the embedded phase space provides a faithful representation of thedynamics�

Constructing a real�time two�dimensional embedded phase space isstraightforward� An embedded signal is obtained by sending the orig�inal signal through a delay line� The current signal and the delayedsignal are sent to the oscilloscope� thereby giving us a real�time repre�sentation of the phase space dynamics from our black box�


Figure �� Trajectory of the Du�ng oscillator in �a� phase space and�b� the embedded phase space with delay time � � ��

�� Power Spectrum

A signal from a nonlinear process is a function of time which we willcall F �t� in this section� It is possible to develop signatures for periodic�quasiperiodic� and chaotic signals by analyzing the periodic propertiesof F �t�� These signatures� which are based on Fourier analysis� arevaluable experimental aids in identifying di�erent types of attractors�

Fourier Series

To analyze the periodic properties of F �t� it is useful to uncover a func�tional representation of the signal in terms of the orthogonal functionscos�t�L� and sin�t�L� of period L� To determine the Fourier se�

ries for F �t� we must �nd the constants ak and bk so that the followingidentity holds ��

F �t� �a�

� a� cos

Lt � b� sin

Lt

� a� cos

Lt � b� sin

Lt � � � � � ��

Fourier showed that the constants ak and bk in equation �� arecomputed from F �t� by means of the integral formulas

ak �

L

Z L

�F �t� cos

�

Lkt�dt k � ��


Figure �� Digitized time series�

and

bk �

L

Z L

�F �t� sin

�

Lkt�dt k � ��

Finite Fourier Series

In equations �� we are assuming that F �t� is a continuousfunction of t� Equations �� and �� dictate that to calculate thekth constants� ak and bk in the Fourier series �eq� �� we substituteF �t� into the previous equations and integrate� In experiments withdigitized data� the signal we actually work with is an equally spaceddiscrete set of points� F �tp�� measured at the set of times ftpg �Fig�� Therefore� we need to develop a discrete analog to the Fourierseries� the �nite Fourier series�

Let us consider an even number of points per period� N � Then theN sample points are measured at

��L

N�

L

N� � � � �

�N � ��L

N�

or more succinctly�

tp �pL

N� p � �� N � ��

The �nite Fourier series for a function sampled at F �tp� is

F �t� �A�

�

N��Xk�

�Ak cos

Lkt � Bk sin

Lkt�

�AN

cos

LNt� ��


where

Ak ��

N

�N��Xp�

F �tp� cos

Lktp� k � �� N ��

and

Bk ��

N

�N��Xp�

F �tp� sin

Lktp� k � �� N � ��

In the above formulas� we assumed that the function is sampledat N points� that the value at the point N � � is L� and that theendpoints � and L satisfy the periodic boundary condition�

F �� F �L��

If the latter assumption does not hold� the convention is to average thevalues at the endpoints so that the value at F �� is taken to be

F �� F �L�

�

In this case the formulas for the coe�cients are

Ak ��

N

��F ��

�

�N��Xp�

F �tp� cos

Lktp �

F �L�

! ��

and

Bk ��

N

�N��Xp�

F �tp� sin

Lktp� ��

Equations �� and �� can be viewed as a transform or mapping�That is� given a set of numbers F �tp�� these relations generate two newsets of numbers� A�k� and B�k�� It is easy to program this transform�The code to calculate the discrete Fourier transform of F �tp� involvesa double loop �see eqs� �� and �� the inner loop cycles throughthe index k� and the outer loop covers the index p� Each loop hasorder N steps� so the total number of computations is of order N��The discrete Fourier transform is� therefore� quite slow for large N�say N �� see Appendix D for programs to calculate discreteFourier transforms�� Fortunately� there exists an alternative method


for calculating the Fourier transform called the fast Fourier transformor FFT� The FFT is an N lnN computation� which is much faster thanthe discrete Fourier for large data sets� A detailed explanation of theFFT along with C code examples is found in Numerical Recipes ��

Power Spectrum

The amplitude coe�cients Ak and Bk give us a measure of how well thesignal �ts the kth sinusoidal term� A plot of Ak versus k or Bk versusk is called a frequency spectrum� The �normalized� power spectrumamplitude is de�ned by

Hk �qA�k � B�

k� ��

A plot of Hk versus k is called the power spectrum� This graphicalrepresentation tells us how much of a given frequency is in the originalsignal�

Experimental power spectra can be obtained in at least three ways��i� obtain a spectrum analyzer or signal analyzer� which is a commercialinstrument dedicated to displaying real�time power spectra� �ii� obtaina signal analyzer card for a microcomputer �this is usually less expensivethan option �i�� or �iii� digitize your data and write an FFT for yourcomputer �this option is the cheapest�� Having successfully procuredspectra capabilities� we now move on to describing how to use them�

Spectral Signatures

The spectral signatures for periodic� quasiperiodic� and chaotic motionare illustrated in Figure �� The power spectrum of a period one orbitis dominated by one central peak� call it �� The power spectrum of aperiod two orbit also has a sharp peak at �� and additional peaks at thesubharmonic �� and the ultrasubharmonic �� These new spectral

All things are fair in love� war� and experimental physics� Methods to obtaina spectrum analyzer may include� �a� locating and �nationalizing� a spectrumanalyzer from a nearby laboratory� or �b� locating and appropriating a spectrumanalyzer on grounds of �national security�� Another option is to use an older modelspectrum analyzer� such as a Tektronix �L spectrum analyzer which plugs into anolder model Tek scope�


Figure �� Time series and power spectra� �a� periodic �period one��b� periodic �period two�� c� quasiperiodic� �d� chaotic�


peaks are the �sidebands about the primary frequency that come intoexistence through� say� a period doubling bifurcation of the period oneorbit� More generally� the power spectrum of a period n orbit consistsof a collection of discrete peaks showing the primary frequency and itsovertones� In periodic motion� all the peaks are rationally related tothe primary peak �resonance��

Quasiperiodic motion is characterized by the coexistence of two in�commensurate frequencies� Thus� the power spectrum for a quasiperi�odic motion is made up from at least two primary peaks� �� and ��which are not rationally related� Additionally� each of the primarypeaks can have a complicated overtone spectrum� The mixing of theovertone spectra from �� and �� usually allows one to distinguish pe�riodic motion from quasiperiodic motion� A quick examination of thetime series� or the Poincar�e section� can also help to distinguish periodicmotion from quasiperiodic motion�

The power spectrum of a chaotic motion is easy to distinguish fromperiodic or quasiperiodic motion� Chaotic motion has a broad�bandpower spectrum with a rich spectral structure� The broad�band natureof the chaotic power spectrum indicates the existence of a continuumof frequencies� A purely random or noisy process also has a broad�bandpower spectrum� so we need to develop methods to distinguish noisefrom chaos� In addition to its broad�band feature� a chaotic powerspectrum can also have many broad peaks at the nonlinear resonancesof the system� These nonlinear resonances are directly related to theunstable periodic orbits embedded within the chaotic attractor� So thepower spectrum of a chaotic attractor does provide some limited infor�mation concerning the dynamics of the system� namely� the existence ofunstable periodic orbits �nonlinear resonances� that strongly in�uencethe recurrence properties of the chaotic orbit�

Additionally� in the periodic and quasiperiodic regimes� new humpsand peaks can appear in a power spectrum whenever the system isnear a bifurcation point� These spectral features are called transient

precursors of a bifurcation� A detailed theory of these precursors withmany practical applications has been developed by Wiesenfeld and co�workers ��


�� Attractor Identi�cation

So far we have discussed four measurements that allow us to visualizeand identify the attractor coming from a nonlinear process�

time series

power spectra

phase space portrait� or reconstructed phase space

experimental Poincar�e sections

All these qualitative techniques can be set up with instruments thatare commonly available in any laboratory�

To get a time series we hook up the output signal from the nonlinearprocess to an input channel of the oscilloscope and use the time baseof the scope to generate the temporal dimension of the plot� To obtaina power spectrum� we use a spectrum analyzer or digitize the data anduse an FFT� An experimental phase space portrait can be plotted onan oscilloscope either by recording two system variables directly� suchas �x� y� or �x� x�� or from the delayed variable �x�t�� x�t� � �� where� is the delay time� Lastly� in a forced system� the Poincar�e sectionis obtained from the phase space trajectory by strobing it once eachforcing period using the �z blanking on the back of the oscilloscope�

Now to identify an attractor� we monitor these four diagnostic toolsas we vary a system parameter� A bifurcation point is easy to iden�tify by using these tools� and the existence of a particular bifurcationsequence� say a sequence of period doubling bifurcations� is a strongindicator for the possible existence of chaotic motion�

The use of these diagnostic tools is illustrated schematically in Fig�ure �� for the period doubling route to chaos� For the parametervalues �� and �� an examination of any one of these diagnosticsis su�cient to identify the existence of a periodic motion� as well as itsperiod� For � �c� these four diagnostics� as well as the fact that thestrange Poincar�e section arose from a sequence of bifurcations from aperiodic state� support the claim that the motion is chaotic�

In particular� the Poincar�e section is useful for distinguishing low�dimensional chaos from noise� The chaotic Poincar�e section illustrated


Figure �� Period doubling route to chaos� �a� period one� �b� periodtwo� �c� period four� and �d� chaotic� �Adapted from Tredicce andAbraham ��


Figure �� Poincar�e maps from periodic� quasiperiodic� chaotic� andnoisy �random� processes�

in Figure �� resembles the familiar one�humped map studied in Chap�ter � In contrast� the Poincar�e map for a noisy signal from a stochasticprocess �lls the whole oscilloscope screen with a random collection ofdots �see Fig� �� Thus motion on a strange attractor has a strongspatial correlation not present in a purely random signal� In the nextsection we will quantify this observation by introducing the correlationdimension� a measure that allows us to distinguish whether the signalis coming from a low�dimensional strange attractor or from noise�

The quasiperiodic route to chaos is illustrated in Figure �� Inthis case� the phase portrait for a quasiperiodic motion would resemblea Lissajous pattern� which may be hard to distinguish from a slowlyevolving chaotic orbit� The Poincar�e map� on the other hand� is usefulin distinguishing between these two cases� The sequence of dots formingthe Poincar�e map in the quasiperiodic regime lie on a closed curve thatis easy to distinguish from the spread of points in the Poincar�e map fora strange attractor�

�� Correlation Dimension

One di�erence between a chaotic signal from a strange attractor and asignal from a noisy random process is that points on the chaotic attrac�tor are spatially organized� One measure of this spatial organization isthe correlation integral�

C�� limn��

�

n�� number of pairs i� j whose distance jyi � yjj � ��


Figure �� Quasiperiodic route to chaos� �a� equilibrium� �b� peri�odic� �c� quasiperiodic� and �d� chaotic� �Adapted from Tredicce andAbraham ��


where n is the total number of points in the time series� This correlationfunction can be written more formally by making use of the Heavisidefunction H�z��

C�� limn��

�

n�

nXi�j�

H�� jyi � yjj��

where H�z� � � for positive z� and � otherwise� Typically� the vectoryi used in the correlation integral is a point in the embedded phasespace constructed from a single time series according to

yi � �xi� xi�r� xi��r� � � � � xi��m��r�� i � ��

For a limited range of � it is found that

C��

that is� the correlation integral is proportional to some power of � ��This power � is called the correlation dimension� and is a simple mea�sure of the �possibly fractal� size of the attractor�

The correlation integral gives us an e�ective procedure for assigninga fractal dimension to a strange set� This fractal dimension is a simpleway to distinguish a random signal from a signal generated by a strange�possibly chaotic� set� In principle� a random process has an �in�nite correlation dimension� Intuitively� this is because an orbit of a randomprocess is not expected to have any spatial structure� In contrast� thecorrelation dimension for a closed curve �a periodic orbit� is �� and fora two�dimensional surface �such as quasiperiodic motion on a torus�is � A strange �fractal� set can have a correlation dimension that isnot an integer� For instance� the strange set arising at the end of theperiod doubling cascade found in the quadratic map has a correlationdimension of �� indicating that the dimension of this strangeattractor is somewhere between that of a �nite collection of points�� and a curve ��

There exist some technical issues associated with the calculation ofa correlation dimension � from a time series that have to do with thechoice of the embedding dimension m and the delay time r� Theseissues are dealt with more fully in references �� and �� There now

�� String

exist several computer codes in the public domain that have� to a largeextent� automated the calculation of � from a single time series x�ti��Such a time series could come from a simulation or from an experiment�See� for example� the BINGO code by Albano �� or the e�cient algo�rithm discussed by Grassberger �� Thus� the correlation dimensionis now a standard tool in a nonlinear dynamicist�s toolbox that helpsone distinguish between noise and low�dimensional chaos�


�� J� R� Tredicce and N� B� Abraham� Experimental measurements to identifyand�or characterize chaotic signals� in Lasers and quantum optics� edited byL� M� Narducci� E� J� Quel� and J� R� Tredicce� CIF Series Vol� �� WorldScienti�c� New Jersey� �� N� Gershenfeld� An experimentalist�s introduc�tion to the observation of dynamical systems� in Directions in Chaos� Vol� ��edited by Hao B��L� �World Scienti�c� New Jersey� �� pp� �� N� B�Abraham� A� M� Albano� and N� B� Tu�llaro� Introduction to measures ofcomplexity and chaos� in NATO ASI Series B� Physics� Proceedings of theinternational workshop on Quantitative Measures of Dynamical Complexity inNonlinear Systems� Bryn Mawr� PA� USA� June �� edited by N� B�Abraham� A� M� Albano� T� Passamante� and P� Rapp� �Plenum Press� NewYork� �� J� Crutch�eld� D� Farmer� N� Packard� R� Shaw� G� Jones� and R�Donnelly� Power spectral analysis of a dynamical system� Phys� Lett� ��A ��

�� C� Gough� The nonlinear free vibration of a damped elastic string� J� Acoust�Soc� Am� ��

�� The original version of this apparatus was constructed and examined for possi�ble chaotic motions by T� C� A� Molteno at the University of Otago� Dunedin�New Zealand� For more details� see T� C� A� Molteno and N� B� Tu�llaro�Torus doubling and chaotic string vibrations� Experimental results� J� SoundVib� ��

�� R� C� Cross� A simple measurement of string motion� Am� J� Phys� ��

�� D� Martin� Decay rates of piano tones� J� Acoust� Soc� Am� ��

�� R� Hanson� Optoelectronic detection of string vibrations� The Physics Teacher�March ��


�� N� B� Tu�llaro� Nonlinear and chaotic string vibrations� Am� J� Phys� �� N� B� Tu�llaro� Torsional parametric oscillations in wires� Eur�J� Phys� ��

�� For theoretical and experimental details on chaos in an elastic beam� see F� C�Moon� Chaotic vibrations� An introduction for applied scientists and engineers�John Wiley $ Sons� New York� �� In particular� Appendix C containsa detailed description of the construction of an inexpensive beam experimentshowing chaos� Some discussion of the Du�ng equation as it relates to themotion of a beam can also be found in J� M� Thompson and H� B� Steward�Nonlinear dynamics and chaos �John Wiley $ Sons� New York� ��

�� J� Elliot� Intrinsic nonlinear e�ects in vibrating strings� Am� J� Phys� �� J� Elliot� Nonlinear resonance in vibrating strings� Am� J�Phys� � ��

�� See Chapter � of A� Nayfeh and D� Mook� Nonlinear oscillators �Wiley�Interscience� New York� �� A more elementary account of perturbativesolutions to the Du�ng equation is presented by A� Nayfeh� Introduction toperturbation techniques �John Wiley $ Sons� New York� ��

�� A classic in nonlinear studies is C� Hayashi�s Nonlinear oscillations in physicalsystems �Princeton University Press� Princeton� NJ� �� The method ofharmonic balance is used extensively by Hayashi� see pages �� and �� for a discussion and examples of this technique applied to theDu�ng oscillator�

�� A real gem of mathematical exposition is V� I� Arnold�s Ordinary di�erentialequations �MIT Press� Cambridge� MA� �� The basic theorems of ordinarydi�erential equations are proved in Chapter �� Arnold�s text is the best intro�ductory exposition to the geometric study of di�erential equations available� Amust read for all students of dynamical systems�

�� S� Wiggins� Introduction to applied nonlinear dynamical systems and chaos�Springer�Verlag� New York� �� pp� ��

�� N� B� Tu�llaro and G� A� Ross� Ode�A program for the numerical solution ofordinary di�erential equations� Bell Laboratories Technical Memorandum �� program and documentation are in the public domain� Alsosee Appendix C�

�� U� Parlitz and W� Lauterborn� Superstructure in the bifurcation set of theDu�ng equation� Phys� Lett� ��A �� U� Parlitz and W�Lauterborn� Resonance and torsion numbers of driven dissipative nonlinearoscillators� Z� Naturforsch� ��a� ��

�� String

�� The method of slowly varying amplitude is also known as the averaging methodof Krylov�Bogoliubov�Mitropolsky� see E� A� Jackson� Perspectives in nonlin�ear dynamics� Vol� � �Cambridge University Press� New York� �� pp� �� In the optics community� it is also known as the rotating�wave ap�proximation� see P� W� Milonni� M��L� Shih� and J� R� Ackerhalt� Chaos inlaser�matter interactions �World Scienti�c� Singapore� �� pp� ��

�� This is a bit sneaky� A � A� here because there is no damping term causinga frequency shift in the approximation to �rst order� In general A �� A�� andthen the slowly varying amplitude approximation results in both a detuningand a phase shift� Both data are encoded by A � C�

�� A marvelous pictorial introduction to dynamical systems theory is the uniqueseries of books by R� Abraham and C� Shaw� Dynamics�The geometry ofbehavior� Vol� �� Aerial Press� Santa Cruz� CA� �� Volume �� pp� �� contains a good description of the geometry of the phase space associatedwith hysteresis in the Du�ng oscillator� and Volume �� pp� �� shows howhomoclinic tangles are formed�

�� J� M� Johnson and A� K� Bajaj� Amplitude modulated and chaotic dynamicsin resonant motion of strings� J� Sound Vib� ��

�� J� Miles� Resonant� nonplanar motion of a stretched string� J� Acoust� Soc�Am� ��

�� O� M� O�Reilly� The chaotic vibration of a string� Ph�D� thesis� Cornell Univer�sity� ��

�� P� Bak� The devil�s staircase� Physics Today �� P� Bak� T�Bohr� and M� H� Jensen� Mode�locking and the transition to chaos in dissipativesystems� Physica Scripta� Vol� T�� P� Cvitanovi�c� B� Shraiman�and B� S!oderberg� Scaling laws for mode lockings in circle maps� Physica Scripta��

�� For a fuller account of torus attractors and quasiperiodic motion in a nonlinearsystem� see Chapter of P� Berg�e� Y� Pomeau� and C� Vidal� Order withinchaos �John Wiley $ Sons� New York� �� An elementary discussion offractal dimensions and the correlation integral is found on pages ��

�� A� Arn�edo� P� Coullet� and E� Spiegel� Cascade of period doublings of tori� Phys�Lett� ��A �� K� Kaneko� Doubling of torus� Prog� Theor� Phys�� F� Argoul and A� Arn�edo� From quasiperiodicityto chaos� an unstable scenario via period doubling bifurcation or tori� J� deM�ecanique Th�eorique et Appliqu�ee� Numer�ero sp�ecial� ��

Problems ��

�� W� Ditto� M� Spano� H� Savage� S� Rauseo� J� Heagy� and E� Ott� Experimentalobservation of a strange nonchaotic attractor� Phys� Rev� Lett� ��

�� N� Packard� J� Crutch�eld� J� Farmer� and R� Shaw� Geometry from a timeseries� Phys� Rev� Lett� �� Also see Chapter � of D� Ruelle�Chaotic evolution and strange attractors �Cambridge University Press� NewYork� �� For a geometric approach to the embedding problem see Th�Buzug� T� Reimers� and G� P�ster� Optimal reconstruction of strange attractorsfrom purely geometrical arguments� Europhys� Lett� ��

�� R� Hamming� An introduction to applied numerical analysis �McGraw�Hill�New York� ��

�� W� Press� B� Flannery� S� Teukolsky� and W� Vetterling� Numerical recipes inC �Cambridge University Press� New York� ��

�� K� Wiesenfeld� Virtual Hopf phenomenon� A new precursor of period doublingbifurcations� Phys� Rev� A �� K� Wiesenfeld� Noisy precursorsof nonlinear instabilities� J� Stat� Phys� �� P� Bryant and K�Wiesenfeld� Suppression of period doubling and nonlinear parametric e�ectsin periodically perturbed systems� Phys� Rev� A �� B�McNamara and K� Wiesenfeld� Theory of stochastic resonance� Phys� Rev� A�� K� Wiesenfeld and N� B� Tu�llaro� Suppression of perioddoubling in the dynamics of the bouncing ball� Physica ��D� ��

�� P� Grassberger and I� Procaccia� Characterization of strange attractors� Phys�Rev� Lett� �� P� Grassberger and I� Procaccia� Measuring thestrangeness of strange attractors� Physica �D� ��

�� The BINGO program runs on the IBM�PC� For a copy contact A� Albano�Department of Physics� Bryn Mawr College� Bryn Mawr� PA ��

�� P� Grassberger� An optimized box�assisted algorithm for fractal dimensions�Phys� Lett� A �� This brief paper provides a Fortran programfor calculating the correlation integral� For another new approach to calculatingthe correlation integral see Xin�Jun Hou� Robert Gilmore� Gabriel B� Mindlin�and Hern�an Solari� An e�cient algorithm for fast O�N � ln�N �� box counting�Phys� Lett� ��

Problems


�� String

�� Use Amp'ere�s law to �nd the magnetic �eld at a radial distance r from a longstraight current�carrying wire�

�� The Joule heating law says that the power dissipated by a current�carryingobject is

P � V I � I�R� ��

where V is the voltage� I is the current� and R is the resistance� Furthermore�for small temperature changes %Temp� the fractional change of length of asolid obeys

%L

L� �%Temp� ��

where � is the linear coe�cient of thermal expansion of the material� Discussthe relevance of these two physical laws on the string apparatus�

Section ��

�� From equation �� show that the location of the two equilibrium points is

approximately given by r� �lq

l��l�l�

�

�� Solve the di�erential equation !x � � (x � ��x and graph x�t� versus t for afew values of � Why is called the damping coe�cient�

�� Solve the di�erential equation �r � � �r� r � �x� y�� x� y�� Draw a plotof r�t� � �x�t�� y�t�� for a �xed � x� y��

�� Verify that the variables in equation �� are dimensionless�

�� Verify that equation �� is a solution to equation �� with f �t� � �� and K � � �discard second�order terms in �� Draw the solution� r�t�� in thex�y plane�

�� Derive equation �� from Figure �� Hint� To account for the factor of �realize that each spring makes a separate contribution to the restoring force�

Section ��

�� a� Show that the equation of motion for a simple pendulum in dimensionlessvariables is

!�� sin� � ��

�b� Write this as a �rst�order system with �� v� � S� �R� Using equation�� nd the potential energy function and a few integral curves for apendulum�

Problems ��

�c� Show that the pendulum has equilibrium points at �� and �� anddiscuss the stability of these �xed points by relating them to the con�gura�tions of the physical pendulum� Are there any saddle points� Are there anycenters�

�d� Draw a schematic of the phase plane� Identify the separatrix in this phaseportrait� Orbits inside the separatrix are called oscillations� Why� Orbitsoutside the separatrix are called rotations� Why�

�e� Add a dissipative term � (�� to get a damped pendulum and discuss howthis changes the phase portrait� In particular� discuss the relation of theinsets and outsets of the equilibrium points with the basins of attraction� Arethere any attractors� Are there any repellers�

�f� Now add a forcing term f cos��t� and write the di�erential equations forthe system in the extended phase space �� v� �� R�S��S�� where � � �t�De�ne a global cross section for the forced damped pendulum �see eq� ��

Section ��

�� Verify that equation �� is a solution to equation �� discard higher�order terms in ��

�� Plot the linear response curve a�� eq� �� for a few representative valuesof F and ��

�� This exercise illustrates the method of harmonic balance� Assume an approx�imate solution of the form

x� � X sin � � Y cos �

to the Du�ng equation ��

�a� Substitute x� into equation �� and equate terms containing sin� andcos � separately to zero�

�b� Show thatAX � ��Y � �� X � AY � F�

where

A � ��

��R�� R� � X� � Y ��

�c� Show that�A� � ��R� � F ��

�d� Plot the response �R� versus �� for � � �� and several valuesof F � Plot the amplitude characteristic �R� versus F � for � � �� and � � �� Indicate the unstable solution with a dashed line� Hint� It iseasier to plot R� as the independent variable�

�� String

Section ��

�� Verify equations �� for the free whirling motions of a string�

�� Write a program to iterate the circle map �eq� �� and explore its solutionsfor di�erent values of K and )�

Section ��

�� For a time series fxig� in which the xi are sampled at evenly spaced times�write a transformation between the phase space variable yi � �xi�xi��ti�ti�� and the embedded phase space variable yi � xi�r� with r � �� Why isthe embedded phase space rotated like it is in Figure �� Why does oneusually choose r � ��

�� Write a program based on equations �� to calculate the discreteFourier amplitude coe�cients and the power spectrum �eq�� for a dis�crete time series� Test the program on some sample functions for periodicmotion �e�g�� x�t� � cos��t� � cos��t�� and quasiperiodic motions �e�g��x�t� � cos��t� � cos�

p��t��

Chapter �

Dynamical Systems Theory

�� Introduction

This chapter is an eclectic mix of standard results from the mathemat�ical theory of dynamical systems along with practical results� terminol�ogy� and notation useful in the analysis of a low�dimensional dynamicalsystem �� The examples in this chapter are usually con�ned to two�dimensional maps and three�dimensional �ows�

In the �rst three chapters we presented nonlinear theory by wayof examples� We now present the theory in a more general setting�Many of the fundamental ideas have already been illustrated in theone�dimensional setting� For example� in one�dimension we found thatthe stability of a �xed point is determined by the derivative at the�xed point� The same result holds in higher dimensions� However� theactual computational machinery needed is far more intricate becausethe derivative of an n�dimensional map is an n� n matrix�

Another key idea we have already introduced is hyperbolicity� hyper�bolic sets� and symbolic analysis �see section �� In this chapter wepresent a detailed study of the Smale horseshoe� which is the canonicalexample of a �hyperbolic chaotic invariant set� A thorough under�standing of this example is essential to the analysis of a chaotic repelleror attractor� We conclude this chapter with a discussion of sensitivedependence on initial conditions� This brings us back to the Lyapunovexponent� which we de�ne for a general n�dimensional dynamical sys�

��

�� CHAPTER �� DYNAMICAL SYSTEMS THEORY

tem�

�� INTRODUCTION ��

Flows

Lorenz Equation

��x� y� z� � R� � R�

(x � ��y � x�(y � �x� y � xz(z � ��z � xy

��

parameters��

Du�ng Equation

��x� v� �� R� � S� � R� � S�

(x � v(v � ��x � �x� � �v� � f cos��(� � �

��

nonautonomous form!x� � (x� x� �x� � f cos��t�parameters�� f� �

Forced Damped Pendulum

�� v� �� S� � S� �R� S� � S� �R

(� � v(v � �"�v � � sin��# � f cos��(� � �

��

parameters�� f� �

Modulated Laser

��u� z� �� R� � S� � R� � S�

(u � "z � f cos��#u(z � �� z�� z�u(� � �

��

parameters�� f� �

Maps

Quadratic Map

f�x� � R� R

xn�� xn�� xn�parameter

Sine Circle Map

f�� S� � S�

�n�� n � )� K�� sin��n�

parameters)�K

H�enon Map

f�x� y� � R� � R�

xn�� x�n � �ynyn�� xn

parameters��

Baker�s Map

f�x� y� � I � I � I � I

xn�� xn mod �

yn��

�yn for � � xn �� yn for �� xn � �

��

parameter� ��

IHJM Optical Map "�#

f�z� � C� C

zn�� Bzn exphi��

��jznj�

�iparameters�� B


�� Flows and Maps

Most of the dynamical systems studied in this book are either three�dimensional �ows or one� or two�dimensional maps� Common examplesof maps and �ows are listed on the previous page�

Flows are speci�ed by di�erential equations �section �� Simi�larly� maps are speci�ed by di�erence equations�

xn�� f�xn��

Maps can also be written as x �� f�x�� The notation �� is readas �maps to� The forward orbit of x is O��x� � ffn�x� � n �g�where fn � f � � � � � f is the nth composite of f � and f� is the identityfunction� If the inverse f�� is well de�ned� then the backward orbit ofx is O��x� � ff�n�x� � n �g� Finally� the orbit of x is the sequenceof all positions visited by x� O�x� � O��x�

SO��x��

�� Flows

A �rst�order system of di�erential equations is written as

dx

dt� f�x� t��

where x � �x�� x�� xn� are the dependent variables� t is the inde�pendent variable time� and � is the set of parameters for the system�Sometimes the parameter dependence is denoted by a subscript as inf��x� t��

A vector �eld is formally de�ned by a map F � A � Rn � Rn

that assigns a vector F�x� to each point x in its domain A� Moregenerally� a vector �eld on a manifold M is given by a map that assignsa vector to each point in M � For most of this chapter we only need towork with Rn� A system governed by a time�independent vector �eldis called autonomous� otherwise it is called nonautonomous� As we sawin section �� any nonautonomous vector �eld can be converted toan autonomous vector �eld of a higher dimension�

The �ow� �� of the vector �eld F is analytically de�ned by�

�

�t��x� t� � F��x� t��

��x� �� x� ��

�� FLOWS AND MAPS ��

Figure �� a� The �ow of a solution curve� �b� The �ow of a collec�tion of solution curves resulting in a continuous transformation of themanifold�

The position x is the initial condition or initial state� The initial con�dition is also written as x� when it is speci�ed at t � �� A solution

curve� trajectory� or integral curve of the �ow is an individual solutionof the above di�erential equation based at x�� We often explicitly notethe time�dependence of the position �the solution curve to the abovedi�erential equation� by writing x�t� when we want to indicate the po�sition of the trajectory at a time t �� The collection of all states of adynamical system is called the phase space�

The term ��ow describing the evolution of the system in phasespace comes by analogy from the motion of a real �uid �ow� The�ow ��x� t� is regarded as a function of the initial condition x and thesingle parameter time t� The �ow � tells us the position of the initialcondition x after a time t� As illustrated in Figure ��a�� the positionof the point on the �ow line through x is carried or �ows to the point��x� t�� A geometric description of a �ow says that it is a one�parameterfamily of di�eomorphisms of a manifold� That is� the �ow lines of the�ow� which are solution curves of the di�erential equation� provide acontinuous transformation of the manifold into itself �Fig� ��b��

The �ow is often written as �t�x� to highlight its dependence onthe single parameter t� The �ow �t is also known as the evolution

operator� Composition of the evolution operator is de�ned in a naturalway� starting at the state x at time s � �� it �rst �ows to the point


�s � ��x� s� and then onto the point �t�s � ��x� s�� t�� The evolutionoperator satis�es the group properties

�i� �� identity� �ii� �t�s � �t � �s� ��

which are taken as the de�ning relations of a �ow for an abstract dy�

namical system� If the system is nonreversible we speak of a semi�ow�A semi�ow �ows forward in time� but not backward�

An explicit example of an evolution operator is given by solving thelinear di�erential equation $x � �x� Here the vector �eld is found fromthe equivalent �rst�order system � x � v� v � �x� so that the vector�eld is

F�x� v� � �v��x��

This vector �eld generates a �ow that is a simple rotation about theorigin �see Fig� �� The evolution operator is given by the rotationmatrix�

�t�x� �

�x�t�v�t�

��

�cos�t� sin�t�

� sin�t� cos�t�

� �xv

��

�� Poincar e Map

A continuous �ow can generate a discrete map in at least two ways� bya time�T map and by a Poincar�e map� A time�T map results when a�ow is sampled at a �xed time interval T � That is� the �ow is sampledwhenever t � nT for n � �� and so on�

The more important way �as described� for instance� in Gucken�heimer and Holmes �� in which a continuous �ow generates a discretemap is via a Poincar�e map� Let � be an orbit of a �ow �t in Rn� Asillustrated in Figure �� it is often possible to �nd a local cross sec�

tion ' � Rn about �� which is of dimension n � �� The cross sectionneed not be planar� however� it must be transverse to the �ow� All theorbits in the neighborhood of � must pass through '� The technicalrequirement is that F�x� �N�x� �� for all x � '� where N�x� is the

�The components of the map � should be written as ��x� �v�� where the super�script indicates the dependent coordinate� It is a common convention� though� tosuppress the � and mix the dependent variables with the coordinate functions sothat "�x�t�� v�t�# � "x�t�� v�t�#�

�� FLOWS AND MAPS ��

Figure �� Construction of a Poincar�e map from a local cross section�

unit normal vector to ' at x� Let p be a point where � intersects '�and let q � ' be a point in the neighborhood of p� Then the Poincar�e

map �or �rst return map� is de�ned by

P � ' � '� P �q� � ��q��

where � � � �q� is the time taken for an orbit starting at q to returnto '� It is useful to de�ne a Poincar�e map in the neighborhood of aperiodic orbit� If the orbit � is periodic of period T � then � �p� � T �A periodic orbit that returns directly to itself is a �xed point of thePoincar�e map� Moreover� an orbit starting at q close to p will have areturn time close to T �

For forced systems� such as the Du�ng oscillator studied in sec�tion �� a global cross section and Poincar�e map are easy to de�nesince the phase space topology is R� � S�� All periodic orbits of aforced system have a period that is an integer multiple of the forcingperiod� In this situation� it is sensible to pick a planar global crosssection that is transverse to S� �see Fig� �� The return time for thiscross section is independent of position and equals the forcing period�In this special case� the Poincar�e map is equivalent to a time�T map�

The Poincar�e map for the example of the rotational �ow generatedby F�x� v� � �v��x� is particularly trivial� A good cross section isde�ned by the positive half of the x�axis� ' � f�x� v�jv � � and x �g�All the orbits of the �ow are �xed points in the cross section� so thePoincar�e map is just the identity map� P �x� � x�


Figure �� a� Construction of a suspension of a map� �b� The sus�pension is not unique� an arbitrary number of twists can be added�

See Appendix E� H�enon�s Trick� for a discussion of the numericalcalculation of a Poincar�e map from a cross section�

�� Suspension of a Map

A discrete map can also be used to generate a continuous �ow� Acanonical construction for this is the so�called suspension of a map ��which is in a sense the inverse of a Poincar�e map�� Given a discrete mapf of an n�dimensional manifold M � it is always possible to construct a�ow on an n��dimensional manifold formed by the Cartesian productR with the original manifold� � � M � R � M � This suspensionprocess is illustrated in Figure ��a� where we show a mapping andthe �ow formed by �suspending this map� Each sequence of pointsof the map becomes an orbit of the �ow with the property that iffn�x� � �t�x� then fn��x� � �T�t�x�� The original map is recoveredfrom the suspended �ow by a time�T map� The suspension construction

�The suspension is de�ned globally� while the cross section for a Poincar�emap is only de�ned locally� Thus� these two constructions are not completelycomplementary�

�� ASYMPTOTIC BEHAVIOR AND RECURRENCE ��

is far from unique� For instance� as illustrated in Figure ��b�� wecould add an arbitrary number of complete twists to this particularsuspension and still get an identical time�T map� The number of fulltwists in the suspended �ow is called the global torsion� and it is atopological invariant of the �ow independent of the underlying map�

�� Creed and Quest

The close connection between maps and �ows�Poincar�e maps andsuspensions�gives rise to �The Discrete Creed �

Anything that happens in a �ow also happens in a �lower�dimensional� discrete dynamical system �and conversely��

�The creed is stated in the mode of the sunshine patriot� leaving plentyof room to duck as necessary �� This creed is implicit in Poincar�e�soriginal work� but was �rst clearly enunciated by Smale�

The essence of dynamical systems studies is stated in �The Dynam�ical Quest �

Where do orbits go� and what do they do%see when theyget there"

The dynamical quest emphasizes the topological �i�e�� qualitative� char�acterization of the long�term behavior of a dynamical system�

�� Asymptotic Behavior and Recurrence

In this section we present some more mathematical vocabulary thathelps to re�ne our notions of invariant sets� limit sets �attractors andrepellers�� asymptotic behavior� and recurrence� For the most part westate the fundamental de�nitions in terms of maps� The correspond�ing de�nitions for �ows are completely analogous and can be found inWiggins ��

Recurrence is a key theme in the study of dynamical systems� Thesimplest notion of recurrence is periodicity� Recall that a point of a mapis periodic of period n if there exists an integer n such that fn�p� � pand f i�p� �� p� � � i � n� This notion of recurrence is too restricted


since it fails to account for quasiperiodic motions or strange attractors�We will therefore explore more general notions of recurrence� At theend we will argue that the �chain recurrent set is the best de�nitionof recurrence that captures most of the interesting dynamics�

�� Invariant Sets

Formally� a set S is an invariant set of a �ow if for any x� � S we have��x�� t� � S for all t � R� S is an invariant set of a map if for anyx� � S� fn�x�� S for all n� We also speak of a positively invariant set

when we restrict the de�nition to positive times� t � or n ��Invariant sets are important because they give us a means of de�

composing phase space� If we can �nd a collection of invariant sets�then we can restrict our attention to the dynamics on each invariantset and then try to sew together a global solution from the invariantpieces� Invariant sets also act as boundaries in phase space� restrictingtrajectories to a subset of phase space�

�� Limit Sets� �� and Nonwandering

We begin by introducing the ��limit set� which starts us down the roadtoward de�ning an attractor� Let p be a point of a map f � M �M ofthe manifold M � Then the ��limit set of p is

��p� � fa �M j there exists a sequence ni �� such that fni�p� � ag�

Conversely� by going backwards in time we get the ��limit set of p�

��p� � fr �M j there exists a sequence ni �� such that f�ni�p� � rg�

These limit sets are the closure of the ends of the orbits� For ex�ample� if p is a periodic point� then ��p� � O��p�� It is not di�cult toshow that ��p� is a closed subset of M and that it is invariant underf � i�e�� f��p�� p�� Finally� a point p � M is called recurrent if itis part of the ��limit set� i�e�� p � ��p��

The forward limit set� L�� is de�ned as the union of all ��limit sets�likewise the backward limit set� L�� is de�ned as the union of all ��limit


sets�L��f� �

�p�M

��p� and L��f� ��p�M

��p��

The forward and backward limit sets are not necessarily closed�This observation motivates yet another useful notion of recurrence� thenonwandering set� A point p �M wanders if there exist a neighborhoodU of p and an m � such that fn�U� � U � � for all n m� Anonwandering point is one that does not wander� This brings us to thenonwandering set� (� which is a closed� invariant �under f� subset ofM �

(�f� � fq �M j q is nonwanderingg�The dynamical decomposition of a set into its wandering and non�

wandering parts separates� in mathematical terms� a dynamical sys�tem into its transient behavior�the wandering set�and long�term orasymptotic behavior�the nonwandering set�

The ��limit set and the nonwandering set do not address the ques�tion of the stability of an asymptotic motion� To get to the idea of anattractor� we begin with the idea of an attracting set� A closed invari�ant set A � M is an attracting set if there is some neighborhood U ofA such that for all x � U and all n ��

fn�x� � U and fn�x� � A�

Moreover� the domain or basin of attraction of A is given by�n��

fn�U��

The attracting set can consist of a collection of di�erent sets that aredynamically disconnected� For instance� a single attracting set couldconsist of two separate periodic orbits� To overcome this last di�culty�we will say that an attractor is an attracting set that contains a dense

orbit� Conversely� a repeller is de�ned as a repelling set with a denseorbit�

Although this is a reasonable de�nition mathematically� we will seethat this is not the most useful de�nition of an attractor for physicalapplications or numerical simulations� In these circumstances it willturn out that a more useful �albeit mathematically naive� de�nition of


an attractor or repeller is the closure of a certain collection of periodicorbits� The proper de�nition of an attractor is yet another hot spot inthe creative tension between the rigor demanded by a mathematicianand the utility required by a physicist�

�� Chain Recurrence

In the previous section we attempted to capture all the recurrent be�havior of the mapping f � M �M � Let

Per�f� � fperiodic points of fgL��f� �

Sp�M ��p�

(�f� � fnonwandering pointsgClearly�

Per�f� � L��f� � (�f��

There is no universally accepted notion of a set that contains all therecurrence� but this set ideally ought to be closed and invariant� Per�f�is too small�periodicity is too limited a kind of recurrence� L��f�is not necessarily closed� (�f�� while closed and invariant� has thedrawback that�

(�f j�� (�

The mapping f j� � ( � ( makes sense� of course� because of (�sinvariance�

In certain contexts the chain recurrent set� which is bigger than (�has all the desirable properties� According to the �chain recurrent pointof view� all the interesting dynamics take place in the chain recurrentset ��

Let � �� An ��pseudo orbit is a �nite sequence such that

d�f�xi�� xi�� i � �� n � ��

where d�� is a metric on the manifold M � An ��pseudo orbit canbe thought of as a �computer�generated orbit because of the slightroundo� error the computer makes at each stage of an iteration �seeFig� ��a��


Figure �� a� An ��pseudo orbit� or computer orbit� of a map� �b� achain recurrent point�

A point x � M is chain recurrent if� for all � �� there exists an��pseudo orbit x�� x�� xn such that x � x� � xn �see Fig� ��b��The chain recurrent set is de�ned as

R�f� � fchain recurrent pointsg�

The chain recurrent set R�f� is closed and f �invariant� Moreover� ( �R� For proofs� see reference �� or ��

As an example consider the quadratic map� f��x� � �x�� x�� for� �

p� and M � �� It is easy to see from graphical

analysis that if x �� then f��g is the attracting limit set� Whenx � �� the limit set & is a Cantor set �see section �� and thedynamics on & are topologically conjugate to a full shift on two symbols�so the periodic orbits are dense in &� The chain recurrent set R�f� �&Sf��g�

As another example� consider the circle map f � S� � S� shownin Figure �� in which the only �xed points are x and y� The arrowsindicate the direction a point goes in when it is iterated� It is easy tosee that

fx� yg � Per�f� � L� � L� � (�

� CHAPTER �� DYNAMICAL SYSTEMS THEORY

Figure �� Circle map with two �xed points�

However� R�f� � S� because an ��pseudo orbit can jump across the�xed points� Chain recurrence is a very weak form of recurrence�

�� Expansions and Contractions

In this section we consider how two�dimensional maps and three�dimensional �ows transform areas and volumes� Does a map locallyexpand or contract a region" Does a �ow locally expand or contract avolume in phase space" To answer each of these questions we need tocalculate the Jacobian of the map and the divergence of the �ow ��In this section we are concerned with showing how to do these calcula�tions� In section �� where we introduce the tangent map� we providesome geometric insight into these calculations�

�� Derivative of a Map

To �x notation� let f be a map fromRn to Rm speci�ed by m functions�

f � �f�� fm��

of n variables� Recall that the derivative� of a map f � Rn � Rm at x�is written as T � Df�x�� and consists of an m � n matrix called thematrix of partial derivatives of f at x��

Df�x��

�""�

f� x�

� � � f� xn

��

fm x�

� � � fm xn

##! � ��

�Some books call this the di�erential of f and denote it by df�x��

�� EXPANSIONS AND CONTRACTIONS ��

Figure �� Deformation of an in�nitesimal region under a map�

The derivative of f at x� represents the best linear approximation to fnear to x��

As an example of calculating a derivative� consider the function fromR� to R� that transforms polar coordinates into Cartesian coordinates�

f�r� �� f��r� �� f��r� �� r cos �� r sin ��

The derivative of this particular transformation is

Df�r� ��

�� f�

r f� �

f� r

f� �

! �

�cos � �r sin �sin � r cos �

��

�� Jacobian of a Map

The derivative contains essential information about the local dynamicsof a map� In Figure �� we show how a small rectangular region Rof the plane is transformed to f�R� under one iteration of the mapf�u� v� � R� � R� where f��u� v� � x�u� v� and f��u� v� � y�u� v��The Jacobian of f � written ��x� y��u� v�� is the determinant of thederivative matrix Df�x� y� of f ��

��x� y�

��u� v��

�� x u

x v

y u

y v

�� x

�u� �y�v� �x

�v� �y�u

� ��

�See Marsden and Tromba " # for the n�dimensional de�nition�


In the example just considered� �x� y� � �r cos �� r sin �� the Jacobianis

��x� y�

��r� �� r�cos� � � sin� �� r�

The Jacobian of a map at x� determines whether the area about x�expands or contracts� If the absolute value of the Jacobian is less thanone� then the map is contracting� if the absolute value of the Jacobianis greater than one� then the map is expanding�

A simple example of a contracting map is provided by the H�enonmap for the parameter range � � � �� In this case�

f��xn� yn� � xn�� x�n � �yn�

f��xn� yn� � yn�� xn�

and a quick calculation shows

��xn�� yn��

��xn� yn��

The Jacobian is constant for the H�enon map� it does not depend onthe initial position �x�� y�� When iterating the H�enon map� the area ismultiplied each time by �� and after k iterations the size of an initialarea a� is

a � a�j�kj�In particular� if � � � �� then the area is contracting�

�� Divergence of a Vector Field

Recall from a basic course in vector calculus that the divergence of avector �eld represents the local rate of expansion or contraction perunit volume �� So� to �nd the local expansion or contraction of a �owwe must calculate the divergence of a vector �eld� The divergence of athree�dimensional vector �eld F�x� y� z� � �F�� F�� F�� is

div F � r � F ��F�

�x��F�

�y��F�

�z� ��

Let V �� be the measure of an in�nitesimal volume centered at x�Figure �� shows how this volume evolves under the �ow� the diver�

�� EXPANSIONS AND CONTRACTIONS ��

Figure �� Evolution of an in�nitesimal volume along a �ow line�

gence of the vector �eld measures the rate at which this initial volumechanges�

div F�x� ��

V ��

d

dtV �t�jt��

For instance� the divergence of the vector �eld for the Lorenz systemis

r ��"� F�

F�

F�

#! � r �

�"� ��y � x��x� y � xz��z � xy

#! � ��

we �nd that the �ow is globally contracting at a constant rate wheneverthe sum of � and � is positive�

�� Dissipative and Conservative

The local rate of expansion or contraction of a dynamical system can becalculated directly from the vector �eld or di�erence equation withoutexplicitly �nding any solutions� We say a system is conservative if theabsolute value of the Jacobian of its map exactly equals one� or if thedivergence of its vector �eld equals zero�

r � F � ��


for all times and all points� A physical system is dissipative if it isnot conservative�� Most of the physical examples studied in this bookare dissipative dynamical systems� The phase space of a dissipativedynamical system is continually shrinking onto a smaller region of phasespace called the attracting set�

�� Equation of First Variation

Another quantity that can be calculated directly from the vector �eldis the equation of �rst variation� which provides an approximation forthe evolution of a region about an initial condition x� The �ow ��x� t�is a function of both the initial condition and time� We will often beconcerned with the stability of an initial point in phase space� and thuswe are led to consider the variation about a point x while holding thetime t �xed� Let Dx denote di�erentiation with respect to the phasevariables while holding t �xed� Then from the di�erential equation fora �ow �eq� �� we �nd

Dx

�

�t��x� t� � Dx�F��x� t��

which� on applying the chain rule on the right�hand side� yields the

equation of �rst variation�

�

�tDx��x� t� � DF��x� t��Dx��x� t��

This is a linear di�erential equation for the operator Dx�� DF��x� t��is the derivative of F at ��x� t�� If the vector �eld F is n�dimensional�then both DxF�� and Dx� are n� n matrices�

Turning once again to the vector �eld

F�x� v� � �v��x��

we �nd that the equation of �rst variation for this system is�� xx �xv

�vx �vv

! �

��

��

!�� xx �xv

�vx �vv

! �

�Note that this de�nition of dissipative can include expansive systems� Thesewill not arise in the physical examples considered in this book� See Problem ��


The superscript i to �i indicates the ith component of �ow� and thesubscript j to �j denotes that we are taking the derivative with respectto the jth phase variable� For example� �vx �

x�v� The compo�

nents of � are the coordinate positions� so they could be rewritten as��x�t�� v�t�� x�t�� v�t�� The dot� as always� denotes di�erentiationwith respect to time�

�� Fixed Points

An equilibrium solution of a vector �eld x � f�x� is a point !x that doesnot change with time�

f�!x� � ��

Equilibria are also known as �xed points� stationary points� or steady�state solutions� A �xed point of a map is an orbit which returns to itselfafter one iteration�

!x �� f�!x� � !x� ��

We will tend to use the terminology ��xed point when referring toa map and �equilibrium when referring to a �ow� The theory forequilibria and �xed points is very similar� Keep in mind� though� thata �xed point of a map could come from a periodic orbit of a �ow� Thissection brie�y outlines the theory for �ows� The corresponding theoryfor maps is completely analogous and can be found� for instance� inRasband ��

�� Stability

At least three notions of stability apply to a �xed point� local stability�global stability� and linear stability� Here we will discuss local stabilityand linear stability� Linear stability often� but not always� implies localstability� The additional ingredient needed is hyperbolicity� This turnsout to be quite general� hyperbolicity plus a linearization procedure isusually su�cient to analyze the stability of an attracting set� whetherit be a �xed point� periodic orbit� or strange attractor�

The notion of the local stability of an orbit is straightforward� A�xed point is locally stable if solutions based near !x remain close to


Figure �� a� A stable �xed point� �b� An asymptotically stable �xedpoint�

!x for all future times� Further� if the solution actually approachesthe �xed point� i�e�� x�t� � !x as t � �� then the orbit is calledasymptotically stable� Figure ��a� shows a center that is stable� butnot asymptotically stable� Centers commonly occur in conservativesystems� Figure ��b� shows a sink� an asymptotically stable �xedpoint that commonly occurs in a dissipative system� A �xed point isunstable if it is not stable� A saddle and source are examples of unstable�xed points �see Fig� ��

�� Linearization

To calculate the stability of a �xed point consider a small perturbation�y� about !x�

x � !x�t� � y� ��

The Taylor expansion �substituting eq� �� into eq� �� about !xgives

x � !x�t� � y � f�!x�t�� Df�!x�t��y� higher�order terms� ��

It seems reasonable that the motion near the �xed point should begoverned by the linear system

y � Df�!x�t��y� ��


since x�t� � f�!x�t�� If !x�t� � !x is an equilibrium point� then Df�!x�is a matrix with constant entries� We can immediately write down thesolution to this linear system as

y�t� � exp�Df�!x�t�y��

where exp�Df�!x�� is the evolution operator for a linear system� If we letA � Df�!x� denote the constant n�n matrix� then the linear evolutionoperator takes the form

exp�At� � id� At��

*A�t� �

�

�*A�t� � � � � ��

where id denotes the n� n identity matrix�The asymptotic stability of a �xed point can be determined by the

eigenvalues of the linearized vector �eld Df at !x� In particular� we havethe following test for asymptotic stability� an equilibrium solution of anonlinear vector �eld is asymptotically stable if all the eigenvalues ofthe linearized vector �eld Df�!x� have negative real parts�

If the real part of at least one eigenvalue exactly equals zero �andall the others are strictly less than zero� then the system is still linearlystable� but the original nonlinear system may or may not be stable�

�� Hyperbolic Fixed Points� � Saddles�

Sources� and Sinks

Let x � !x be an equilibrium point of a vector �eld� Then !x is calleda hyperbolic �xed point if none of the real parts of the eigenvalues ofDf�!x� is equal to zero� The test for asymptotic stability of the previoussection can be restated as� a hyperbolic �xed point is stable if the realparts of all its eigenvalues are negative� A �xed point of a map ishyperbolic if none of the moduli of the eigenvalues equals one�

The motion near a hyperbolic �xed point can be analyzed andbrought into a standard form by a linear transformation to the eigen�vectors of Df�!x�� Additional analysis� including higher�order terms� isusually needed to analyze the motion near a nonhyperbolic �xed point�

At last� we can precisely de�ne the terms saddle� sink� source� andcenter� A hyperbolic equilibrium solution is a saddle if the real part of


Figure �� Complex eigenvalues for a two�dimensional map with ahyperbolic �xed point� �a� saddle� �b� sink� and �c� source�

at least one eigenvalue of the linearized vector �eld is less than zero andif the real part of at least one eigenvalue is greater than zero� Similarly�a saddle point for a map is a hyperbolic point if at least one of theeigenvalues of the associated linear map has a modulus greater thanone� and if one of the eigenvalues has modulus less than one�

A hyperbolic point of a �ow is a stable node or sink if all the eigen�values have real parts less than zero� Similarly� if all the moduli areless than one then the hyperbolic point of a map is a sink�

A hyperbolic point is an unstable node or source if the real parts ofall the eigenvalues are greater than zero� The moduli of a source of amap are all greater than one�

A center is a nonhyperbolic �xed point for which all the eigenvaluesare purely imaginary and nonzero �modulus one for maps�� For a pic�ture of the elementary equilibrium points in three�dimensional space seeFigure �� of Thompson and Stewart �� The corresponding stabilityinformation for a hyperbolic �xed point of a two�dimensional map issummarized in Figure ��

�� Invariant Manifolds

According to our discussion of invariant sets in section �� we wouldlike to analyze a dynamical system by breaking it into its dynamicallyinvariant parts� This is particularly easy to accomplish with linear

�� INVARIANT MANIFOLDS ��

systems because we can write down a general solution for the �owoperator as etA �see section �� The eigenspaces of a linear �ow ormap �i�e�� the spaces formed by the eigenvectors of A� are invariantsubspaces of the dynamical system� Moreover� the dynamics on eachsubspace are determined by the eigenvalues of that subspace� If theoriginal manifold is Rn� then each invariant subspace is also a Euclideanmanifold which is a subset of Rn� It is sensible to classify each of theseinvariant submanifolds according to the real parts of its eigenvalues� �i�

Es is the subspace spanned by the eigenvectors of A with Re��i� � ��

Ec is the subspace spanned by the eigenvectors of A with Re��i� � ��

Eu is the subspace spanned by the eigenvectors of A with Re��i� ��

Es is called the stable space of dimension ns� Ec is called the centerspace of dimension nc� and Eu is called the unstable space of dimensionnu� If the original linear manifold is of dimension n� then the sum ofthe dimensions of the invariant subspaces must equal n� nu �nc �ns �n� This de�nition also works for maps when the conditions on �i arereplaced by modulus less than one �Es�� modulus equal to one �Ec�� andmodulus greater than one �Eu��

For example� consider the matrix

A �

�B� � �

� � ��

�CA

This matrix has eigenvalues � � �� and eigenvectors �� and the �ow on the invariant manifolds is illustratedin Figure ��

�� Center Manifold Theorem

It is important to keep in mind that we always speak of invariant man�ifolds based at a point� This point is usually a �xed point �x of a �owor a periodic point of a map� In the linear setting� the invariant man�ifold is just a linear vector space� In the nonlinear setting we can alsode�ne invariant manifolds that are not linear subspaces but are still


Figure �� Invariant manifolds for a linear �ow with eigenvalues �i �� Es�� Ec�� Eu��

manifolds� That is� locally they look like a copy of Rn� These invariantmanifolds are a direct generalization of the invariant subspaces of thelinear problem� They are the most important geometric structure usedin the analysis of a nonlinear dynamical system�

The way to generalize the notion of an invariant manifold from thelinear to the nonlinear setting is straightforward� In both the linear andnonlinear settings� the stable manifold is the collection of all orbits thatapproach a point x� Similarly� the unstable manifold is the collection ofall orbits that depart from x� The fact that this notion of an invariantmanifold for a nonlinear system is well de�ned is guaranteed by thecenter manifold theorem ��

Center Manifold Theorem for Flows� Let f��x� be asmooth vector �eld on Rn with f��!x� � � and A � Df�!x��The spectrum �set of eigenvalues� f�ig of A divides intothree sets �s� �c� and �u� where

�i ��

�s� Re��i� � ��c� Re��i� � ��u� Re��i� ��

Let Es� Ec� and Eu be the generalized eigenspaces of �s� �c�and �u� There exist smooth stable and unstable manifolds�called W s and W u� tangent to Es and Eu at !x� and a center


Figure �� Invariant manifolds of a saddle for a two�dimensional map�

manifold W c tangent to Ec at !x� The manifolds W s� W c�and W u are invariant for the �ow� The stable manifoldW s and the unstable manifold W u are unique� The centermanifold W c need not be unique�

W s is called the stable manifold� W c is called the center manifold� andW u is called the unstable manifold� A corresponding theorem for mapsalso holds and can be found in reference �� or �� Numerical methodsfor the construction of the unstable and stable manifolds are describedin reference ��

Always keep in mind that �ows and maps di�er� a trajectory of a�ow is a curve in Rn while the orbit of a map is a discrete sequence ofpoints� The invariant manifolds of a �ow are composed from a union ofsolution curves� the invariant manifolds of a map consist of a union of adiscrete collection of points �Fig� �� The distinction is crucial whenwe come to analyze the global behavior of a dynamical system� Onceagain� we reiterate that the unstable and stable invariant manifoldsare not a single solution� but rather a collection of solutions sharing acommon asymptotic past or future�

An example �from Guckenheimer and Holmes �� where the invari�ant manifolds can be explicitly calculated is the planar vector �eld

x � x� y � �y � x��

This system has a hyperbolic �xed point at the origin where the lin�


Figure �� a� Invariant manifolds �at the origin� for the linear ap�proximation� �b� Invariant manifolds for the original nonlinear system�

earized vector �eld is x � x� y � �y�

The stable manifold of the linearized system is just the y�axis� and theunstable manifold of the linearized system is the x�axis �Fig� ��a��Returning to the nonlinear system� we can solve this system by elimi�nating time�

y

x�

dy

dx��yx

� x or y�x� �x�

��

c

x�

where c is a constant of integration� It is now easy to see �Prob� ��that �Fig� ��b��

W u�� f�x� y� j y �x�

�g and W s�� f�x� y� j x � �g�

�� Homoclinic and Heteroclinic Points

We informally de�ne the unstable manifold and the stable manifold fora hyperbolic �xed point !x of a map f by

W s�!x� � fx j limn��fn�x� � !xg


Figure �� a� Poincar�e map in the vicinity of a periodic orbit�W s�!x� � W u�!x�� b� The map shown with a transversal intersectionat a homoclinic point�

andW u�!x� � fx j limn��f

�n�x� � !xg�These manifolds are tangent to the eigenvectors of f at !x�

We are led to study a two�dimensional map f by considering thePoincar�e map of a three�dimensional �ow in the vicinity of a periodicorbit� This situation is illustrated in Figure �� The map f is orienta�tion preserving� because it comes from a smooth �ow� The periodic or�bit of the �ow gives rise to the �xed point !x of the map� The �xed point!x has a one�dimensional stable manifold W s�!x� and a one�dimensionalunstable manifold W u�!x�� Poincar�e was led to his discovery of chaoticbehavior and homoclinic tangles �see section �� and Appendix H� byconsidering the interaction between the stable and unstable manifoldof !x� One possible interaction is shown in Figure ��a� where the un�stable manifold exactly matches the stable manifold� W s�!x� � W u�!x��

However� such a smooth match is exceptional� The more commonpossibility is for a transversal intersection between the stable and unsta�

�Informally� a map of a surface is orientation preserving if the normal vector tothe surface is not �ipped under the map�


Figure �� a� A homoclinic point� �b� A heteroclinic point�

ble manifold �Fig� ��b�� The location of the transversal intersectionis called a homoclinic point when both the unstable and stable manifoldemanate from the same periodic orbit �Fig� ��a�� The intersectionpoint is called a heteroclinic point when the manifolds emanate from dif�ferent periodic orbits� A heteroclinic point is shown in Figure ��b�where the unstable manifold emanating from !x intersects the stablemanifold of a di�erent �xed point !y�

The existence of a single homoclinic or heteroclinic pointforces the existence of an in�nity of such points� Moreover�it also gives rise to a homoclinic heteroclinic tangle� Thistangle is the geometric source of chaotic motions�

To see why this is so� consider Figure �� A homoclinic pointis indicated at x�� This homoclinic point is part of both the stablemanifold and the unstable manifold�

x� � W s�!x� and x� � W u�!x��

Also shown is a point a that lies on the stable manifold behind x� �thedirection is determined by the arrow on the manifold�� i�e�� a � x� onW s� Similarly� the point b lies on the unstable manifold with b � x� onW u� Now� we must try to �nd the location of the next iterate of f�x��subject to the following conditions�

�� The map f is orientation preserving�

� f�x�� W s�!x� and f�x�� W u�!x� �all the iterates of a homo�clinic point are also homoclinic points��


Figure �� The interaction of the stable manifold W s�!x� and theunstable manifold W u�!x� with a homoclinic point x�� The homoclinicpoint x� gets mapped to the homoclinic point f�x�� The orientationof the map is determined by considering where points a and b in thevicinity of x� are mapped�

�� f�a� � f�x�� on W s and f�b� � f�x�� on W u �

A picture consistent with these assumptions is shown in Figure ��The point f�x�� must lie at a new homoclinic point �that is� at a newintersection point� ahead of x�� The �rst candidate for the locationof f�x�� is the next intersection point� indicated at d� However� f�x��could not be located here because that would imply that the map fis orientation reversing �see Prob� �� The next possible location�which does satisfy all the above conditions� is indicated by f�x�� Morecomplicated constructions could be envisioned that are consistent withthe above conditions� but the solution shown in Figure �� is the sim�plest�

Now f�x�� is itself a homoclinic point� And the same argumentapplies again� the point f��x�� must lie closer to !x and ahead of f�x��Fig� ��a�� In this way a single homoclinic orbit must generatean in�nite number of homoclinic orbits� This sequence of homoclinicpoints asymptotically approaches !x�

Since f arises from a �ow� it is a di�eomorphism and thus invertible�


Figure �� a� Images and preimages of a homoclinic point� �b� Ahomoclinic tangle resulting from a single homoclinic point�

Therefore� exactly the same argument applies to the preimages of !x��That is� f�n�x�� approaches !x via the unstable manifold� The endresult of this construction is the violent oscillation of W s and W u inthe region of !x� These oscillations form the homoclinic tangle indicatedschematically in Figure ��b��

The situation is even more complicated than it initially appears�The homoclinic points are not periodic orbits� but Birkho� and Smithshowed that each homoclinic point is an accumulation point for an in��nite family of periodic orbits �� Thus� each homoclinic tangle has anin�nite number of homoclinic points� and in the vicinity of each homo�clinic point there exists an in�nite number of periodic points� Clearly�one major goal of dynamical systems theory� and nonlinear dynamics�is the development of techniques to dissect and classify these homo�clinic tangles� In section �� we will show how the orbit structure of ahomoclinic tangle is organized by using a horseshoe map� In Chapter �we will continue this topological approach by showing how knot theorycan be used to unravel a homoclinic tangle�

�� EXAMPLE� LASER EQUATIONS ��

�� Example� Laser Equations

We now consider a detailed example to help reinforce the barrage ofmathematical de�nitions and concepts in the previous sections� Ourexample is taken from nonlinear optics and is known as the laser rateequation ��

F �

�F�

F�

��

� u z

��

�zu

�� z�� z�u

��

In this model u is the laser intensity and z is the population inversion�The parameters �� and �� are damping constants� When certain lasersare turned on they tend to settle down to a constant intensity light out�put �constant u� after a series of damped oscillations �ringing� aroundthe stable steady state solution� This behavior is predicted by the laserrate equation�

To calculate the stability of the steady states we need to know thederivative of F�

DF �

�� F�

u F� z

F� u

F� z

�A �

�� z u

�� z� �� u�

�A �

�� Steady States

The steady states are found by setting F � ��

zu � ��

�� z�� z�u � ��

These equations have two equilibrium solutions� The �rst� which welabel a� occurs at

u � � � z ��

�� a �

��

�

��

��

The second� which we label b� occurs at

z � � � u � �� b � ��


The location in the phase plane of these equilibrium points is shown inFigure ��

The motion in the vicinity of each equilibrium point is analyzed by�nding the eigenvalues � and eigenvectors v�

�v � A � v� ��

of the derivative matrix of F� A � DF� at each �xed point of the �ow�At the point a we �nd

DFja �

��

��

��

� ��

�A �

�� Eigenvalues of a �� Matrix

To calculate the eigenvalues of DF� we recall that the general solutionfor the eigenvalues of any � real matrix�

A �

�a�� a��a�� a��

��

are given by

��

�tr�A� �

p+��

�

�tr�A��

p+��

where

tr�A� � a�� a��

det�A� � a��a�� a��a��

+�A� � �tr�A�� det�A��

Applying these formulas to DFja we �nd

tr�DFja� � ��

det�DFja� � ��

+�DFja� � ��

��

�� EXAMPLE� LASER EQUATIONS �

and the eigenvalues for the �xed point a are

��

��and ��

The eigenvalue �� is positive� and indicates an unstable direction� theeigenvalue �� is negative and indicates a stable direction� The �xedpoint a is a hyperbolic saddle�

�� Eigenvectors

The stable and unstable directions� and hence the stable space Es�a�and the unstable space Eu�a�� are determined by the eigenvectors ofDFja� The stable direction is calculated from

�

��

��

��

��

��

��

� ��

�A �

��

��

Solving this system of simultaneous equations for � and � gives theunnormalized eigenvector for the unstable space as

v��

��

�m�

�� m �

��

�

Similarly� the stable space is found from the eigenvalue equation for ��

��

��

��

��

��

��

� ��

�A �

��

��

Solving this set of simultaneous equations shows that the stable spaceis just the z�axis�

v��

��

��

In fact� the z�axis is invariant for the whole �ow �i�e�� if z � � then u � �for all t� so the z�axis is the global stable manifold at a� Es�a� � W s�a��

CHAPTER �� DYNAMICAL SYSTEMS THEORY

Figure �� Phase portrait for the laser rate equation�

�� Stable Focus

To analyze the dynamics in the vicinity of the �xed point b we need to�nd the eigenvalues and eigenvectors of

DFjb �

��

��

The eigenvalues of DFjb are

�� i� and �� i��

where

� ��

and � �

p��

The �xed point b is a stable focus since the real parts of the eigenvaluesare negative� This focus represents the constant intensity output of alaser� and the oscillation about this steady state is the ringing a laserinitially experiences when it is turned on�

The global phase portrait� pieced together from local informationabout the �xed points� is pictured in Figure ��

�� SMALE HORSESHOE �

�� Smale Horseshoe

In section �� we stressed the importance of analyzing the orbit struc�ture arising within a homoclinic tangle� From a topological and physicalpoint of view� analyzing the orbit structure primarily means answeringtwo questions�

�� What are the relative locations of the periodic orbits"

� How are the stable and unstable manifolds interwoven within ahomoclinic tangle"

By studying the horseshoe example we will see that these questions areintimately connected�

In sections �� and �� we answered the �rst question for theone�dimensional quadratic map by using symbolic dynamics� For thespecial case of a chaotic hyperbolic invariant set �to be discussed insection �� Smale found an answer to both of the above questions formaps of any dimension� Again� the solution involves the use of symbolicdynamics�

The prototypical example of a chaotic hyperbolic invariant set is theSmale horseshoe �� A detailed knowledge of this example is essentialfor understanding chaos� The Smale horseshoe �like the quadratic mapfor � � is an example of a chaotic repeller� It is not an attractor�Physical applications properly focus on attractors since these are di�rectly observable� It is� therefore� sometimes believed that the chaotichorseshoe has little use in physical applications� In Chapter � we willshow that such a belief could not be further from the truth� Remnantsof a horseshoe �sometimes called the proto�horseshoe �� are buriedwithin a chaotic attractor� The horseshoe �or some other variant ofa hyperbolic invariant set� acts as the skeleton on which chaotic andperiodic orbits are organized� To quote Holmes� �Horseshoes in a senseprovide the ,backbone� for the attractors �� Therefore� horseshoesare essential to both the mathematical and physical analysis of a chaoticsystem�

CHAPTER �� DYNAMICAL SYSTEMS THEORY

�� From Tangles to Horseshoes

The horseshoe map is motivated by studying the dynamics of a map inthe vicinity of a periodic orbit with a homoclinic point� Such a systemgives rise to a homoclinic tangle� Consider a small box �ABCD� in thevicinity of a periodic orbit as seen from the surface of section� Thissituation is illustrated in Figure ��a�� The box is chosen so that theside AD is part of the unstable manifold� and the sides AB and DCare part of the stable manifold� We now ask how this box evolves underforward and backward iterations� The unstable and stable manifoldsof the periodic orbit are invariant� Therefore� when the box is iterated�any point of the box that lies on an invariant manifold must alwaysremain on this invariant manifold�

If we iterate points in the box forward� then we generally end up �af�ter a �nite number of iterations� with the �horseshoe shape �C �D�A�B��shown in Figure ��b�� The initial segment AD� which lies on the un�stable manifold� gets mapped to the segment A�D�� which is also partof the unstable manifold� Similarly� if we iterate the box backwardwe �nd a backward horseshoe perpendicular to the forward horseshoe�Fig� ��c�� The box of initial points gets compressed along the un�stable manifold W u and stretched along the stable manifold W s� Aftera �nite number of iterations� the forward image of the box will inter�sect the backward image of the box� Further iteration produces moreintersections�

Each new region of intersection contains a periodic orbit �see Fig�� as well as segments of the unstable and stable manifolds� That is�the horseshoe can be viewed as generating the homoclinic tangle� Smalerealized that this type of horseshoe structure occurs quite generally ina chaotic system� Therefore� he decided to isolate this horseshoe mapfrom the rest of the problem ��

A schematic for this isolated horseshoe map is presented in Figure��d�� Like the quadratic map� it consists of a stretch and a fold�The horseshoe map can be thought of as a �thickened quadratic map�Unlike the quadratic map� though� the horseshoe map is invertible� Thefuture and past of all points are well de�ned� However� the itineraryof points that get mapped out of the box are ignored� We are onlyconcerned with points that remain in the box under all future and past


Figure �� Formation of a horseshoe inside a homoclinic tangle�

iterations� These points form the invariant set�

�� Horseshoe Map

A mathematical discussion of the horseshoe map is provided by De�vaney �� or Wiggins �� Here� we will present a more descriptiveaccount of the horseshoe that closely follows Wiggins�s discussion�

The forward iteration of the horseshoe map is shown in Figure


Figure �� a� Forward iteration of the horseshoe map� �b� Backwarditeration of the horseshoe map�


��a�� The horseshoe is a mapping of the unit square D�

f � D � R�� D � f�x� y� � R�j� x �� y �g�which contracts the horizontal directions� expands in the vertical direc�tion� and then folds� The mapping is only de�ned on the unit square�Points that leave the square are ignored� Let the horizontal strip H�

be all points on the unit square with � y �� and let horizontalstrip H� be all points with � � �� y �� Then a linear horseshoe

map is de�ned by the transformation

f�H��

�xy

��

��

��xy

��

f�H��

�xy

��

� ��

��xy

��

��

��

where � � � � �� and � � The horseshoe map takes the horizontalstrip H� to the vertical strip V� � f�x� y�j� x �g� and H� to thevertical strip V� � f�x� y�j�� x �g�

f�H�� V� and f�H�� V��

The strip H� is also rotated by �� The inverse of the horseshoe mapf�� is shown in Figure ��b�� The inverse map takes the verticalrectangles V� and V� to the horizontal rectangles H� and H��

The invariant set & of the horseshoe map is the collection of allpoints that remain in D under all iterations of f �

& � � � � f��D��f��D�

�D�f�D�

�f��D� � � � �

��n��

fn�D��

This invariant set consists of a certain in�nite intersection of horizontaland vertical rectangles� To keep track of the iterates of the horse�shoe map �the rectangles�� we will need the symbols si � f�� g withi � �� The symbolic encoding of the rectangles works muchthe same way as the symbolic encoding of the quadratic map �see sec�tion ��

The �rst forward iteration of the horseshoe map produces two ver�tical rectangles called V� and V�� V� is the vertical rectangle on the


Figure �� a� Forward iteration of the horseshoe map and symbolicnames� �b� Backward iteration� �c� Symbolic encoding of the invariantpoints constructed from the forward and backward iterations�


left and V� is the vertical rectangle on the right� The next step is toapply the horseshoe map again� thereby producing f��D�� As shownin Figure ��a�� V� and V� produce four vertical rectangles labeled�from left to right� V�� V�� V�� and V�� Applying the map yet againproduces eight vertical strips labeled V�� V�� V�� V�� V�� V��V�� and V�� In general� the nth iteration produces n rectangles�The labeling for the vertical strips is recursively de�ned as follows� ifthe current strip is left of the center� then a � is added to the front ofthe previous label of the rectangle� if it falls to the right� a � is added�So� for instance� the rectangle labeled V� starts on the right� The rect�angle labeled V�� originates from strip V�� but it currently lies on theleft� Lastly� the strip V�� starts on the right� then goes to the left�and then returns to the right again� To each vertical strip we associatea symbolic itinerary�

Vs��s��s��s�i��s�n �

which gives the approximate orbit �left or right� of a vertical strip aftern iterations� The minus sign in the symbolic label indicates that thesymbol s�i arises from considering the ith preimage of the particularvertical strip under f � Also note that the vertical strips get progres�sively thinner� so that after n iterations� each strip has a width of �n�

The backward iterates produce n horizontal strips at the nth it�eration� The height of each of these horizontal strips is ��n� Fromthe two horizontal strips H� and H�� the inverse map f�� producesfour horizontal rectangles labeled �from bottom to top� as H�� H��H�� and H�� Fig� ��b�� This in turn produces eight horizontalrectangles� H�� H�� H�� H�� H�� H�� H�� and H�� Eachhorizontal strip can be uniquely labeled with a sequence of ��s and ��s�

Hs�s�s��si��sn��

where the symbol s� indicates the current approximate location �bot�tom or top� of the horizontal rectangle� The fact that the labelingscheme is unique follows from the de�nition of f and the observationthat all of the horizontal rectangles are disjoint� Unlike the verticalstrips� the indexing for the horizontal strips starts at � and is positive�The need for this indexing convention will become apparent when wespecify the labeling of points in the invariant set�


Now� the invariant set & of the horseshoe map f is given by the in��nite intersection of all the horizontal and vertical strips� The invariantset is a fractal� in fact� it is a product of two Cantor sets� The map fgenerates a Cantor set in the horizontal direction� and the inverse mapf�� generates a Cantor set in the vertical direction� The invariant setis� in a sense� the product of these two Cantor sets�

�� Symbolic Dynamics

We can identify points in the invariant set according to the followingscheme� After one forward iteration and one backward iteration� the in�variant set is located within the four shaded rectangular regions shownin Figure ��c�� After two forward iterations and two backward it�erations� the invariant set is a subset of the �� shaded regions� Theshaded regions are the intersection of the horizontal and vertical strips�To each shaded region we associate a bi�in�nite symbol sequence�

� � � s�n � � � s��s��s��s�s�s� � � � sn � � � �constructed from the label of the vertical and horizontal strips forminga point in the invariant set� The right�hand side of the symbolic name�s�s�s� � � � sn � � �� is the label from the horizontal strip Hs�s�s�sn� Theleft�hand side of the symbolic name� � � � s�n � � � s��s��s�� is the labelfrom the vertical strip written backwards� Vs��s��s�� s�i��s�n� For in�stance� the shaded region labeled L in Figure ��c� has a symbolicname �� The �� to the right of the dot indicates horizontalstrip H�� The �� backwards� to the left of the dot indicatesthat the shaded region comes from the vertical strip V��

We hone in closer and closer to the invariant set by iterating thehorseshoe map both forward and backward� Moreover� the above la�beling scheme generates a symbolic name� or symbolic coordinate� foreach point of the invariant set� This symbolic name contains informa�tion about the dynamics of the invariant point�

To see how this works more formally� let us call ' the symbol spaceof all bi�in�nite sequences of ��s and ��s� A metric on ' between thetwo sequences

s � �� s�n � � � s��s�s� � � � sn � � ��

�� SMALE HORSESHOE ��

!s � �� !s�n � � � !s��!s�!s� � � � !sn � � ��

is de�ned by

d�s� !s� ��X

i��

ijij

where i �

�� if si � !si�� if si �� !si�

��

Next we de�ne a shift map � on ' by

��s� � �� s�n � � � s��s��s�s� � � � sn � � �� i�e�� s�i � si��

The shift map is continuous and it has two �xed points consistingof a string of all ��s or all ��s� A period n orbit of � is written ass�n � � � s��s�s� � � � sn�� where the overbar indicates that the symbolicsequence repeats forever� A few of the periodic orbits and their �shiftequivalent representations are listed below�

Period � � �� Period � �� Period � � ��

��

and so on� In addition to periodic orbits of arbitrarily high period� theshift map also possesses an uncountable in�nity of nonperiodic orbits aswell as a dense orbit� See Devaney �� or Wiggins �� for the details�

In section �� we showed that the shift map on the space of one�sided symbol sequences is topologically semiconjugate to the quadraticmap� A similar results holds for the shift map on the space of bi�in�nitesequences and the horseshoe map� namely� there exists a homeomor�phism � � & � ' connecting the dynamics of f on & and � on ' suchthat � � f � � � ��

&f�� &

� � � �'

�� '

The correspondence between the shift map on ' and the horseshoe mapon the invariant set & is pretty easy to see �again� for the mathematical


Figure �� Equivalence between the dynamics of the horseshoe mapon the unit square and the shift map on the bi�in�nite symbol space�

details see Wiggins �� The invariant set consists of an in�nite inter�section of horizontal and vertical strips� These intersection points arelabeled by their symbolic itinerary� and the horseshoe map carries onepoint of the invariant set to another precisely by a shift map� Consider�for instance� the period two orbit

��

The shift map sends �� to �� and back again� This correspondsto an orbit of the horseshoe map that bounces back and forth betweenthe points labeled �� and �� in & �see Fig� �� The topologicalconjugacy between � and f allows us to immediately conclude that�like the shift map� the horseshoe map has

�� a countable in�nity of periodic orbits �and all the periodic orbitsare hyperbolic saddles��

� an uncountable in�nity of nonperiodic orbits�

�� a dense orbit�

The shift map �and hence the horseshoe map� exhibits sensitive de�pendence on initial conditions �see section �� As stated above� it

�� SMALE HORSESHOE ��

possesses a dense orbit� These two properties are generally taken asde�ning a chaotic set�

�� From Horseshoes to Tangles

Forward and backward iterations of the horseshoe map generate thelocations of periodic points to a higher and higher precision� That is� byiterating the horseshoe map� we can specify the location of a periodicorbit within a homoclinic tangle �of the horseshoe� to any degree ofaccuracy� For instance� after one iteration� we know the approximatelocation of a period two orbit� It lies somewhere within the shadedregions labeled �� and �� in Figure ��c�� After two iterations� weknow its position even better� It lies somewhere within the shadedregions labeled �� and �� Fig� ��c��

The forward iterates of the horseshoe map produce a �snake thatapproaches the unstable manifold W u of the periodic point� The back�ward iterates produce another snake that approaches the stable mani�fold W s of the periodic point at the origin� Thus� iterating a horseshoegenerates a tangle� The relative locations of horizontal and verticalbranches of this tangle are the same as those that occur in a homoclinictangle with a horseshoe arising in a particular �ow� This is illustratedin Figure �� We can name the branches of the tangle with the samelabeling scheme we used for the horseshoe� For a horseshoe� the label�ing scheme is easy to see once we notice that both the horizontal andvertical branches are labeled according to the alternating binary treeintroduced in section ��

The labeling of the horizontal branches is determined by the symbolss�s�s� � � � � For instance� the horizontal label for branch H�� can bedetermined by reading down the alternating binary tree as illustratedin Figure �� A second alternating binary tree is used to determinethe labeling for the vertical branches� The labeling for the branch V��is indicated in Figure �� The labeling scheme for the horizontal andvertical branches at �rst appears complicated� However� the branchnames are easy to write down once we realize that they can be readdirectly from the alternating binary tree�


Figure �� Homoclinic �horseshoe� tangle and the labeling scheme forhorizontal and vertical branches from a pair of alternating binary trees�

�� HYPERBOLICITY ��

Figure �� Examples of horseshoe�like maps that generate hyperbolicinvariant sets�

�� Hyperbolicity

The horseshoe map is just one of an in�nity of possible return maps�chaotic forms� that can be successfully analyzed using symbolic dy�namics� A few other possibilities are shown in Figure �� Each dif�ferent return map generates a di�erent homoclinic tangle� but all thesetangles can be dissected using symbolic dynamics� All these maps aresimilar to the horseshoe because they are topologically conjugate to anappropriate symbol space with a shift map� All these maps possess aninvariant Cantor set &� These invariant sets all possess a special prop�erty that ensures their successful analysis using symbolic dynamics�namely� hyperbolicity�

Recall our de�nition of a hyperbolic point� A �xed point of a mapis hyperbolic if none of the moduli of its eigenvalues exactly equalsone� The notion of a hyperbolic invariant set is a generalization of ahyperbolic �xed point� Informally� to de�ne a hyperbolic set we extendthis property of �no eigenvalues on the unit circle to each point of theinvariant set� In other words� there is no center manifold for any pointof the invariant set� Technically� a set & arising in a di�eomorphismf � R� � R� is a hyperbolic set if

�� there exists a pair of tangent lines Es�x� and Eu�x� for each x � &which are preserved by Df�x��


� Es�x� and Eu�x� vary smoothly with x�

�� there is a constant � � such that kDf�x�wk �kwk for allw � Eu�x� and kDf��x�wk �kwk for all w � Es�x��

A more complete mathematical discussion of hyperbolicity can be foundin Devaney �� or Wiggins �� A general mathematical theory for thesymbolic analysis of chaotic hyperbolic invariant sets is described byDevaney �� and goes under the rubric of �subshifts and transitionmatrices�

Like the horseshoe� the chaotic hyperbolic invariant sets encoun�tered in the mathematics literature are often chaotic repellers� In phys�ical applications� on the other hand� we are more commonly faced withthe analysis of nonhyperbolic chaotic attractors� The extension of sym�bolic analysis from the �mathematical� hyperbolic regime to the �morephysical� nonhyperbolic regime is still an active research question ��

�� Lyapunov Characteristic Exponent

In section �� we informally introduced the Lyapunov exponent asa simple measure of sensitive dependence on initial conditions� i�e��chaotic behavior� The notion of a Lyapunov exponent is a general�ization of the idea of an eigenvalue as a measure of the stability of a�xed point or a characteristic exponent �� as the measure of the sta�bility of a periodic orbit� For a chaotic trajectory it is not sensible toexamine the instantaneous eigenvalue of a trajectory� The next bestquantity� therefore� is an eigenvalue averaged over the whole trajectory�The idea of measuring the average stability of a trajectory leads us tothe formal notion of a Lyapunov exponent� The Lyapunov exponent isbest de�ned by looking at the evolution �under a �ow� of the tangentmanifold� That is� �sensitive dependence on initial conditions is mostclearly stated as an observation about the evolution of vectors in thetangent manifold rather than the evolution of trajectories in the �owof the original manifold M �

The tangent manifold at a point x� written as TMx� is the collectionof all tangent vectors of the manifold M at the point x� The tangentmanifold is a linear vector space� The collection of all tangent manifolds

�� LYAPUNOV CHARACTERISTIC EXPONENT ��

Figure �� Evolution of vectors in the tangent manifold under a �ow�

is called the tangent bundle� For instance� for a surface embedded in R�

the tangent manifold at each point of the manifold is a tangent plane�More generally� if the original manifold is of dimension n� then the tan�gent manifold is a linear vector space also of dimension n� For furtherbackground material on manifolds� tangent manifolds� and �ows� seeArnold ��

The integral curves of a �ow on a manifold provide a smooth foli�

ation of that manifold� in the following manner� A point x � M goesto the point �t�x� �M under the �ow �see Fig� �� Now we make akey observation� the tangent vectors w � TMx are also carried by the�ow �this is called a �Lie dragging � so that we can set up a uniquecorrespondence between the tangent vectors in TMx and the tangentvectors in TM�t�x�� Namely� for each w � TMx� there exists a uniquevector D�t�w� � TM�t�x�� The Lyapunov characteristic exponent of a�ow is de�ned as

��x�w� � limt��

�

tln

�kD�t�x��w�kkwk

��

That is� the Lyapunov characteristic exponent measures the averagegrowth rate of vectors in the tangent manifold� The correspondingLyapunov characteristic exponent of a map is ��

��x�w� � limn��

�

nln k�Df�x��n�w�k� ��

�� Dynamical Systems Theory

where�Df�x��n � Df�fn��x�� Df�x��w��

A �ow is said to have sensitive dependence on initial conditions if theLyapunov characteristic exponent is positive� From a physical point ofview� the Lyapunov exponent is a very useful indicator distinguishinga chaotic from a nonchaotic trajectory ��


�� Good general references for the material in this chapter are listed here� Forordinary di�erential equations� see V� I� Arnold� Ordinary di�erential equa�tions �MIT Press� Cambridge� MA� �� Suspensions are discussed in Z�Nitecki� Di�erentiable Dynamics �MIT Press� Cambridge� MA� �� andS� Wiggins� Introduction to applied nonlinear dynamical systems and chaos�Springer�Verlag� New York� �� The standard reference for applied dy�namical systems theory is J� Guckenheimer and P� Holmes� Nonlinear oscil�lations� dynamical systems� and bifurcations of vector �elds� second printing�Springer�Verlag� New York� ��

�� IHJM stands for Ikeda� Hammel� Jones� and Moloney� See S� K� Hammel� C�K� R� T� Jones� and J� V� Moloney� Global dynamical behavior of the optical�eld in a ring cavity� J� Opt� Soc� Am� B ��

�� The quotes are from Paul Blanchard� Some of the introductory materialis based on notes from Paul Blanchard�s �� Dynamical Systems Course�MA�� We are indebted to Paul and to Dick Hall for explaining to us the�chain recurrent point of view��

�� C� Conley� Isolated Invariant Sets and the Morse Index� CBMS ConferenceSeries �� J� M� Franks� Homology and Dynamical Systems� ConferenceBoard of the Mathematical Sciences Regional Conference Series in MathematicsNumber �� American Mathematical Society� Providence� �� R� Easton�Isolating blocks and epsilon chains for maps� Physica ��D� � ��

�� J� Marsden and A� Tromba� Vector Calculus� third ed� �W� H� Freeman� NewYork� ��

�� S� Rasband� Chaotic dynamics of nonlinear systems �John Wiley� New York�� Chapter discusses �xed point theory for maps�

�� J� Thompson and H� Stewart� Nonlinear dynamics and chaos �John Wiley�New York� ��

Problems ��

�� T� Parker� and L� Chua� Practical numerical algorithms for chaotic systems�Springer�Verlag� New York� ��

�� A nice description of the homoclinic periodic orbit theorem of Birkho� andSmith is presented by E� V� Eschenazi� Multistability and Basins of Attractionin Driven Dynamical Systems� Ph�D� Thesis� Drexel University �� also seesection �� of R� Abraham and C� Shaw� Dynamics�The geometry of behavior�Part three� Global behavior �Aerial Press� Santa Cruz� CA� ��

�� H� G� Solari and R� Gilmore� Relative rotation rates for driven dynamicalsystems� Phys� Rev� A �� L� M� Narducci and N� B�Abraham� Laser physics and laser instabilities �World Scienti�c� New Jersey��

�� S� Smale� The mathematics of time� Essays on dynamical systems� economicprocesses� and related topics �Springer�Verlag� New York� �� J� Yorke andK� Alligood� Period doubling cascades for attractors� A prerequisite for horse�shoes� Comm� Math� Phys� ��

�� P� Holmes� Knotted periodic orbits in the suspensions of Smale�s horseshoe�Extended families and bifurcation sequences� Physica D ��

�� R� L� Devaney� An introduction to chaotic dynamical systems� second edition�Addison�Wesley� New York� ��

�� S� Wiggins� Introduction to applied nonlinear dynamical systems and chaos�Springer�Verlag� New York� ��

�� P� Cvitanovi�c� G� H� Gunaratne� and I� Procaccia� Topological and metricproperties of H�enon�type strange attractors� Phys� Rev� A ��

�� An elementary introduction to Lyapunov exponents is provided by S� Souza�Machado� R� Rollins� D� Jacobs� and J� Hartman� Studying chaotic systemsusing microcomputers and Lyapunov exponents� Am� J� Phys� �� For the state of the art in computing Lyapunov exponents� see P�Bryant� R� Brown� and H� Abarbanel� Lyapunov exponents from observed timeseries� Phys� Rev� Lett� ��

Problems


�� Show that the H�enon map with � � � reduces to an equivalent form of thequadratic map�

� Dynamical Systems Theory

�� Introduce the new variable � �t to the forced damped pendulum andrewrite the equations for the vector �eld so that the �ow is �� v� � S��R��

�� Write the IHJM optical map as a map of two real variables f�x� y� � R� � R�

where z � x� iy� Consider separately the cases where the parameters � andB are real and complex�

Section ��

�� What is the ��limit set of a point x� of a period two orbit of the quadraticmap� O�x�� fx�� x�g�

�� Let f�z� � S� � S�� with z �� e��i�z� Show that if � is irrational� then��p� � S��

�� Give an example of a map f �M �M with L� �� )�

�� Why is the label ) a good name for the nonwandering set�

Section ��

�� Verify that the Jacobian for the H�enon map is

��xn�� yn��

��xn� yn��

�� Compute the divergence of the vector �eld for the damped linear oscillator�!x� � (x� ��x � ��

�� Calculate the divergence of the vector �eld for the Du�ng equation� forceddamped pendulum� and modulated laser rate equation� For what parametervalues are the �rst two systems dissipative or conservative�

�� Calculate the Jacobian for the quadratic map� the sine circle map� the Baker�smap� and the IHJM optical map� For what parameter values is the Baker�smap dissipative or conservative�

�� The de�nition of a �dissipative� dynamical system in section �� actuallyincludes expansive systems� How would you rede�ne the term dissipative tohandle these cases separately�

�� Solve the equation of �rst variation for the vector �eld F�x� v� � �v��x��That is� �nd the time evolution of �xx�t��

xv�t��

vx�t��

vv�t��

Section ��

Problems �

�� Explicitly derive the evolution operator for the vector �eld F�x� v� � �v��x�by solving the di�erential equation for the harmonic oscillator in dimensionlessvariables�

�� Show that the following �� Jordan matrices have the indicated linear evo�lution operators�

�a�

A �

� � ��

�� etA �

�e��t �� e��t

��

�b�

A �

��

�� etA � e�t

�cos�t � sin�tsin�t cos�t

��

�c�

A �

� ��

�� etA � e�t

�� t �

��

�� Find an example of a di�erential equation with an equilibrium point which islinearly stable but not locally stable� Hint� See Wiggins� reference "�#�

�� Find the �xed points of the sine circle map and the Baker�s map�

�� Find all the �xed points of the linear harmonic oscillator !x� x�� (x � � andevaluate their local stability for � � � and � �� When are the �xed pointsasymptotically stable�

�� Find the equilibrium points of the unforced Du�ng equation and the dampedpendulum� and analyze their linear stability�

Section ��

�� CalculateWu�� andW s�� for the planar vector �eld (x � x� (y � �y�x��

�� Draw a picture from Figure �� to show that if f�x�� is located at d� thenthe map is orientation reversing�

Section ��

�� Follow section �� to construct the phase portrait for the laser rate equationwhen �a� �� and �b� ��

Section ��

�� Calculate the inverse of the horseshoe map f�� described in section ��

Dynamical Systems Theory

Chapter �

Knots and Templates

�� Introduction

Physicists are confronted with a fundamental challenge when studyinga nonlinear system� to wit� how are theory and experiment to be com�pared for a chaotic system" What properties of a chaotic system shoulda physical theory seek to explain or predict" For a nonchaotic system�a physical theory can attempt to predict the long�term evolution ofan individual trajectory� Chaotic systems� though� exhibit sensitivedependence on initial conditions� Long�term predictability is not anattainable or a sensible goal for a physical theory of chaos� What is tobe done"

A consensus is now forming among physicists which says that aphysical theory for low�dimensional chaotic systems should consist oftwo interlocking components�

�� a qualitative encoding of the topological structure of the chaoticattractor �symbolic dynamics� topological invariants�� and

� a quantitative description of the metric structure on the attractor�scaling functions� transfer operators� fractal measures��

A physicist�s �dynamical quest consists of �rst dissecting the topolog�ical form of a strange set� and second� �dressing this topological formwith its metric structure�

�

CHAPTER �� KNOTS AND TEMPLATES

In this chapter we introduce one beautiful approach to the �rstpart of a physicist�s dynamical quest� that is� unfolding the topologyof a chaotic attractor� This strategy takes advantage of geometricalproperties of chaotic attractors in phase space�

A low�dimensional chaotic dynamical system with one unstable di�rection has a rich set of recurrence properties that are determined bythe unstable saddle periodic orbits embedded within the strange set�These unstable periodic orbits provide a sort of skeleton on whichthe strange attractor rests� For �ows in three dimensions� these pe�riodic orbits are closed curves� or knots� The knotting and linking ofthese periodic orbits is a bifurcation invariant� and hence can be usedto identify or ��ngerprint a class of strange attractors� Although achaotic system de�es long�term predictability� it may still possess goodshort�term predictability� This short�term predictability fundamentallydistinguishes �low�dimensional chaos from our notion of a �random process� Mindlin and co�workers �� building on work initiated bySolari and Gilmore �� recently developed this basic set of observa�tions into a coherent physical theory for the topological characterizationof chaotic sets arising in three�dimensional �ows� The approach advo�cated by Mindlin� Solari� and Gilmore emphasizes the prominent roletopology must play in any physical theory of chaos� In addition� it sug�gests a useful approach toward developing dynamical models directlyfrom experimental time series�

In recent years the ergodic theory of di�erentiable dynamical sys�tems has played a prominent role in the description of chaotic physicalsystems �� In particular� algorithms have been developed to computefractal dimensions �� metric entropies �� and Lyapunov exponents�� for a wide variety of experimental systems� It is natural to considersuch ergodic measures� especially if the ultimate aim is the characteri�zation of turbulent motions� which are� presumably� of high dimension�However� if the aim is simply to study and classify low�dimensionalchaotic sets� then topological methods will certainly play an importantrole� Topological signatures and ergodic measures usually present dif�ferent aspects of the same dynamical system� though there are someunifying principles between the two approaches� which can often befound via symbolic dynamics �� The metric properties of a dynam�ical system are invariant under coordinate transformations� however�

�� INTRODUCTION �

they are not generally stable under bifurcations that occur during pa�rameter changes� Topological invariants� on the other hand� can bestable under parameter changes and therefore are useful in identifyingthe same dynamical system at di�erent parameter values�

The aim of this chapter is to develop topological methods suitablefor the classi�cation and analysis of low�dimensional nonlinear dynam�ical systems� The techniques illustrated here are directly applicable toa wide spectrum of experiments including lasers �� uid systems suchas those giving rise to surface waves �� the bouncing ball systemdescribed in Chapter �� and the forced string vibrations described inChapter ��

The major device in this analysis is the template� or knot holder�of the hyperbolic chaotic limit set� The template is a mathematicalconstruction �rst introduced by Birman and Williams �� and furtherdeveloped by Holmes and Williams �� Roughly� a template is anexpanding map on a branched surface� Templates are useful becauseperiodic orbits from the �ow of a chaotic hyperbolic dynamical systemcan be placed on a template in such a way as to preserve their originaltopological structure� Thus templates provide a visualizable model forthe topological organization of the limit set� Templates can also bedescribed algebraically by �nite matrices� and this in turn gives us akind of homology theory for low�dimensional chaotic limit sets�

As recently described by Mindlin� Solari� Natiello� Gilmore� andHou �� templates can be reconstructed from a moderate amount ofexperimental data� This reconstructed template can then be used bothto classify the strange attractor and to make speci�c predictions aboutthe periodic orbit structure of the underlying �ow� Strictly speaking�the template construction only applies to �ows in the three�sphere�S�� although it is hoped that the basic methodology illustrated by thetemplate theory can be used to characterize �ows in higher dimensions�

The strategy behind the template theory is the following� For anonlinear dynamical system there are generally two regimes that arewell understood� the regime where a �nite number of periodic orbitsexist and the hyperbolic regime of fully developed chaos� The essen�tial idea is to reconstruct the form of the fully developed chaotic limitset from a non�fully developed �possibly nonhyperbolic� region in pa�rameter space� Once the hyperbolic limit set is identi�ed� then the

� CHAPTER �� KNOTS AND TEMPLATES

topological information gleaned from the hyperbolic limit set can beused to make predictions about the chaotic limit set in other �possiblynonhyperbolic� parameter regimes� since topological invariants such asknot types� linking numbers� and relative rotation rates are robust un�der parameter changes�

In the next section we follow Auerbach and co�workers to show howperiodic orbits are extracted from an experimental time series �� These periodic cycles are the primary experimental data that the tem�plate theory seeks to organize� Section �� de�nes the core mathemat�ical ideas we need from knot theory� knots� braids� links� Reidemeistermoves� and invariants� In section �� we describe a simple� but phys�ically useful� topological invariant called a relative rotation rate� �rstintroduced by Solari and Gilmore �� Section �� discusses templates�their algebraic representation� and their symbolic dynamics� Here� wepresent a new algebraic description of templates in terms of �framedbraids� a representation suggested by Melvin �� This section alsoshows how to calculate relative rotation rates from templates� Section�� provides examples of relative rotation rate calculations from twocommon templates� In section �� we apply the template theory tothe Du�ng equation� This �nal section is directly applicable to ex�periments with nonlinear string motions� such as those described inChapter � ��

�� Periodic Orbit Extraction

Periodic orbits are available in abundance from a single chaotic timeseries� To see why this is so consider a recurrent three�dimensional �owin the vicinity of a hyperbolic periodic orbit �Fig� �� Since the�ow is recurrent� we can choose a surface of section in the vicinity ofthis �xed point� This section gives a compact map of the disk ontoitself with at least one �xed point� In the vicinity of this �xed point achaotic limit set �a horseshoe� containing an in�nite number of unstableperiodic orbits can exist� A single chaotic trajectory meanders aroundthis chaotic limit set in an ergodic manner� passing arbitrarily closeto every point in the set including its starting point and each periodic

point�

�� PERIODIC ORBIT EXTRACTION �

Figure �� A recurrent �ow around a hyperbolic �xed point�

Figure �� a� A close recurrence of a chaotic trajectory� �b� By gentlyadjusting a segment of the chaotic trajectory we can locate a nearbyperiodic orbit� �Adapted from Cvitanovi�c ��

Now let us consider a small segment of the chaotic trajectory thatreturns close to some periodic point� Intuitively� we expect to be ableto gently adjust the starting point of the segment so that the segmentprecisely returns to its initial starting point� thereby creating a periodicorbit �Fig� ��

Based on these simple observations� Auerbach and co�workers ��showed that the extraction of unstable periodic orbits from a chaotictime series is surprisingly easy� Very recently� Lathrop and Kostelich�� and Mindlin and co�workers �� successfully applied this orbitextraction technique to a strange attractor arising in an experimental�ow� the Belousov�Zhabotinskii chemical reaction�


As discussed in Appendix H� the idea of using the periodic orbitsof a nonlinear system to characterize the chaotic solutions goes back toPoincar�e �� In a sense� Auerbach and co�workers made the inverseobservation� not only can periodic orbits be used to describe a chaotictrajectory� but a chaotic trajectory can also be used to locate periodicorbits�

A real chaotic trajectory of a strange attractor can be viewed asa kind of random walk on the unstable periodic orbits� A segmentof a chaotic trajectory approaches an unstable periodic orbit along itsstable manifold� This approach can last for several cycles during whichtime the system has good short�term predictability� Eventually� though�the orbit is ejected along the unstable manifold and proceeds until itis captured by the stable manifold of yet another periodic orbit� Aconvenient mathematical language to describe this phenomenon is theshadowing theory �� of Conley and Bowen� Informally� we say thata short segment of a chaotic time series shadows an unstable periodicorbit embedded within the strange attractor� This shadowing e�ectmakes the unstable periodic orbits �and hence� the hyperbolic limitset� observable� In this way� horseshoes and other hyperbolic limit setscan be �measured�

This also suggests a simple test to distinguish low�dimensional chaosfrom noise� Namely� a time series from a chaotic process should havesubsegments with strong recurrence properties� In section �� wegive examples of recurrence plots that show these strong recurrenceproperties� These recurrence plots give us a quick test for determiningif a time series is from a low�dimensional chaotic process�

�� Algorithm

Let the vector xi be a point on the strange attractor where xi in oursetting is three�dimensional� and the components are either measureddirectly from experiment �e�g�� x� x� $x� or created from an embed�ding xi � �si� si�� si�� of fsigni�� the scalar time series data gen�erated from an experimental measurement of one variable� In thethree�dimensional phase space the saddle orbits generally have a one�dimensional repelling direction and a one�dimensional attracting direc�tion� When a trajectory is near a saddle it approximately follows the

�� PERIODIC ORBIT EXTRACTION �

motion of the periodic orbit� For a data segment xi�xi��xi�� neara periodic orbit� and an � �� we can �nd a smallest n such thatkxi�n�xik � �� We will often write n � kn�� In a forced system� n� issimply the number of samples per fundamental forcing period and k isthe integer period� If the system is unforced� then n� can be found byconstructing a histogram of the recurrence times as described by Lath�rop and Kostelich �� Candidates for periodic orbits embedded inthe strange attractor are now given by all xi which are �k� �� recurrentpoints� where k � n�n� is the period of the orbit�

In practice we simply scan the time series for close returns �strongrecurrence properties��

kxi�n � xik � ��

for k � �� to �nd the period one� period two� period three orbits�and so on� When the data are normalized to the maximum of the timeseries� then � � �� appears to work well� The recurrence criterion � isusually relaxed for higher�order orbits and can be made more stringentfor low�order orbits�

In a �ow with a moderate number of samples per period� say n� �� the recurrent points tend to be clustered about one another� Thatis� if xi is a recurrent point� then xi��xi�� are also likely to berecurrent points for the same periodic orbit� simply because the peri�odic orbit is being approached from its attracting direction� We calla cluster of such points a window� The saddle orbit is estimated� orreconstructed� by choosing the orbit with the best recurrence prop�erty �minimum �� in a window� Alternatively� the orbit can also beapproximated by averaging over nearby segments� which is the moreappropriate procedure for maps ��

Data segments of strong recurrence in a chaotic time series are easilyseen in recurrence plots� which are obtained by plotting kxi�n�xik as afunction of i for �xed n � kn�� Di�erent recurrence plots are necessaryfor detecting period one �k � �� orbits� period two �k � � orbits� andso on� Windows in these recurrence plots are clear signatures of near�periodicity over several complete periods in the corresponding segmentof the time series data� The periodic orbits are reconstructed by choos�ing the orbit at the bottom of each window� Windows in recurrence

�� CHAPTER �� KNOTS AND TEMPLATES

plots also provide a quick test showing that the time series is not beinggenerated by noise� but may be coming from a low�dimensional chaoticprocess�

�� Local Torsion

In addition to reconstructing the periodic orbits� we can also extractfrom a single chaotic time series the linearized �ow in the vicinity of eachperiodic orbit� In particular� it is possible to calculate the local torsion�that is� how much the unstable manifold twists about the �xed point�Let vu be the eigenvector corresponding to the largest eigenvalue �uabout the periodic orbit� That is� vu is the local linear approximationof the unstable manifold of the saddle orbit� To �nd the local torsion�we need to estimate the number of half�twists vu makes about theperiodic orbit� The operator ST��

ST�� exp

�Z T

�DFd�

��

gives the evolution of the variational vector� This variational vectorwill generate a strip under evolution� and the number of half�turns �thelocal torsion� is nothing but one�half the linking number �de�ned insection �� of the central line and the end of the strip de�ned by ��

A � fx��t� � �vu�t�� t � Tg� ��

where x� is the curve corresponding to the periodic orbit�The evolution operator can be estimated from a time series by �rst

noting that

exp

�Z T

�DFd�

�

exp�DF jT+t�� exp�DF jT��T+t�� exp�DF j�+t��

Let xi be a recurrent point� and let fxjgnj� be a collection of pointsin some prede�ned neighborhood about xi� Then DF �xi� is a � � �matrix� and we can use a least�squares procedure on points from thetime series in the neighborhood of xi to approximate both the Jaco�bian and the tangent map �� The local torsion �the number of

�� PERIODIC ORBIT EXTRACTION ��

half�twists about the periodic orbit� is a rather rough number calcu�lated from the product of the Jacobians and hence is insensitive to thenumerical details of the approximation�

�� Example� Du�ng Equation

To illustrate periodic orbit extraction we again consider the Du�ngoscillator� in the following form�

x� � x��

x� � ��x� � x� � x�� f cos�x� � ��

x� � ��

with the control parameters � � �� f � �� and � �� At these parameter values a strange attractor exists� but theattractor is probably not hyperbolic�

The Du�ng equation is numerically integrated for �� periods with�� steps per period� Data are sampled and stored every � steps� sothat � points are sampled per period� A short segment of the chaoticorbit from the strange attractor� projected onto the �x�� x�� phase space�is shown in Figure ��

Periodic orbits are reconstructed from the sampled data using astandard Euclidean metric� The program used to extract the periodicorbits of the Du�ng oscillator is listed in Appendix F� The distancesd�xi�n � xi� are plotted as a function of i for �xed n � kn�� withn� � �� Samples of these recurrence plots are shown in Figure ��for k � and k � �� The windows at the bottom of these plotsare clear signatures of near�periodicity in the corresponding segmentof the time series data� The segment of length n with the smallestdistance �the bottom of the window� was chosen to represent the nearbyunstable periodic orbit� Some of the unstable periodic orbits that werereconstructed by this procedure are shown in Figure �� We believethat the orbits shown in Figure �� correctly identify all the period oneand some of the period three orbits embedded in the strange attractorat these parameter values� In addition� a period two orbit and higher�order periodic orbits can be extracted �see Table �� However� thismay not be all the periodic orbits in the system since some of them


Figure �� Two�dimensional projection of a data segment from thestrange attractor of the Du�ng oscillator�

�� PERIODIC ORBIT EXTRACTION ��

Figure �� Recurrence plots� Distances d�xi�n � xi� plotted as a func�tion of i for �xed n � kn�� with n� � �� The windows in these plotsshow near�periodicity over at least a full period in the correspondingsegment of the time series data� The bottom of each window is a goodapproximation to the nearby unstable periodic orbit� �a� k � � �b�k � ��

may have basins of attraction separate from the basin of attraction forthe strange attractor�

The periodic orbits give a spatial outline of the strange attractor�By superimposing the orbits in Figure �� onto one another� we recoverthe picture of the strange attractor shown in the upper left�hand cornerof Figure �� Each periodic orbit indicated in Figure �� is actually aclosed curve in three�space �we only show a two�dimensional projectionin Fig� ��

Now we make a simple but fundamental observation� Each of theseperiodic orbits is a knot� and the periodic orbits of the strange attractorare interwoven �tied together� in a very complicated pattern� Our goalin this chapter is to understand the knotting of these periodic orbits�and hence the spatial organization of the strange attractor� We beginby reviewing a little knot theory�


Figure �� Some of the periodic orbits extracted from the chaotictime series data of Figure �� a� symmetric period one orbit� �b�� c�asymmetric pair of period one orbits� �d�� e� symmetric period threeorbits� �f�� g� asymmetric period three orbits�

�� KNOT THEORY ��

Figure �� Planar diagrams of knots� �a� the trivial or unknot� �b��gure�eight knot� �c� left�handed trefoil� �d� right�handed trefoil� �e�square knot� �f� granny knot�

Figure �� Link diagrams� �a� Hopf link� �b� Borromean rings� �c�Whitehead link�

�� Knot Theory

Knot theory studies the placement of one�dimensional objects calledstrings �� in a three�dimensional space� A simple and accuratepicture of a knot is formed by taking a rope and splicing the endstogether to form a closed curve� A mathematician�s knot is a non�self�intersecting smooth closed curve �a string� embedded in three�space� Atwo�dimensional planar diagram of a knot is easy to draw� As illustratedin Figure �� we can project a knot onto a plane using a solid �broken�line to indicate an overcross �undercross�� A collection of knots is calleda link �Fig� ��

The same knot can be placed in space and drawn in planar dia�gram in an in�nite number of di�erent ways� The equivalence of twodi�erent presentations of the same knot is usually very di�cult to see�


Figure �� Equivalent planar diagrams of the trefoil knot�

Figure �� Two knots whose equivalence is hard to demonstrate�

Classi�cation of knots and links is a fundamental problem in topology�Given two separate knots or links we would like to determine whentwo knots are the same or di�erent� Two knots �or links� are said tobe topologically equivalent if there exists a continuous transformationcarrying one knot �or link� into another� That is� we are allowed todeform the knot in any way without ever tearing or cutting the string�For instance� Figure �� shows two topologically equivalent planar di�agrams for the trefoil knot and a sequence of �moves showing theirequivalence� The two knots shown in Figure �� are also topologicallyequivalent� However� proving their equivalence by a sequence of movesis a real challenge�

A periodic orbit of a three�dimensional �ow is also a closed noninter�secting curve� hence a knot� A periodic orbit has a natural orientationassociated with it� the direction of time� This leads us to study orientedknots� Formally an oriented knot is an embedding S� � R� where S�

is oriented� Informally� an oriented knot is just a closed curve with anarrow attached to it telling us the direction along the curve�


Figure �� Crossing conventions� �a� positive �b� negative�

The importance of knot theory in the study of three�dimensional�ows comes from the following observation� The periodic orbits of athree�dimensional �ow form a link� In the chaotic regime� this linkis extraordinarily complex� consisting of an in�nite number of periodicorbits �knots�� As the parameters of the �ow are varied the componentsof this link� the knots� may collapse to points �Hopf bifurcations� orcoalesce �saddle�node or pitchfork bifurcations�� But no component of

the link can intersect itself or any other component of the link� becauseif it did� then there would be two di�erent solutions based at the sameinitial condition� thus violating the uniqueness theorem for di�erentialequations� The linking of periodic orbits �xes the topological structureof a three�dimensional �ow �� Moreover� as we showed in the previoussection� periodic orbits and their linkings are directly available fromexperimental data� Thus knot theory is expected to play a key role inany physical theory of three�dimensional �ows�

�� Crossing Convention

In our study of periodic orbits we will work with oriented knots andlinks� To each crossing C in an oriented knot or link we associate asign ��C� as shown in Figure �� A positive cross �also known asright�hand cross� or overcross� is written as ��C� � �� A negativecross �also known as a left�hand cross� or undercross� is written as��C� � �� This de�nition of crossing is the opposite of the Artincrossing convention adopted by Solari and Gilmore ��


Figure �� Reidemeister moves� �a� untwist� �b� pull apart� �c� middleslide�

�� Reidemeister Moves

Reidemeister observed that two di�erent planar diagrams of the sameknot represent topologically equivalent knots under a sequence of justthree primary moves� now called Reidemeister moves of type I� II� andIII� These Reidemeister moves� illustrated in Figure �� simplify thestudy of knot equivalence by reducing it to a two�dimensional problem�The type I move untwists a section of a string� the type II move pulls

apart two strands� and the type III move acts on three strings sliding themiddle strand between the outer strands� The Reidemeister moves canbe applied in an in�nite number of combinations� So knot equivalencecan still be hard to show using only the Reidemeister moves�

�� Invariants and Linking Numbers

A more successful strategy for classifying knots and links involves theconstruction of topological invariants� A topological invariant of a knotor link is a quantity that does not change under continuous deforma�tions of the strings� The calculation of topological invariants allowsus to bypass directly showing the geometric equivalence of two knots�since distinct knots must be di�erent if they disagree in at least onetopological invariant� What we really need� of course� is a complete

set of calculable topological invariants� This would allow us to de��nitely say when two knots or links are the same or di�erent� Unfor�tunately� no complete set of calculable topological invariants is known


Figure �� Linking numbers� �a� one� �b� zero�

for knots� However� mathematicians have been successful in develop�ing some very �ne topological invariants capable of distinguishing largeclasses of knots ��

The linking number is a simple topological invariant de�ned for alink on two oriented strings � and �� Intuitively� we expect the Hopflink in Figure ��a� to have linking number �� since the two stringsare linked once� Similarly� the two strings in Figure ��b� are unlinkedand should have linking number �� The linking number� which agreeswith this intuition� is de�ned by

lk��

XC

��C��

In words� we just add up the crossing numbers for each cross betweenthe two strings � and � and divide by two� The calculation of linkingnumbers is illustrated in Figure �� Note that the last example isa planar diagram for the Whitehead link showing that �links can belinked even when their linking number is zero ��

The linking number is an integer invariant� More re�ned algebraicpolynomial invariants can be de�ned such as the Alexander polynomialand the Jones polynomial �� We will not need these more re�nedinvariants for our work here�

�� Braid Group

Braid theory plays a fundamental role in knot theory since any orientedlink can be represented by a closed braid �Alexander�s Theorem ��The identi�cation between links� braids� and the braid group allows


Figure �� Examples of linking number calculations� �Adapted fromKau�man ��

us to pass back and forth between the geometric study of braids �andhence knots and links� and the algebraic study of the braid group� Forsome problems the original geometric study of braids is useful� Formany other problems a purely algebraic approach provides the onlyintelligible solution�

A geometric braid is constructed between two horizontal level lineswith n base points chosen on the upper and lower level lines� From up�per base points we draw n strings or strands to the n lower base points�Fig� ��a�� Note that the strands have a natural orientation fromtop to bottom� The trivial braid is formed by taking the ith upper basepoint directly to the ith lower base point with no crossings between thestrands �Fig� ��b�� More typically� some of the strands will inter�sect� We say that the i� �st strand passes over the ith strand if thereis a positive crossing between the two strands �see Crossing Conven�tion� section �� As illustrated in Figure ��a�� an overcrossing�or right�crossing� between the i� �st and ith string is denoted by thesymbol bi� The inverse b��i represents an undercross �or left�cross�� i�e��the i � �st strand goes under the ith strand �Fig� ��b��

By connecting opposite ends of the strands we form a closed braid�


Figure �� a� A braid on n�strands� �b� A trivial braid�

Figure �� Braid operators� �a� bi� overcross� �b� b��i � undercross�


Figure �� a� Braid of a trefoil knot� �b� Braid of a Hopf link�

Figure �� Braid on four strands whose braid word is b�b�� b��

Each closed braid is equivalent to a knot or link� and conversely Alexan�der�s theorem states that any oriented link is represented by a closedbraid� Figure ��a� shows a closed braid on two strands that is equiv�alent to the trefoil knot� Figure ��b� shows a closed braid on threestrands that is equivalent to the Hopf link� The representation of alink by a closed braid is not unique� However� only two operations onbraids �the Markov moves� are needed to prove the identity betweentwo topologically equivalent braids ��

A general n�braid can be built up from successive applications ofthe operators bi and b��i � This construction is illustrated for a braidon four strands in Figure �� The �rst crossing between the secondand third strand is positive� and is represented by the operator b��The next crossing is negative� b�� and is between the third and fourthstrands� The last positive crossing is represented by the operator b��Each geometric diagram for a braid is equivalent to an algebraic braid


Figure �� Braid relations�

word constructed from the operators used to build the braid� The braidword for our example on four strands is b�b

�� b��

Two important conventions are followed in constructing a braidword� First� at each level of operation �bi� b

��i � only the ith and i� �st

strands are involved� There are no crossings between any other strandsat a given level of operation� Second� it is not the string� but the basepoint which is numbered� Each string involved in an operation incre�ments or decrements its base point by one� All other strings keep theirbase points �xed�

The braid group on n�strands� Bn� is de�ned by the operators fbi� i �� n� �g� The identity element of Bn is the trivial n�braid� How�ever� as previously mentioned� the expression of a braid group element�that is� a braid word� is not unique� The topological equivalence be�tween seemingly di�erent braid words is guaranteed by the braid rela�

tions �Fig� ��

bibj � bjbi� ji� jj � ��

bibi��bi � bi��bibi��

The braid relations are taken as the de�ning relations of the braidgroup� Each topologically equivalent class of braids represents a col�lection of words that are di�erent representations for the same braidin the braid group� In principle� the braid relations can be used toshow the equivalence of any two words in this collection� Finding apractical solution to word equivalence is called the word problem� Theword problem is the algebraic analog of the geometric braid equivalenceproblem�


�� Framed Braids

In our study of templates we will need to consider braids with �fram�ing� A framed braid is a braid with a positive or negative integerassociated to each strand� This integer is called the framing� We thinkof the framing as representing an internal structure of the strand� Forinstance� if each strand is a physical cable then we could subject thiscable to a torsional force causing a twist� In this instance the framingcould represent the number of half�twists in the cable�

Geometrically� a framed braid can be represented by a ribbon graph�Take a braid diagram and replace each strand by a ribbon� To see theframing we twist each ribbon by an integer number of half�turns� Ahalf�twist is a rotation of the ribbon through radians� A positive half�twist is a half�twist with a positive crossing� the rightmost half of theribbon crosses over the leftmost half of the ribbon� Similarly� a negativehalf�twist is a negative crossing of the ribbon� Figure �� shows howthe framing is pictured as the number and direction of internal ribboncrossings�

This concludes our brief introduction to knot theory� We now turnour attention to discussing how our rudimentary knowledge of knottheory and knot invariants is used to characterize the periodic andchaotic behavior of a three�dimensional �ow�

�� Relative Rotation Rates

Solari and Gilmore �� introduced the �relative rotation rate in anattempt to understand the organization of periodic orbits within a �ow�The phase of a periodic orbit is de�ned by the choice of a Poincar�esection� Relative rotation rates make use of this choice of phase andare topological invariants that apply speci�cally to periodic orbits inthree�dimensional �ows� Our presentation of relative rotation ratesclosely follows Eschenazi�s ��

As usual� we begin with an example� Figure �� shows a periodtwo orbit� a period three orbit� and their intersections with a surface ofsection� The relative rotation rate between an orbit pair is calculated

�� RELATIVE ROTATION RATES ��

Figure �� Geometric representation of a framed braid as a ribbongraph� The integer attached to each strand is the sum of the half�twistsin the corresponding branch of the ribbon graph�

beginning with the di�erence vector formed at the surface of section�

+r � �xA � xB� yA � yB��

where �xA� yA� and �xB� yB� are the coordinates of the periodic orbitslabeled A of period pA and B of period pB� In general there will bepA �pB choices of initial conditions from which to form +r at the surfaceof section� To calculate the rotation rate� consider the evolution of +rin pA � pB periods as it is carried along by the pair of periodic orbits�The di�erence vector +r will make some number of rotations beforereturning to its initial con�guration� This number of rotations dividedby pApB is the relative rotation rate� Essentially� the relative rotationrate describes the average number of rotations of the orbit A about theorbit B� or the orbit B about the orbit A� In the example shown inFigure �� the period three orbit rotates around the period two orbittwice in six periods� or one�third average rotations per period�

The general de�nition proceeds as follows� Let A and B be twoorbits of periods pA and pB that intersect the surface of section at


Figure �� Rotations between a period two and period three orbitpair� The rotation of the di�erence vector between the two orbits iscalculated at the surface of section� This vector is followed for � � � �full periods� and the number of average rotations of the di�erence vectoris the relative rotation rate between the two orbits� �Adapted fromEschenazi ��

�� RELATIVE ROTATION RATES ��

�a�� a�� apA�� and �b�� b�� bpB�� The relative rotation rate Rij�A�B�is

Rij�A�B� ��

pApB

Zd�arctan�+r��+r��

or in vector notation�


pApB

Zn � �+r� d�+r��

+r �+r � ��

The integral extends over pA � pB periods� and n is the unit vectororthogonal to the plane spanned by the vectors +r and d+r� Theindices i and j denote the initial conditions ai and bj on the surface ofsection� In the direction of the �ow� a clockwise rotation is positive��

The self�rotation rate Rij�A�A� is also well de�ned by equation�� if we establish the convention that Rii�A�A� � �� The rela�tive rotation rate is clearly symmetric� Rij�A�B� � Rji�B�A�� It alsocommonly occurs that di�erent initial conditions give the same relativerotation rates� Further properties of relative rotation rates� includinga discussion of their multiplicity� have been investigated by Solari andGilmore ��

Given a parameterization for the two periodic orbits� their relativerotation numbers can be calculated directly from equation �� bynumerical integration �see Appendix F�� There are� however� severalalternative methods for calculating Rij�A�B�� For instance� imaginearranging the periodic orbit pair as a braid on two strands� This isillustrated in Figure �� where the orbits A and B are partitioned intosegments of length pA and pB each starting at ai�bj� and ending atai��bj�� We keep track of the crossings between A and B with thecounter �ij�

�ij �

��

�� if Ai crosses over Bj from left to right�� if Ai crosses over Bj from right to left

� if Ai does not cross over Bj ��

�As previously mentioned� the crossing convention and this de�nition of therelative rotation rate are the opposite of those originally adopted by Solari andGilmore "�#�


Figure �� The orbit pair of Figure �� arranged as a braid on twostrands� The relative rotation rates can be computed by keeping trackof all the crossings of the orbit A over the orbit B� Each crossing addsor subtracts a half�twist to the rotation rate� The linking number iscalculated from the sum of all the crossings of A over B� �Adaptedfrom Eschenazi ��

Then the relative rotation rates can be computed from the formula


pApB

Xn

�i�n�j�n� n � �� pApB� ��

Using this same counter� the linking number of knot theory is

lk�A�B� �Xi�j

�ij� i � �� pA and j � �� pB� ��

The linking number is easily seen to be the sum of the relative rotationrates�

lk�A�B� �Xij

Rij�A�B��

An intertwining matrix is formed when the relative rotation rates forall pairs of periodic orbits of a return map are collected in a �possiblyin�nite�dimensional� matrix� Intertwining matrices have been calcu�lated for several types of �ows such as the suspension of the Smalehorseshoe �� and the Du�ng oscillator �� Perhaps the simplest wayto calculate intertwining matrices is from a template� a calculation wedescribe in section ��

Intertwining matrices serve at least two important functions� First�they help to predict bifurcation schemes� and second� they are used toidentify a return mapping mechanism�

�� TEMPLATES ��

Again� by uniqueness of solutions� two orbits cannot interact througha bifurcation unless all their relative rotation rates are identical withrespect to all other existing orbits� With this simple observation inmind� a careful examination of the intertwining matrix often allowsus to make speci�c predictions about the allowed and disallowed bi�furcation routes� An intertwining matrix can give rise to bifurcation�selection rules� i�e�� it helps us to organize and understand orbit ge�nealogies� A speci�c example for a laser model is given in reference��

Perhaps more importantly� intertwining matrices are used to iden�tify or �ngerprint a return mapping mechanism� As described in sec�tion �� low�order periodic orbits are easy to extract from both ex�perimental chaotic time series and numerical simulations� The relativerotation rates of the extracted orbits can then be arranged into an in�tertwining matrix� and compared with known intertwining matrices toidentify the type of return map� In essence� intertwining matrices canbe used as signatures for horseshoes and other types of hyperbolic limitsets�

If the intertwining comes from the suspension of a map then� asmentioned in section �� see Fig� �� the intertwining matrix withzero global torsion is usually presented as the �standard matrix� Aglobal torsion of �� adds a full twist to the suspension of the returnmap� and this in turn adds additional crossings to each periodic orbit inthe suspension� If the global torsion is an integer GT � then this integeris added to each element of the standard intertwining matrix�

Relative rotation rates can be calculated from the symbolic dynam�ics of the return map �� or directly from a template if the returnmap has a hyperbolic regime� We illustrate this latter calculation insection ��

�� Templates

In section �� we provide the mathematical background surroundingtemplate theory� This initial section is mathematically advanced� Sec�tion �� contains a more pragmatic description of templates and canbe read independently of section ��


Before we begin our description of templates� we �rst recall that thedynamics on the attractor can have very complex recurrence propertiesdue to the existence of homoclinic points �see section �� Poincar�e�soriginal observation about the complexity of systems with transversehomoclinic intersections is stated in more modern terms as ��

Katok�s Theorem� A smooth �ow �t on a three�manifold� with pos�itive topological entropy� has a hyperbolic closed orbit with atransverse homoclinic point�

Templates help to describe the topological organization of chaotic�ows in three�manifolds� In our description of templates we will workmainly with forced systems� so the phase space topology is R� � S��However� the use of templates works for any three�dimensional �ow�

Periodic orbits of a �ow in a three�manifold are smooth closed curvesand are thus oriented knots� Recall yet again that once a periodic orbitis created �say through a saddle�node or �ip bifurcation� its knot typewill not change as we move through parameter space� Changing theknot type would imply self�intersection� and that violates uniquenessof the solution� Knot types along with linking numbers and relativerotation rates are topological invariants that can be used to predictbifurcation schemes �� or to identify the dynamics behind a system�� The periodic orbits can be projected onto a plane and ar�ranged as a braid� Strands of a braid can pass over or under oneanother� where our convention for positive and negative crossings isgiven in section �� We next try to organize all the knot informationarising from a �ow� and this leads us to the notion of a template� Ourinformal description of templates follows the review article of Holmes��

�� Motivation and Geometric Description

Before giving the general de�nition of a template� we begin by illus�trating how templates� or knot holders� can arise from a �ow in R�� Inaccordance with Katok�s theorem� let O be a closed hyperbolic orbitwith a transversal intersection and a return map resembling a Smalehorseshoe �Figure ��a�� For a speci�c physical model� the form ofthe return map can be obtained either by numerical simulations or by


Figure �� a� Suspension of a Smale horseshoe type return mapfor a system with a transversal homoclinic intersection� This returnmap resembles the orientation preserving H�enon map� �b� The dots�both solid and open� will be the boundary points of the branches ina template� The pieces of the unstable manifold W u on the intervals�a� !a� and �b�!b� generate two ribbons�

analytical methods as described by Wiggins �� The only periodicorbit shown in Figure ��b� is given by a solid dot �� The pointsof transversal intersection indicated by open dots �� are not periodicorbits� but�according to the Smale�Birkho� homoclinic theorem ��they are accumulation points for an in�nite family of periodic orbits �seesection ��

The periodic orbits in the suspension of the horseshoe map havea complex knotting and linking structure� which was �rst explored byBirman and Williams �� using the template construction�

Let us assume that our example is from a forced system� Thenthe simplest suspension consistent with the horseshoe map is shownin Figure ��a�� However� this is not the only possible suspension�We could put an arbitrary number of full twists around the homoclinicorbit O� The number of twists is called the global torsion� and it is atopological invariant of the �ow� In Figure ��b� the suspension ofthe horseshoe with a global torsion of �� is illustrated by representing


Figure �� Suspension of intervals on W u�O�� a� global torsion is ��b� global torsion is ��

the lift� of the boundaries on W u�O� of the horseshoe as a braid of tworibbons�

Note that adding a single twist adds one to the linking numberof the boundaries of the horseshoe� and this in turn adds one to therelative rotation rates of all periodic orbits within the horseshoe� Thatis� a change in the global torsion changes the linking and knot types�but it does so in a systematic way� In the horseshoe example the globaltorsion is the relative rotation rate of the period one orbits�

To �nish the template construction we identify certain orbits inthe suspension� Heuristically� we project down along the stable mani�folds W s�O� onto the unstable manifold W u�O�� i�e�� we �collapse ontoW u�O�� For the Smale horseshoe� this means that we �rst identifythe ends of the ribbons �now called branches� at 'T in Figure ��a��and next identify '� and 'T � The resulting braid template for thehorseshoe is shown in Figure ��b�� The template itself may nowbe deformed to several equivalent forms �not necessarily resembling abraid� including the standard horseshoe template illustrated in Figure

�For our purposes� a lift is a suspension consisting of �ow with a simple twist�


Figure �� Template construction� �a� identify the branch ends at'T � i�e�� collapse onto W u�O�� and �b� identify '� and 'T to get a�braid template�

�� For this particular example it is easy to see that such a projection is

one�to�one on periodic orbits� Each point of the limit set has a distinctsymbol sequence and thus lies on a distinct leaf of the stable manifoldW s�O�� This projection takes each leaf of W s�O� onto a distinct pointof W u�O�� In particular� for each periodic point of the limit set thereis a unique point on W u�O��

Each periodic orbit of the map corresponds to some knot in thetemplate� Since the collapse onto W u�O� is one�to�one� we can use thestandard symbolic names of the horseshoe map to name each knot inthe template �sections �� Each knot will generate a symbolicsequence of �s and �s indicated in Figure �� Conversely� each periodicsymbolic sequence of �s and �s �up to cyclic permutation� will generatea unique knot� The three simplest periodic orbits and their symbolicnames are illustrated in Figure ��

If a template has more than two branches� then we number thek branches of the template with the numbers f�� k � �g� Inthis way we associate a symbol from the set f�� k � �g to eachbranch� A periodic orbit of period n on the template generates a se�quence of n symbols from the set f�� k��g as it passes throughthe branches� Conversely� each periodic word �up to cyclic permuta�


Figure �� Standard Smale horseshoe template� Each periodic �in��nite� symbolic string of �s and �s generates a knot�

tions� generates a unique knot on the template� The template itselfis not an invariant object� However� from the template one can easilycalculate invariants such as knot types and linking numbers� In thissense it is a knot holder�

The branches of a template are joined �or glued� at the branch

lines� In a braid template� all the branches are joined at the samebranch line� Figure ��b� is an example of a braid template� andFigure �� is an example of a �nonbraided� template holding the sameknots� Forced systems always give rise to braid templates� We willwork mostly with full braid templates� i�e�� templates which describea full shift on k symbols� Franks and Williams have shown that anyembedded template can be arranged� via isotopy� as a braid template��

The template construction works for any hyperbolic �ow in a three�manifold� To accommodate the unforced situation we need the follow�ing more general de�nition of a template�

De�nition� A template is a branched surface T and a semi�ow !�t onT such that the branched surface consists of the joining chartsand the splitting charts shown in Figure ��


Figure �� Some periodic orbits held by the horseshoe template� Notethat the orbits � and � are unlinked� but the orbits � and �� are linkedonce�

The semi�ow fails to be a �ow because inverse orbits are not uniqueat the branch lines� In general the semi�ow is an expanding map sothat some sections of the semi�ow may also spill over at the branchlines� The properties that the template �T� !�t� are required to satisfyare described by the following theorem ��

Birman and Williams Template Theorem �� Given a �ow�t on a three�manifold M� having a hyperbolic chain recurrent setthere is a template �T� !�t�� T �M�� such that the periodic orbitsunder �t correspond �with perhaps a few speci�ed exceptions�one�to�one to those under !�t� On any �nite subset of the periodicorbits the correspondence can be taken via isotopy�

The correspondence is achieved by collapsing onto the �strong� stablemanifold� Technically� we establish the equivalence relation betweenelements in the neighborhood� N � of the chain recurrent set� as follows�


Figure �� Template building charts� �a� joining chart and �b� split�ting chart�

x� � x� if k�t�x�� t�x��k � � as t � �� In other words� x� andx� are equivalent if they lie in the same connected component of somelocal stable manifold of a point x � N �

Orbits with the same asymptotic future are identi�ed regardless oftheir past� By throwing out the history of a symbolic sequence� we canhope to establish an ordering relation on the remaining symbols andthus develop a symbolic dynamics and kneading theory for templatessimilar to that for one�dimensional maps� The symbolic dynamics oforbits on templates� as well as their kneading and bifurcation theory� isdiscussed in more detail in the excellent review article by Holmes ��

The �few speci�ed exceptions will become clear when we considerspeci�c examples� In some instances it is necessary to create a fewperiod one orbits in !�t that do not actually exist in the original �ow �t�These virtual orbits can sometimes be identi�ed with points at in�nityin the chain recurrent set�

Some examples of two�branch templates that have arisen in physicalproblems are shown in Figure �� the Smale horseshoe with global tor�sion � and �� the Lorenz template� and the Pirogon� The Lorenz tem�plate is the �rst knot holder originally studied by Birman and Williams�� It describes some of the knotting of orbits in the Lorenz equationfor thermal convection� The horseshoe template describes some of theknots in the modulated laser rate equations mentioned in section ��


Figure �� Common two�branch templates� �a� Smale horseshoe withglobal torsion �� b� Lorenz �ow� showing equivalence to Lorenz mask��c� Pirogon� �d� Smale horseshoe with global torsion �� Adaptedfrom Mindlin et al� ��


Figure �� Representation of a template as a framed braid usingstandard insertion�

�� Algebraic Description

In addition to the geometric view of a template� it is useful to developan algebraic description� Braid templates are described by three piecesof algebraic data� The �rst is a braid word describing the crossingstructure of the k branches of the template� The second is the fram�ing describing the twisting in each branch� that is� the local or branch

torsion internal to each branch� The third piece of data is the �lay�ering information or insertion array� which determines the order inwhich branches are glued at the branch line� We now develop someconventions for drawing a geometric template� In the process we willsee that the �rst piece of data� the braid word� actually contains thelast piece of data� the insertion array� Thus we conclude that a tem�plate is just an instance of a framed braid� a braid word with framing�see sections ��

Drawing Conventions

A graph of a template consists of two parts� a ribbon graph and alayering �or insertion� graph �Fig� �� In the upper section of atemplate we draw the ribbon graph� which shows the intertwining ofthe branches as well as the internal twisting �local torsion� within each


Figure �� The layering graph can always be moved to standard form�back to front�

branch� The lower section of the template shows the layering informa�tion� that is� the order in which branches are glued at the branch line�By convention� we usually con�ne the expanding part of the semi�owon the template to the layering graph� the branches of the layeringgraph get wider before they are glued at the branch line� We also oftendraw the local torsion as a series of half�twists at the top of the ribbongraph�

In setting up the symbolic dynamics on the templates �that is� innaming the knots� we follow two important conventions� First� at thetop of the ribbon graph we label each of the k branches from left to right

with a number from the labeling set �� i� � � � � k�� Second� fromnow on we will always arrange the layering graph so that the branchesof the template are ordered back to front from left to right� This secondconvention is called the standard insertion�

The standard insertion convention follows from the following ob�servation� Any layering graph can always be isotoped to the standardform by a sequence of branch moves that are like type II Reidemeistermoves� This is illustrated in Figure �� where we show a layeringgraph in nonstandard form and a simple branch exchange that puts itinto standard form�

The adoption of the standard insertion convention allows us to dis�pense with the need to draw the layering graph� The insertion infor�mation is now implicitly contained in the lower ordering �left to right�


back to front� of the template branches� We see that the template iswell represented by a ribbon graph or a framed braid� We will oftencontinue to draw the layering graph� However� if it is not drawn thenwe are following the standard insertion convention� A template withstandard insertion is a framed braid�

These conventions and the framed braid representation of a tem�plate are illustrated in Figure �� for a series of two�branch templates�We show the template� its version following standard insertion� and theribbon graph �framed braid� without the layering graph from which wecan write a braid word with framing� We also show the �braid linkingmatrix for the template� which is introduced in the next section�

Braid Linking Matrix

A nice characterization of some of the linking data of the knots heldby a template is given by the braid linking matrix� In particular� inthe second half of section �� we show how to calculate the relativerotation rates for all pairs of periodic orbits from the braid linkingmatrix� The braid linking matrix is a square symmetric k � k matrixde�ned by�

B �

��

bii � the sum of half�twists in the ith branch�bij � the sum of the crossings between the

ith and the jth branches of the ribbon graphwith standard insertion�

��

The ith diagonal element of B is the local torsion of the ith branch�The o��diagonal elements of B are twice the linking numbers of theribbon graph for the ith and jth branches� The braid linking matrixdescribes the linking of the branches within a template and is closelyrelated to the linking of the period one orbits in the underlying �ow�� For the example shown in Figure �� the braid linking matrix is

B �

�B� ��

� ��

�CA �

�The braid linking matrix is equivalent to the orbit matrix and insertion arraypreviously introduced by Mindlin et al� "�#� See Melvin and Tu�llaro for a proof"�#�


Figure �� Examples of two�branched templates� their correspondingribbon graphs �framed braids� with standard insertion� and their braidlinking matrices�


The braid linking matrix also allows us to compute how the strandsof the framed braid are permuted� At the top of the ribbon graphthe branches of the template are ordered �� i� � � � � k � �� At thebottom of the ribbon graph each strand occupies some possibly newposition� The new ordering� or permutation �B� of the strands is givenby

�B�i� � i � - odd entries bij with j � i� - odd entries bij with j i�

��

Informally� to calculate the permutation on the ith strand� we examinethe ith row of the braid linking matrix� adding the number of oddentries to the right of the ith diagonal element to i� and subtractingthe number of odd entries to the left�

For example� for the template shown in Figure �� we �nd that�B�� B�� and �B�� Thepermutation is �B � �� That is� the �rst strand goes to the secondposition� and the third strand goes to the �rst position� The secondstrand goes to the front third position�

�� Location of Knots

Given a knot on a template� the symbolic name of the knot is deter�mined by recording the branches over which the knot passes� Given atemplate and a symbolic name� how do we draw the correct knot on atemplate"

There are two methods for �nding the location of a knot on a tem�plate with k branches� The �rst is global and consists of �nding the lo�cations of all knots up to a length �period� n by constructing the appro�priate k�ary tree with n levels� The second method� known as �knead�ing theory� is local� Kneading theory is the more e�cient method ofsolution when we are dealing with just a few knots�

Trees

A branch of a template is called orientation preserving if the local tor�sion �the number of half�twists� is an even integer� Similarly� a branchis called orientation reversing if the local torsion is an odd integer� Aconvenient way to �nd the relative location of knots on a k�branch


template is by constructing a k�ary tree which encodes the orderingof points on the orientation preserving and reversing branches of thetemplate�

The ordering tree is de�ned recursively as follows� At the �rst level�n � �� we write the symbolic names for the branches from left to rightas �� k�� The second level �n � � is constructed by recordingthe symbolic names at the �rst level of the tree according to the orderingrule� if the ith branch of the template is orientation preserving thenwe write the branch names in forward order �� k � �� if theith branch is orientation reversing then we write the symbolic namesat the �rst level in reverse order �k � �� k � � � � � � � �� The n � �stlevel is constructed from the nth level by the same ordering rule� ifthe ith symbol �branch� at the nth level is orientation preserving thenwe record the ordering of the symbols found at the nth level� if theith symbol labels an orientation reversing branch then we reverse theordering of the symbols found at the nth level�

This rule is easier to use than to state� The ordering rule is illus�trated in Figure �� for a three�branch template� Branch � is orienta�tion reversing� branches � and are both orientation preserving� Thus�we reverse the order of any branch at the n � �st level whose nth levelis labeled with �� In this example we �nd

� � �

��

��

��

��

and so on�To �nd the ordering of the knots on the template we read down the

k�ary tree recording the branch names through which we pass �see Fig�� The ordering at the nth level of the tree is the correct orderingfor all the knots of period n on the template� Returning to our example�we �nd that the ordering up to period two is � � �� The symbol � is read �precedes and indicatesthe ordering relation found from the ordering tree �the order inducedby the template��


Figure �� Example� location of period two knots �n � ��


To draw the knots with the desired symbolic name on a template�we use the ordering found at the bottom of the k�ary tree� Lastly�we draw connecting line segments between �shift equivalent periodicorbits as illustrated in Figure �� For instance� the period two orbit� is composed of two shift equivalent string segments � and �� whichbelong to the branches � and respectively�

Kneading Theory

The limited version of kneading theory �� needed here is a simple rulewhich allows us to determine the relative ordering of two or more orbitson a template� From an examination of the ordering tree� we see thatthe ordering relation between two itineraries s � fs�� s�� si� � � � � sngand s� is given by s � s� if si � s�i for � i � n� and sn � s�n when thenumber of symbols in fs�� sn��g representing orientation reversingbranches is even� or sn s�n when the number of symbols representingorientation reversing branches is odd�

As an example� consider the orbits �� and �� on the templateshown in Figure �� We �rst construct all cyclic permutations ofthese orbits�

��

Next we sort these permutations in ascending order�

��

Last� we note that the only orientation reversing branch is �� so weneed to reverse the ordering of the points �� and �� yielding

��

which agrees with the ordering shown on the ordering tree in Figure��

�� Calculation of Relative Rotation Rates

Relative rotation rates can be calculated from the symbolic dynamicsof the return map or directly from the template� We now illustrate this


latter calculation for the zero global torsion lift of the horseshoe� Wethen describe a general algorithm for the calculation of relative rotationrates from the symbolics�

Horseshoe Example

Consider two periodic orbits A and B of periods pA and pB� At thesurface of section� a periodic orbit is labeled by the set of initial condi�tions �x�� x�� xn�� each xi corresponding to some cyclic permutationof the symbolic name for the orbit� That is� it amounts to a choiceof �phase for the periodic orbit� For instance� the period three orbit�� on the standard horseshoe template gives rise to three symbolicnames �� When calculating relative rotation rates it isimportant to keep track of this phase since di�erent permutations cangive rise to di�erent relative rotation rates�

To calculate the relative rotation rate between two periodic orbitswe �rst represent each orbit by some symbolic name �choice of phase��Next� we form the composite template of length pA � pB� This is il�lustrated for the period three orbit �� and the period one orbit ��in Figure ��a�� The two periodic orbits can now be extracted fromthe composite template and presented as two strands of a pure braidof length pA � pB with the correct crossing data �Figure ��b�� Theself�rotation rate is calculated in a similar way� The case of the periodtwo orbit in the horseshoe template is illustrated in Figure ��c�d��

General Algorithm

Although we have illustrated this process geometrically� it is completelyalgorithmic and algebraic� For a general braid template� the relativeordering of the orbits at each branch line is determined from the sym�bolic names and kneading theory� Given the ordering at the branchlines� and the form of the template� all the crossings between orbitsare determined� and hence so are the relative rotation rates� Gilmoredeveloped a computer program �� that generates the full spectrumof relative rotation rates when supplied with only the periodic orbitmatrix and insertion array �for a de�nition of periodic orbit matrix�also known as the template matrix� see ref� �� i�e�� purely algebraic


Figure �� Relative rotation rates from the standard horseshoe tem�plate� �a� composite template for the orbits �� and �� b� the pe�riodic orbits represented as pure braids� �c� composite template forcalculating self�rotation rate of �� d� pure braid of �� and �� e� theintertwining matrix for the orbits �� and ��


data� Here we describe an alternative algorithm based on the framedbraid presentation and the braid linking matrix� This algorithm is im�plemented as a Mathematica package� listed in Appendix G�

To calculate the relative rotation rate between two orbits we needto keep track of three pieces of crossing data� �� crossings betweentwo knots within the same branch �recorded by the branch torsion�� crossings between two knots on separate branches �recorded by thebranch crossings�� and �� any additional crossings occurring at theinsertion layer �calculated from kneading theory�� One way to organizethis crossing information is illustrated in Figure �� which shows thebraid linking matrix for a three�branch template and two words forwhich we wish to calculate the relative rotation rate�

Formulas can be written down describing the relative rotation ratecalculation �� but we will instead try to describe in words the �rel�ative rotation rate arithmetic that is illustrated in Figure �� Tocalculate the relative rotation rate between two orbits we use the fol�lowing sequence of steps�

�� Write the braid linking matrix for the template with standard insertion �re�arrange branches as necessary until you reach back to front form��

�� In the row labeled � write the word w� until the length of the row equalsthe least common multiple �LCM � between the lengths of w� and w�� dothe same with word w� in row ��

�� Above these rows create a new row �called the zeroth level� formed by thebraid linking matrix elements b�i�i � where i indexes the rows�

�� Find all identical blocks of symbols� that is� all places where the symbolicsin both words are identical �these are boxed in Figure �� Wrap aroundfrom the end of the rows to the beginning of the rows if appropriate�

� The remaining groups of symbols are called unblocked regions� for the un�blocked regions write the zeroth�level value mod � �if even record a zero� ifodd record a one� directly below the word rows at the �rst level�

�� For the blocked regions sum the zeroth�level values above the block �i�e�� addup all the entries at the zeroth level that lie directly above a block� and writethis sum mod � at the second level�

� For the unblocked regions look for a sign change �orientation reversingbranches� from one pair of symbols to the next �i�e�� i �i but �i�u � �i�u�

�� INTERTWINING MATRICES ��

or �i � �i but �i�u �i�u� and write a � at the second level if there is asign change� or write a � if there is no change of sign� The counter u givesthe integer distance to the next unblocked region� Wrap around from theend of the rows to the beginning of the rows if necessary�

� Group all terms in the �rst and second levels as indicated in Figure ��Add all terms in each group mod � and write at the third level�

�� Sum all the terms at the zeroth level� and write the sum to the right of thezeroth row�

�� Sum all the terms at the third level� and write the sum to the right of thethird row�

�� To calculate the relative rotation rate� add the sums of the zeroth level and

the third level� and divide by �� the LCM found in step ��

The rules look complicated� but they can be mastered in just afew minutes� after which time the calculation of relative rotation ratesbecomes just an exercise in the rotation rate arithmetic�

�� Intertwining Matrices

With the rules learned in the previous section we can now calculate rel�ative rotation rates and intertwining matrices directly from templates�For reference we present a few of these intertwining matrices below�

�� Horseshoe

The intertwining matrix for the zero global torsion lift of the Smalehorseshoe is presented in Table ��

�� Lorenz

The intertwining matrix for the relative rotation rates from the Lorenztemplate is presented in Table �� Adding a global torsion of one �afull twist� adds two to the braid linking matrix� and it adds one to eachrelative rotation rate� i�e�� each entry of the intertwining matrix�


Figure �� Example of the relative rotation rate arithmetic�

�� INTERTWINING MATRICES ��

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Table �� Horseshoe intertwining matrix�

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Table �� Lorenz intertwining matrix�


Figure �� Template for the Du�ng oscillator for the parameterregime explored in section ��

�� Du�ng Template

In this �nal section we will apply the periodic orbit extraction tech�nique and the template theory to a chaotic time series from the Du�ngoscillator� For the parameter regime discussed in section �� Gilmoreand Solari �� argued on theoretical grounds that the template forthe Du�ng oscillator is that shown in Figure �� The resulting in�tertwining matrix up to period three is presented in Table �� Allthe relative rotation rates calculated from the periodic orbits extractedfrom the chaotic time series agree �see Appendix G� with those foundin Table �� which were calculated from the braid linking matrix forthe template shown in Figure ��

However� not all orbits �up to period three� were found in the chaotic

�� DUFFING TEMPLATE ��

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Table �� Du�ng intertwining matrix� calculated from the templatefor the Du�ng oscillator� All the periodic orbits were extracted froma single chaotic time series except for those in italics� the so�called�pruned orbits�

� Templates

time series� In particular� the orbits �� and � did not appearto be present� Such orbits are said to be �pruned�

The template theory helps to organize the periodic orbit structurein the Du�ng oscillator and other low�dimensional chaotic processes�Mindlin and co�workers �� have carried the template theory furtherthan our discussion here� In particular� they show how to extract notonly periodic orbits� but also templates from a chaotic time series�Thus� the template theory is a very promising �rst step in the develop�ment of topological models of low�dimensional chaotic processes�


�� G� B� Mindlin� X��J� Hou� H� G� Solari� R� Gilmore� and N� B� Tu�llaro� Classi��cation of strange attractors by integers� Phys� Rev� Lett� ��

�� G� B� Mindlin� H� G� Solari� M� A� Natiello� R� Gilmore� and X��J� Hou� Topo�logical analysis of chaotic time series data from the Belousov�Zhabotinskii re�action� J� Nonlinear Sci� � ��

�� N� B� Tu�llaro� H� Solari� and R� Gilmore� Relative rotation rates� Fingerprintsfor strange attractors� Phys� Rev� A ��

�� H� G� Solari and R� Gilmore� Relative rotation rates for driven dynamicalsystems� Phys� Rev� A ��

�� H� G� Solari and R� Gilmore� Organization of periodic orbits in the drivenDu�ng oscillator� Phys� Rev� A ��

�� J��P� Eckmann and D� Ruelle� Ergodic theory of chaos and strange attractors�Rev� Mod� Phys� ��

�� P� Grassberger and I� Procaccia� Measuring the strangeness of strange attrac�tors� Physica �D� ��

�� T� C� Halsey� M� H� Jensen� L� P� Kadano�� I� Procaccia� and B� I� Shraiman�Fractal measures and their singularities� The characterization of strange sets�Phys� Rev� A �� erratum� Phys� Rev� A ��

�� I� Procaccia� The static and dynamic invariants that characterize chaos andthe relations between them in theory and experiments� Physica Scripta T��


�� J��P� Eckmann� S� O� Kamphorst� D� Ruelle� and S� Ciliberto� Lyapunov expo�nents from time series� Phys� Rev� A ��

�� J� M� Franks� Homology and dynamical systems� Conference Board of theMathematical Sciences Regional Conference Series in Mathematics Number�� American Mathematical Society� Providence� ��

�� F� Simonelli and J� P� Gollub� Surface wave mode interactions� E�ects of sym�metry and degeneracy� J� Fluid Mech� ��

�� J� Birman and R� Williams� Knotted periodic orbits in dynamical systems I�Lorenz�s equations� Topology �� J� Birman and R� Williams�Knotted periodic orbits in dynamical systems II� Knot holders for �bered knots�Contemporary Mathematics ��

�� P� Holmes� Knots and orbit genealogies in nonlinear oscillators� in New direc�tions in dynamical systems� edited by T� Bedford and J� Swift� London Math�ematical Society Lecture Notes �� Cambridge University Press� Cambridge�� pp� � �� P� J� Holmes and R� F� Williams� Knotted periodic orbitsin suspensions of Smale�s horseshoe� Torus knots and bifurcation sequences�Archives of Rational Mechanics Annals ��

�� D� Auerbach� P� Cvitanovi�c� J��P� Eckmann� G� Gunaratne� and I� Procaccia�Exploring chaotic motion through periodic orbits� Phys� Rev� Lett� ��

�� D� P� Lathrop and E� J� Kostelich� Characterization of an experimental strangeattractor by periodic orbits� Phys� Rev� A � ��

�� P� Melvin and N� B� Tu�llaro� Templates and framed braids� Phys� Rev� A �� R��

�� N� B� Tu�llaro� Chaotic themes from strings� Ph�D� Thesis� Bryn Mawr College��

�� P� Cvitanovi�c� Chaos for cyclists� in Noise and chaos in nonlinear dynamicalsystems� edited by F� Moss �Cambridge University Press� Cambridge� ��

�� H� Poincar�e� Les m�ethodes nouvelles de la m�ecanique c�eleste� Vol� ��Gauthier�Villars� Paris� �� reprinted by Dover� �� English translation�New methods of celestial mechanics �NASA Technical Translations� �� SeeVolume I� Section ��

�� R� Easton� Isolating blocks and epsilon chains for maps� Physica �� D� � ��

�� Knots and Templates

�� D� L� Gonz�alez� M� O� Magnasco� G� B� Mindlin� H� Larrondo and L� Romanelli�A universal departure from the classical period doubling spectrum� Physica ��D� ��

�� V� F� R� Jones� Knot theory and statistical mechanics� Sci� Am� �November��

�� M� Wadati� T� Deguchi� and Y� Akutsu� Exactly solvable models and knottheory� Phys� Reps� �� $ �� L� H� Kau�man� On knots�Princeton University Press� Princeton� NJ� ��

�� E� V� Eschenazi� Multistability and basins of attraction in driven dynamicalsystems� Ph�D� Thesis� Drexel University ��

�� J� Franks and R� F� Williams� Entropy and knots� Trans� Am� Math Soc� ��

�� S� Wiggins�Global bifurcations and chaos� analytical methods �Springer�Verlag�New York� ��

�� J� Guckenheimer and P� Holmes�Nonlinear oscillations� dynamical systems andbifurcation of vector �elds �Springer�Verlag� New York� ��

�� R� Gilmore� Relative rotation rates from templates� a Fortran program� privatecommunication �� Address� Dept� of Physics� Drexel University� Philadel�phia� PA ��

Problems


�� Calculate the linking numbers for the links shown in Figures ��a� and �c��Choose di�erent orientations for the knots and recalculate�

�� Calculate the linking numbers between the three orbits shown in Figure ��

�� Calculate the linking numbers between the orbits �� and �� in Figure ��

�� Write the braid words for the braids shown in Figure ��

�� Verify that the braid group �section �� is� in fact� a group�

�� Find an equivalent braid to the braid shown in Figure �� and write downits corresponding braid word� Use the braid relations to show the equivalenceof the two braids�

Problems ��

�� Write down the braid words for the braids shown in Figure �� Use thebraid relations to demonstrate the equivalence of the braids as shown�

�� Write down the braid word for the braid in Figure ��

Section ��

�� Calculate the relative rotation rates from both the A orbit and the B orbit inFigure �� That is� verify the equivalence of R�A�B� and R�B�A� in thisinstance� Use the geometric method illustrated in Figure �� which is takenfrom Figure �� Attempt the same calculation directly from Figure ��

�� Show that the sum of the relative rotation rates is the linking number �seereference "�# for more details��

�� Show the equivalence of equations � �� and � ��

Section � �

�� Draw three di�erent three�branch templates and sketch their associated returnmaps� Assume a linear expansive �ow on each branch� Construct their braidlinking matrices�

�� Show that the Lorenz template arises from the suspension of a discontinuousmap� Consider the evolution of a line segment connecting the two branchesat the middle�

�� Draw the template in Figure �� as a ribbon graph and as a framed braid�What is its braid linking matrix�

�� Verify that the strands of a braid are permuted according to equation � ��

�� Verify the relative rotation rates in Figure ��e� by the relative rotationrate arithmetic described in section � �� Add the orbit �� to the table�

Section ��

�� Calculate the intertwining matrix for the orbits shown in Figure �� up toperiod two�

Section ��

�� Verify the �� row in Table �� by the relative rotation arithmetic�

�� Appendix A

Appendix A� Bouncing Ball Code

The dynamical state of the bouncing ball system is speci�ed by three variables�the ball�s height y� its velocity v� and the time t� The time also speci�es the table�svertical position s�t�� Let tk be the time of the kth table�ball collision� and vk theball�s velocity immediately after the kth collision� The system evolves according tothe velocity and phase equations�

vk � (s�tk� � ��fvk�� g"tk � tk��#� (s�tk�g�s�tk� � s�tk�� vk��"tk � tk��#� �

�g"tk � tk��#

��

which are collectively called the impact map� The velocity equation states thatthe relative ball�table speed just after the kth collision is a fraction � of its valuejust before the kth collision� The phase equation determines tk �given tk�� byequating the table position and the ball position at time tk� tk is the smalleststrictly positive solution of the phase equation� The simulation of the impact mapon a microcomputer presents no real di�culties� The only somewhat subtle pointarises in �nding an e�ective algorithm for solving the phase equation� which is animplicit algebraic equation in tk�

We choose to solve for tk by the bisection method � because of its great stabilityand ease of coding� Other zero��nding algorithms� such as Newton�s method� arenot recommended because of their sensitivity to initial starting values� All bisectionmethods must be supplied with a natural step size for the method at hand� The stepinterval must be large enough to work quickly� yet small enough so as not to includemore than one zero on the interval� Our solution to the problem is documented inthe function �ndstep � of the C program below� In essence� our step��nding methodworks as follows�

If the relative impact velocity is large then tk and tk�� will not be close� sothe step size need only be some suitable fraction of the forcing period� On theother hand� if tk and tk�� are close then the step size needs to be some fraction ofthe interval� We approximate the interval between tk and tk�� by noting that therelative velocity between the ball and the table always starts out positive and mustbe zero before they collide again� Using the fact that the time between collisions issmall� it is easy to show that

k A� cos��k�� vk��A�� sin��k�� g

�

where �k�� is the phase of the previous impact and k is the time it takes the relativevelocity to reach zero� k provides the correct order of magnitude for the step size�This algorithm is coded in the following C program�

�For a discussion of the bisection method for �nding the real zeros of a continuousfunction� see R� W� Hamming� Introduction to applied numerical analysis �McGraw�Hill� New York� �� p� ��

Bouncing Ball Code ��

�� bb�c bouncing ball program

copyright �� by Nicholas B� Tufillaro

Date written� �� July ��

Date last modified� � August ��

This program simulates the dynamics of a bouncing ball subject to

repeated collisions with an oscillating wall�

The INPUT is�

delta v� Amplitude Frequency damping cycles

where

delta� the initial position of ball between � and �� this is the

phase of forcing frequency that you start at phase mod ��PI

v�� the initial velocity of the ball� this must be greater than

the initial velocity of the wall�

A� Amplitude of the oscillating wall

Freq� Frequency of the oscillating wall in hertz

damp� �� the impact parameter describing the energy loss per

collision� No energy loss conservative case when d � ��

maximum dissipation occurs at d � ��

cycles�the total length the simulation should be run in terms of

the number of forcing oscillations�

Units� CGS assumed

Compile with� cc bb�c �lm �O �o bb

Bugs�

��

�include stdio�h��include math�h�

�� CONSTANTS CGS Units ��

�define STEPSPERCYCLE ��

�define TOLERANCE �e��

�define MAXITERATIONS ��

�define PI ��

�define G �� earth�s gravitational constant ��

�define STUCK ��

�define EIGHTH ��

�� Macros ��

�define maxA� B A � B � A � B

�define minA� B A B � A � B

�� Appendix A

�� Comments of variables

t is time since last impact

ti is time of last impact

tau is ti � t� time since start of simulation

xi is position of ball at last impact

vi is velocity of ball at last impact

w is velocity of wall

v is velocity of ball

��

�� Global Variables ��

double delta� �� initial phase position of ball ��

double v�� initial ball velocity ��

double A� �� table amplitude ��

double freq� �� frequency of forcing ��

double damp� �� impact parameter ��

double omega� �� angular frequency ��PI�f � ��PI�T ��

double T� �� period of forcing ��

double cycles� �� length of simulation ��

�� Functions ��

double s� w� x� v� d� acc�

double find step�

double checkstep�

double root�

double fmod� asin�

main

fint stuckcount�

double t� ti� tau� dt� tstop� xi� vi� tj� xj� vj�

double t alpha� t beta� t ph� �� variables for sticking case ��

�� read input parameters ��

scanf �lf�lf�lf�lf�lf�lf � �delta� �v�� A� �freq� �damp� �cycles�

T � ��freq�

tstop � T�cycles�

omega � ��PI�freq�

delta � delta�freq�

if v� w�� fprintf Error� Initial velocity less than wall velocity nn �printf Wall velocity� �g nn � w��


exit�

g

t � �� ti � �� tau � ti � t� xi � stau� vi � v��

dt � find stepti� xi� vi�

t � dt� tau � ti � t� stuckcount � ��

while tau � tstop ft �� dt� tau � ti � t�

if impact tau� t� xi� vi ft � roott�dt� t� t� xi� vi� ti�

tj � ti � t�

xj � stj�

vj � ��damp�wtj � damp�vt�vi�

t � �� xi � xj� vi � vj� ti � tj�

dt � find stepti� xi� vi�

if dt �� STUCK f �� sticking solution ��

if A�omega�omega G fprintf Stuck forever with tablenn �exit�

giffabsG�A�omega�omega ��

t alpha � ��omega�asinG�A�omega�omega�

else

t alpha � T��

t beta � ��T�t alpha�

dt � t beta�t alpha�STEPSPERCYCLE�

t ph � fmodti�delta�T�

if�t ph � t alpha �� t ph t beta fstuckcount ��

if t alpha t ph

tj � T � ti � t alpha � t ph�

else

tj � ti � t alpha � t ph�

xj � stj� vj � wtj�

t � �� xi � xj� vi � vj� ti � tj�

gifstuckcount �� f

printf �g �gnn � fmodtj�T�� vt�vj�

printf Ball Stuck Twicenn �exit�

ggiftau � tstop��

�� Appendix A

printf �g �gnn � fmodtj�T�� vt�vj�

dt � checkstepdt�ti�xi�vi�

t � dt� tau � ti � dt�

gg

g

double stau �� wall position ��

double tau�

freturnA�sinomega�tau�delta��

g

double wtau �� wall velocity ��

double tau�

freturnA�omega�cosomega�tau�delta�

g

double acctau �� table acceleration ��

double tau�

freturn�A�omega�omega�sinomega�tau�delta�

g

double xt�xi�vi �� ball position ��

double t� xi� vi�

freturnxi�vi�t��G�t�t�

g

double vt�vi �� ball velocity ��

double t� vi�

freturnvi�G�t�

g

double dtau�t�xi�vi �� distance between ball and wall ��

double tau� t� xi� vi�

freturnxt�xi�vi�stau�

g

int impacttau�t�xi�vi �� find when ball is below wall ��


double tau�t�xi�vi�

freturndtau�t�xi�vi � ��

g

�� pick a good step for finding zero ��

double find stepti�xi�vi

double ti�xi�vi�

fdouble t max� tstep�

ifvi�wti � �� f �� should be alpha � �� or sticking ��

returnSTUCK�

gif�accti � G ��

t max � fabswti�vi��accti�G�

else

t max � T�STEPSPERCYCLE�

ift max T�TOLERANCE freturnSTUCK�

gtstep � minEIGHTH�t max�T�STEPSPERCYCLE�

returntstep�

g

double roota�b�t�xi�vi�ti

double a�b�

double t� xi� vi� ti�

fdouble m�

int count�

count � ��

while� fcount ��

ifcount � MAXITERATIONS fprintf ERROR� infinite loop in rootnn �exit�

gifdti�a�a�xi�vi�dti�b�b�xi�vi � �� f

printf root finding error� no zero on intervalnn �exit�

gm � a�b��

ifdti�m�m�xi�vi �� jj b�a T�TOLERANCE

�� Appendix A

returnm�

else if dti�a�a�xi�vi�dti�m�m�xi�vi ��

b � m�

else

a � m�

gg

double checkstepdt�ti�xi�vi

double dt� ti� xi� vi�

fint count�

forcount�� dti�dt�dt�xi�vi �� count fdt � EIGHTH�dt�

if count � �� fprintf Error� Can�t calculate dtnn �exit�

ggreturndt�

g

Cubic Oscillator ��

Appendix B� Exact Solutions for a Cubic Oscillator

The free conservative cubic oscillator� equation �� arose in the single�modemodel of string vibrations� and admits exact solutions in two special circumstances�The �rst is the case of circular motion at a constant radius R� In Figure �� we couldimagine circular orbits arising when the restoring force on the string just balancesthe centrifugal force� Plugging the ansatz

r � �R cos�t� R sin�t� �B��

into equation �� we see that it is indeed a solution provided the frequency isadjusted to

��c � �� KR�� B��

The second solution appears when we consider planar motion� If all the motionis con�ned to the x�z plane� then the system is a single degree of freedom oscillator�whose equation of motion in the dimensionless form obtained from equation ��is

x�� x� �x� � �� B��

The exact solution to equation �B�� is "�#

��

�� E��

�K

�a�

a� � b�

�� F

�arccos

x

a�

a�

a� � b�

�� B��

where F �� is an elliptic integral of the �rst kind and K�� F �� "�#� E isthe energy constant

E ��

�x��

�

�x� �

�

�x� �B� �

and

b�� a� ��

��p

� � ��E � �� B��

As in circular motion� the frequency in planar motion is again shifted to a new valuegiven by

�p ��

�

�� E��

I"a��a� � b��#� �B��

The exact solutions for circular and planar motion are useful benchmarks for testinglimiting cases of more general� but not necessarily exact� results�

An exact solution for the more general case of an anharmonic oscillator�

x�� a� � a�x� a�x� � a�x

� � �

is provided by Reynolds "�#�

�� Appendix B


�� K� Banerjee� J� Bhattacharjee� and H� Mani� Classical anharmonic oscillators�Rescaling the perturbation series� Phys� Rev� A � ��

�� For an elementary discussion of elliptic functions� see Appendix F of E� A� Jack�son� Perspectives of nonlinear dynamics� Vol� � �Cambridge University Press�New York� �� pp� ��

�� M� J� Reynolds� An exact solution in nonlinear oscillators� J� Phys� A� Math�Gen� �� L��L�� Also� see N� Euler� W��H� Steeb� and K� Cyrus� Onexact solutions for damped anharmonic oscillators� J� Phys� A� Math� Gen� ��L�� L��

Ode ��

Appendix C� Ode Overview

Ode renders a numerical solution to the initial value problem for many familiesof �rst�order di�erential equations� It is a programming language that resemblesthe mathematical language so that the problem posed by a system is easy to state�thereby making a numerical solution readily available� Ode solves higher�ordersystems since a simple procedure converts an nth�order equation into n �rst�orderequations� Three distinct numerical methods are implemented at present� Runge�Kutta�Fehlberg �default�� Adams�Moulton� and Euler� The Adams�Moulton andRunge�Kutta routines are available with adaptive step size "�#� The Ode User�sManual provides both a tutorial on applying Ode and a discussion of its design andimplementation "�#�

The user need only be familiar with the fundamentals of the UNIX operatingsystem to access and run this numerical software� Ode provides�

� A simple problem�oriented user interface�

� A table�driven grammar� simplifying extensions and changes to the language�

� A structure designed to ease the introduction of new numerical methods� and

� Remarkable execution speed and capacity for large problems� for an �inter�pretive� system�

Ode currently runs on a wide range of microcomputers and mainframes that supportthe UNIX operating system� Ode was developed at Reed College in the summer of�� under a UNIX operating system and is public domain software� The programis currently in use at numerous educational and industrial sites �Reed College�Tektronix Inc�� U�C� Berkeley� Bell Labs� etc�� and the program and documentationare available from some electronic networks� such as Internet�

Ode solves the initial value problem for a family of �rst�order di�erential equa�tions when provided with an explicit expression for each equation� Ode parses aset of equations� initial conditions� and control parameters� and then provides ane�cient numerical solution� Ode makes the initial value problem easy to express�for example� the Ode program

� an ode to Euler

y � �

y� � y

print y from �

step ��

prints ��A UNIX Shell can be used as a control language for Ode� Indeed� this allows

Ode to be used in combination with other graphical or analytical tools commonlyavailable with the UNIX operating system� For instance� the following shell script"�# could be used to generate a bifurcation diagram for the Du�ng equation�

�� Appendix C

� shell script using Ode to construct a bifurcation diagram

� for the Duffing oscillator�

� Scan in F� ��

� Create a file bif�data with the initial conditions

� before running this shell script�

for I in � � � � � � � � � �

do

for J in � � � � � � � � � �

do

for K in � � � � �

do

tail �� bif�data � lastline�tmp

xo��awk �fprint �g� lastline�tmp�

vo��awk �fprint �g� lastline�tmp�

ode !!marker ��bif�data

alpha � ��

beta � ��

gamma � ��

F � � I� J K

x� � xo

v� � vo

theta� � �

x� � v

v� � �alpha�v � x � beta�x"� � F�costheta

theta� � gamma

F � � I� J K� t � �� theta � ��

print x� v� F every �� from ��PI��

step �� PI�� PI��

marker

done

done

done

In addition to Ode� there exist many other packages that provide numericalsolutions to ordinary di�erential equations� However� if you wish to write your ownroutines see Chapter � of Numerical Recipes "�#�

Ode ��


�� K� Atkinson� An introduction to numerical analysis �John Wiley� New York�� Pages �� contain a review of the literature on the numerical solutionof ordinary di�erential equations�

�� N� B� Tu�llaro and G� A� Ross� Ode�A program for the numerical solution ofordinary di�erential equations� Bell Laboratories Technical Memorandum� TM�� based on the Ode User�s Manual �Reed College AcademicComputer Center� �� At last report the latest version of the source codeand documentation were located on the REED VAX� Contact the Reed CollegeAcademic Computer Center at �� for more details�

�� For a tutorial on the UNIX shell� see S� R� Bourne� The UNIX system �Addison�Wesley� Reading� MA� ��

�� W� Press� B� Flannery� S� Teukolsky� and W� Vetterling� Numerical recipes in C�Cambridge University Press� New York� ��

�� Appendix D

Appendix D� Discrete Fourier Transform

The following C routine calculates a discrete Fourier transform and power spec�trum for a time series� It is only meant as an illustrative example and will not bevery useful for large data sets� for which a fast Fourier transform is recommended�

�� Discrete Fourier Transform and Power Spectrum

Calculates Power Spectrum from a Time Series

Copyright �� Nicholas B� Tufillaro

��

�include !stdio�h�

�include !math�h�

�define PI ��

�define SIZE ��

double ts�SIZE�� A�SIZE�� B�SIZE�� P�SIZE��

main

#

int i� k� p� N� L�

double avg� y� sum� psmax�

�� read in and scale data points ��

i � ��

whilescanf �lf � �y �� EOF #

ts�i� � y��

i ��

$

�� get rid of last point and make sure �

of data points is even ��

ifi��

i ��

else

i ��

L � i� N � L��

�� subtract out dc component from time series ��

fori � �� avg � �� i ! L� ��i #

avg �� ts�i��

$

avg � avg�L�

�� now subtract out the mean value from the time series ��

fori � �� i ! L� ��i #

Discrete Fourier Transform ��

ts�i� � ts�i� � avg�

$

�� o�k� guys� ready to do Fourier transform ��

�� first do cosine series ��

fork � �� k !� N� ��k #

forp � �� sum � �� p ! ��N� ��p #

sum �� ts�p��cosPI�k�p�N�

$

A�k� � sum�N�

$

�� now do sine series ��

fork � �� k ! N� ��k #

forp � �� sum � �� p ! ��N� ��p #

sum �� ts�p��sinPI�k�p�N�

$

B�k� � sum�N�

$

�� lastly� calculate the power spectrum ��

fori � �� i !� N� ��i #

P�i� � sqrtA�i��A�i��B�i��B�i��

$

�� find the maximum of the power spectrum to normalize ��

fori � �� psmax � �� i !� N� ��i #

ifP�i� � psmax

psmax � P�i��

$

fori � �� i !� N� ��i #

P�i� � P�i��psmax�

$

�� o�k�� print out the results� k� Pk ��

fork � �� k !� N� ��k #

printf �d �g%n � k� P�k��

$

$

�� Appendix E

Appendix E� H�enon�s Trick

The numerical calculation of a Poincar�e map from a cross section at �rst appearsto be a rather tedious problem� Consider� for example� calculating the Poincar�e mapfor an arbitrary three�dimensional �ow

dx

dt� f�x� y� z��

dy

dt� g�x� y� z��

dz

dt� h�x� y� z��

with a planar cross section& � �x� y� z � ��

A simpleminded approach to this problem would involve numerically integratingthe equations of motion until the cross section is pierced by the trajectory� Anintersection of the trajectory with the cross section is determined by testing for achange in sign of the variable in question� Once an intersection is found� a bisectionalgorithm could be used to hone�in on the surface of section to any desired degreeof accuracy�

H�enon� however� suggested a very clever procedure that allows one to �nd theintersection point of the trajectory and the cross section in one step "�#� Supposethat between the nth and the n � �st steps we �nd a change of sign in the zcoordinate�

�tn� xn� yn� zn �� and �tn �%t� xn�� yn�� zn��

To �nd the exact value in t at which the z coordinate equals zero we can changet from the independent to a dependent variable� and change z from a dependentvariable to the independent variable� This is accomplished by dividing the �rst twoequations by dt�dz and inverting the last equation�

dx

dz�

f�x� y� z�

h�x� y� z��

dy

dz�

g�x� y� z�

h�x� y� z��

dt

dz�

�

h�x� y� z��

This new system can be numerically integrated forward one step� %z � �zn� withthe initial values xn� yn� tn� A simple numerical integration method� such as aRunge�Kutta procedure� is ideally suited for this single integration step�

As an application of H�enon�s method consider the swinging Atwood�s machine�SAM� de�ned by the conservative equations of motion "�#�

(v ��

� � �

$ru� � cos � � �

%�

(u � ��

r��vu � sin ��

(r � v�(� � u�

H�enon�s Trick ��

A cross section for this system is de�ned by �r� (r� � � �� So we need to change theindependent variable from t to �� When � changes sign� we can �nd the Poincar�emap by numerically integrating the equations�

dv

d��

�

u��

$ru� � cos � � �

%�

du

d��

ur��vu� sin ��

dr

d��

v

udt

d��

�

u�

Pictures of the resulting Poincar�e map are found in reference "�#�


�� M� H�enon� On the numerical computation of Poincar�e maps� Physica �D� �� Also see Hao B��L�� Elementary symbolic dynamics �World Scienti�c�New Jersey� �� pp� ��

�� N� B� Tu�llaro� Motions of a swinging Atwood�s machine� J� Physique ��

�� Appendix F

Appendix F� Periodic Orbit Extraction Code

The �rst C routine below was used to extract the periodic orbits from a chaotictime series arising from the Du�ng oscillator� as described in section �� This codeis speci�c to the Du�ng oscillator and is provided because it illustrates the codingtechniques needed for periodic orbit extraction� The second routine takes a pairof extracted periodic orbits and calculates their relative rotation rates according toequation � �� The input to the relative rotation rate program is a pair of periodicorbits plus a �rst line containing some header information about the input �le�

�� fp�c

Find all Periodic orbits of period P and tolerance TOL�

Copyright �� by Nicholas B� Tufillaro

Department of Physics

Bryn Mawr College� Bryn Mawr� PA �� USA

��



�� This Array Maximum must be greater then ��P�STEPS � � ��

�define AMAX ��

�define DISTX��X��Y��Y�

floatsqrtdoubleX��X��X��X��Y��Y��Y��Y�

mainac� av

char ��av�

#

int m� n� cnt� P� STEPS� LEN� TWOLEN� SHORTERPERIODICORBIT�

float d� ds� phe�AMAX�� x�AMAX�� y�AMAX�� TOL� SHRTOL�

float oldphe�

�� process command line arguments� P TOL ��

��ac� P � atoi��av� ��ac� TOL � floatatof��av�

�� set global variables ��

STEPS � �� LEN � P�STEPS� TWOLEN � ��LEN�

SHRTOL � ��TOL�

�� initialization� get first LEN points ��

forn � �� n ! LEN� ��n #

ifscanf �f �f �f � �phe�n�� x�n�� y�n� �� EOF #

printf Not enough orbits for computations�%n �

exit�

$

$

Periodic Orbit Extraction Code ��

�� find periodic orbits ��

for�� #

forn � LEN� n ! TWOLEN� ��n #

ifscanf �f �f �f � �phe�n�� x�n�� y�n� �� EOF #

exit�

$

$

forn � �� n ! LEN� ��n #

d � DISTx�n�� x�n�LEN�� y�n�� y�n�LEN��

ifd ! TOL #

�� Identify shorter periodic orbits� if any�

This is a kludge�

��

form � �� SHORTERPERIODICORBIT � �� m ! P� ��m #

ds � DISTx�n�� x�n�m�STEPS�� y�n�� y�n�m�STEPS��

ifds ! SHRTOL #

SHORTERPERIODICORBIT � ��

$

$

�� End of kludge� ��

if�SHORTERPERIODICORBIT #

iffabsphe�n��oldphe � �� #

oldphe � phe�n��

printf %n�f �f �f%n%n � �� float P� float cnt�

form � n� m !� n�LEN� ��m #

printf �f �f �f%n � phe�m�� x�m�� y�m��

$

cnt ��

$

$

$

$

forn � �� n ! LEN� ��n #

phe�n� � phe�n�LEN�� x�n� � x�n�LEN�� y�n� � y�n�LEN��

$

$

$

�� Appendix F

�� rrr�c

calculate Relative Rotation Rates of two periodic orbits

of periods PA� PB�

Copyright �� by Nicholas B� Tufillaro

Department of Physics� Bryn Mawr College

Bryn Mawr� PA �� USA

INPUT�

Data file with phase� x� and y listed for points in two

periodic orbits� For the first input line of each

periodic orbit� use �� for phase and give the period in x�

For this first point y is ignored�

��



�define PI ��

�define ARGX�Y floatatan�doubleY�doubleX

main

#

int m� n� M� N� I�� J��

float phe� x� y� PA� PB� A�� B��

int i� j� q� Q�

float rx�� ry�� RR��

ifscanf �f �f �f � �phe� �x� �y��EOF #

printf Error� empty input file%n �

exit�

$

ifphe �� #

printf Error� Input file does not begin with ��%n �

exit�

$

ifphe �� #

PA � x�

forn � �� m � �� n #

ifscanf �f�f�f � �phe� �x� �y��EOF #

printf Error� not enough orbits%n �

exit�

$

ifphe �� #

break�

Periodic Orbit Extraction Code ��

$

iffmodphe�� #

I�m� � n�

m ��

$

A��n� � phe� A��n� � y� A��n� � x�

$

PB � x�

M � n�

forn � �� m � �� scanf �f�f�f � �phe� �x� �y �� EOF� ��n #

ifphe �� #

printf Error� Too many orbits%n �

exit�

$

iffmodphe�� #

J�m� � n�

m ��

$

B��n� � phe� B��n� � y� B��n� � x�

$

N � n�

$

Q � intPA�N��

fori � �� i ! PA� ��i #

forj � �� j ! PB� ��j #

form � I�i�� n � J�j�� q � �� q ! Q� ��q� ��m� ��n #

ifm �� m � M��

ifn �� n � N��

ifm �� M m � ��

ifn �� N n � ��

rx�q� � B��n��A��m�� ry�q� � B��n��A��m��

$

forq � �� q ! Q � �� q #

ifARGrx�q��ry�q� � PI�� ARGrx�q��ry�q� ! PI

�� ARGrx�q��ry�q�� PI ��

ARGrx�q��ry�q�� ! �PI��

RR�i��j� �� PI � ARGrx�q��ry�q��

ARGrx�q��ry�q��

else ifARGrx�q��ry�q� � �PI ��

ARGrx�q��ry�q� ! �PI��

ARGrx�q��ry�q�� PI��

ARGrx�q��ry�q�� ! PI

RR�i��j� �� ARGrx�q��ry�q��

ARGrx�q��ry�q� � ��PI�

�� Appendix F

else

RR�i��j� �� ARGrx�q��ry�q��

ARGrx�q��ry�q��

$

$

$

printf %nROTATION RATES%n%n �

printf PA� �g PB� �g%n%n � PA� PB�

fori � �� i ! PA� ��i #

forj � �� j ! PB� ��j #

printf Index� �d� �d� Rotations� �g Rel� Rot�� g%n �

i�� j�� RR�i��j��PI� RR�i��j��PI�PA�PB�

$

$

printf %n �

$

Relative Rotation Rate Package ��

Appendix G� Relative Rotation Rate Package

The followingMathematica package calculates relative rotation rates and inter�twining matrices by the methods described in section � ��

�

Relative Rotation Rates �� Mathematica Package

Date� ��

Last Modified�

Authors� Copyright ��

A� Lorentz� N� Tufillaro and P� Melvin�

Departments of Mathematics and Physics

Bryn Mawr College� Bryn Mawr� PA �� USA

Bugs�

Many of these symbolic computations are exponential time algorithms�

so they are slow for long periodic orbits and templates with many branches�

A C version of these routines exists which runs considerably

faster� The algorithm for the calculation of the relative

rotation rate from a single word pair� however� is polynomial time�

About the Package�

This collection of routines automates the process for the symbolic

calculation of relative rotation rates� and the intertwining matrix

for an arbitrary template� The template is represented algebraically

by a framed braid matrix� which is specified by the global variable

bm in these routines� This variable must be defined by the user

when these routines are entered�

There are three major routines�

RelRotRate�word pair�� AllRelRotRates�word pair�� and

Intertwine�start row� stop row��

The input for the RelRotRate programs is a word pair� which is just

a list of lists� For example� a valid input for these programs is

# #��$� #��$ $

where the first word of the word pair is �� and the second word

is �� The input for the Intertwine program is just two integers�

start row and stop row � For instance� Intertwine�� would produce

all relative rotation rates for all words of length between � and ��

In addition� the routine

�� Appendix G

GetFunCyc�branches� period�

generates all fundamental cycles of length period for a template with

b branches or a symbolic alphabet of b letters� This routine can

be useful for the cycle expansion techniques see� R� Artuso� E� Aurell�

and P� Cvitanovic� Recycling of strange sets I and II� Nonlinearity

Vol �� Num� �� May �� p� ��

References�

�� G� B� Mindlin� X��J� Hou� H� G� Solari� R� Gilmore� and N� B� Tufillaro�

Classification of strange attractors by integers� Phys� Rev� Lett� ��

��

�� N� B� Tufillaro� H� G� Solari� and R� Gilmore� Relative rotation rates�

fingerprints for strange attractors� Phys� Rev� A ��

�� H� G� Solari and R� Gilmore� Organization of periodic orbits in the

driven Duffing oscillator� Phys� Rev� A ��

�� H� G� Solari and R� Gilmore� Relative rotation rates for driven

dynamical systems� Phys� Rev� A ��

�

�

The template braid matrix is a global variable that should be

defined by the user before this package is used� Comment out the

default setting for the braid matrix�

Examples for the braid matrix are presented below�

Global Variable Abbreviation�

bm braid matrix �� algebraic description of template

�

� Braid Matrix Example� The Horseshoe Template �

�

bm � #

# �� $�

# �� $

$�

�

� Braid Matrix Example� Second Iterate of The Horseshoe Template �

bm � #

# �� $�

# �� $�

# �� $�


# �� $

$�

� Calculate the relative rotation rate for a pair of words�

Variables local to RelRotRate�

wordp wordpair

lcm lowest common multiple of word pair lengths

ewordp expanded word pair

top top list

bottom bottom list

rrr relative rotation rate of word pair

�

RelRotRate�wordp&List� ��

Block�

#

lcm�

ewordp� top� bottom�

rrr�

$�

If�wordp�� wordp�� Return�� Self rotation rate �

ewordp � ExpandWordPair�wordp��

top � GetTop�ewordp��

bottom � GetBottom�ewordp��

lcm � Length�top��

rrr � SumAll�top� � SumAll�bottom�� lcm�

Return�rrr�

�

� Permute the word pair list to generate

all possible relative rotation rates �

AllRelRotRates�wordp&List� ��

Block�

#i� pa� pb� lcm� wordstep� rrr�

firstword� secondword� rrrs$�

firstword � wordp��

secondword � wordp��

pa � Length�firstword��

pb � Length�secondword��

� Appendix G

lcm � LCM�pa� pb��

wordstep � lcm�lcm�pa�pb�

rrrs � #$�

For�i � �� i ! lcm� i �� wordstep�

rrr � RelRotRate�#firstword� secondword$��

rrrs � Append�rrrs� #firstword� secondword� rrr$��

firstword � RotateRight�firstword� wordstep��

��

Return�rrrs��

�

� Calculate the Intertwining Matrix of all orbits of length startrow

to length stoprow � startrow !� stoprow� �

Intertwine�startrow&Integer� stoprow&Integer� ��

Block�

#b� i� j� k� rowsize� rsum� colsize� csum� mult�

cycls � #$� rcycls � #$� ccycls � #$� rrr � #$� rrrs � #$$�

b � Length�bm��

rowsize � �� rsum � ��

For�i � startrow� i !� stoprow� ��i�

cycls � GetFunCyc�b� i�� rowsize � Length�cycls��

rsum �� rowsize�

For�j � �� j !� rowsize� ��j�

rcycls � Append�rcycls� cycls��j��

��

��

colsize � �� csum � ��

For�i � �� i !� stoprow� ��i�

cycls � GetFunCyc�b� i�� colsize � Length�cycls��

csum �� colsize�

For�j � �� j !� colsize� ��j�

ccycls � Append�ccycls� cycls��j��

��

��

rrr � #$� rrrs � #$�

For�i � �� i !� rsum� ��i�

For�j � �� j !� csum� ��j�

rrr � AllRelRotRates�#rcycls��i�� ccycls��j��$��

mult � Length�rrr��

rrrs � #rrr�� rrr��$�

For�k � �� k !� mult� ��k�


rrrs � Append�rrrs� rrr��k��

��

Print�rrrs��

If�rcycls��i�� ccycls��j�� Break��

��

��

�

� Subroutines for major programs� RelRotRate� AllRelRotRates�

Intertwine �

ExpandWordPair�wp&List� ��

Block�

#i� lcm� firstword� secondword$�

firstword � wp�� secondword � wp��

lcm � LCM�Length�firstword�� Length�secondword��

Return�

# Flatten� Table�firstword� #i� lcm�Length�firstword� $��

Flatten� Table�secondword� #i� lcm�Length�secondword� $�� $

��

�

GetTop�ewp&List� ��

Block�

#i� lcm$�

lcm � Length� ewp��

Return�

Table� bm�� ewp��i�� ewp��i�� #i� lcm$ �

��

�

GetBottom�ewp&List� ��

Block�

#i�� s�� j�� prevj�� nextj�� cnt�� lcm�� sgn��

top�#$� ewpd�#$� b��#$� b��#$� b��#$� bot�#$$�

lcm � Length�ewp��

top � GetTop�ewp��

ewpd � ewp�� ewp��

� initialize rows b�� b�� b� �

For�i � �� i ! lcm�� i��

� Appendix G

AppendTo�b�� AppendTo�b�� AppendTo�b��

��

� calculate b� row �

For�i � �� i ! lcm � �� i��

If�ewpd��i�� b��i�� Mod�top��i��

��


For�s � �� ewpd��s�� s��

For�i � �� i ! lcm � �� i��

j � i � s� If�j � lcm� j �� lcm��

prevj � j � �� If�prevj ! �� prevj �� lcm��

If�� ewpd��j��

cnt � ��

For�k � j� ewpd��k�� k��

cnt � cnt � top��k��

If�k �� lcm� k � ��

i��

��

cnt � Mod�cnt� ��

b��prevj�� cnt�

��

��


For�i � �� i ! lcm � �� i��

j � i � s� If�j � lcm� j �� lcm��

nextj � j � �� If�nextj � lcm� nextj �� lcm��

k � nextj�

While�� ewpd��nextj��

��nextj� ��i�

If�nextj � lcm� nextj �� lcm��

��

If�Negative�ewpd��j��ewpd��nextj�� sgn � �� sgn � ��

b��j�� sgn�

��

bot � Mod�b� � b� � b��

Return�bot��

�

SumAll�l&List� ��


Block�#i$�

Return�Sum�l��i�� #i� Length�l�$��

�

� Generates the fundamental cycles of length period for a template

with b branches �

GetFunCyc�b&Integer� period&Integer� ��

Block�

#cycles$�

� get all cycles of length �

cycles � GetAllCyc�b� period�� period � and b branches �

cycles � DelSubCyc�b� cycles�� delete nonfundamental subcycles �

cycles � DelCycPerm�cycles�� delete cyclic permutations

of fundamental cycles �

Return�cycles��

�

� Subroutines for GetFunCyc �

GetAllCyc�branches&Integer� levels&Integer� ��

Block�

#n� m� i� j�

roots� tree� nextlevel� cycle� cycles$�

For�i � �� roots � #$� i ! branches� ��i�

roots � Append�roots� i��

��

� creates full n�ary tree recursively �

tree � #roots$�

For�n � �� n ! levels� ��n�

nextlevel � #$�

For�m � �� m !� branches� ��m�

nextlevel � Append�nextlevel� Last�tree��

��

tree � Append�tree� Flatten�nextlevel��

��

� reads up each branch of tree to root� from left to right �

cycles � #$�

For�i � �� i !� branches"levels� ��i�

cycle � #$�

For�j � levels� j � �� j�

�� Appendix G

k � Ceiling�i�branches"levels�j��

cycle � Prepend�cycle� tree��j��k��

��

cycles � Append�cycles� cycle��

��

Return�cycles��

�

DelSubCyc�b&Integer� allcycls&List� ��

Block�

#i� j� k� levels� period� numofdivs� numofsub� copies� wordpos�

cycles � #$� div � #$� word � #$� droplist � #$�

subcycles � #$� subword � #$� nonfun � #$� funcycls � #$$�

� Initializations and gets divisor list �

cycles � allcycls� levels � Length� cycles��

period � levels� div � Divisors�levels��

� Creates nonfundamental words from periodic orbits created from

divisor list �

numofdivs � Length�div��

For�i � �� i ! numofdivs� ��i� � go throw divisor list �

copies � period�div��i��

subcycles � GetAllCyc�b� div��i��

numofsub � Length�subcycles��

� create subwords of lengths found in divisor list �

For�j � �� subword � #$� j !� numofsub� ��j�

subword � subcycles��j��

� expand subwords to length of periodic orbits �

For�k � �� word � #$� k !� copies� ��k�

word � Flatten�Append�word� subword��

��

� find positions of nonfundamental cycles in all cycles �

wordpos � Flatten�Position�cycles� word��

droplist � Union�Append�droplist� wordpos��

��

��

� this is a kludge to delete nonfundamental cycles �

For�i � �� nonfun � #$� i !� Length�droplist�� i�

nonfun � Append�nonfun� cycles��droplist��i��

��

funcycls � Complement�cycles� nonfun��


Return�funcycls��

�

DelCycPerm�funcycls&List� ��

Block�

#size� i� j� period�

cycs� word $�

cycs � funcycls�

size � Length�cycs�� period � Length�cycs��

For�i � �� i ! size� ��i�

word � cycs��i��

For�j � �� j ! period� ��j�

word � RotateLeft�word��

cycs � Complement�cycs� #word$��

size � size��

��

��

Return�cycs��

�

�� Appendix H

Appendix H� Historical Comments

How hard are the problems posed by classical mechanics� The beast in the ma�chine� the beast we now call chaos� was discovered about a century ago by the Frenchscientist and mathematician Henri Poincar�e during the course of his investigationson �the three�body problem�� the great unsolved problem of classical mechanics�Poincar�e� in his magnumopus on the three�body problem� New Methods of CelestialMechanics� writes��

�� When we try to represent the �gure formed by these two curvesand their intersections in a �nite number� each of which correspondsto a doubly asymptotic solution� these intersections form a type oftrellis� tissue� or grid with in�nitely serrated mesh� Neither of the twocurves must ever cut across itself again� but it must bend back uponitself in a very complex manner in order to cut across all of the meshesin the grid an in�nite number of times�

The complexity of this �gure will be striking� and I shall not even tryto draw it� Nothing is more suitable for providing us with an idea ofthe complex nature of the three�body problem� and of all the problemsof dynamics in general� where there is no uniform integral and wherethe Bohlin series are divergent�

Poincar�e is describing his discovery of homoclinic solutions �homoclinic intersec�tions� or homoclinic tangles�� The existence of these homoclinic solutions �solved�the three�body problem insofar as it proved that no solution of the type envisionedby Jacobi or Hamilton could exist� Volume III of Poincar�e�s New Methods of Celes�tial Mechanics �� from which the above quote is taken� is the foundationalwork of modern dynamical systems theory�

Poincar�e�s great theorem of celestial mechanics� as stated in his prize�winningessay to the King of Sweden� is the following��

The canonical equations of celestial mechanics do not admit �exceptfor some exceptional cases to be discussed separately� any analyticaland uniform integral besides the energy integral�

Simply put� Poincar�e�s theorem says that there does not exist a solution to thethree�body problem of the type assumed by the Hamilton�Jacobi method or anyother method seeking an analytic solution to the di�erential equations of motion�

�H� Poincar�e� Les m�ethodes nouvelles de la m�ecanique c�eleste� Vol� ��Gauthier�Villars� Paris� �� reprinted by Dover� �� English translation�New methods of celestial mechanics �NASA Technical Translations� ��

�R� Abraham and C� Shaw� Dynamics�The geometry of behavior� Vol� ��Aerial Press� Santa Cruz� CA� ��

�H� Poincar�e� Sur le probl'eme des trois corps et les �equations de la dynamique�Acta Math� ��

From Three Bodies to Strange Sets ��

Poincar�e �solved� the three�body problem by showing that no solution exists�atleast not of the type assumed by scientists of his era�

But Poincar�e did much more� Confronted with his discovery of homoclinictangles� Poincar�e went on to reinvent what is meant by a solution� In the process�Poincar�e laid the foundations for several new branches of mathematics includingtopology� ergodic theory� homology theory� and the qualitative theory of di�erentialequations�� Poincar�e also pointed out the possible uses of periodic orbits in tamingthe beast��

�� It seems at �rst that this fact can be of no interest what�ever for practice� In fact� there is a zero probability for the initialconditions of the motion to be precisely those corresponding to a pe�riodic solution� However� it can happen that they di�er very littlefrom them� and this takes place precisely in the case where the oldmethods are no longer applicable� We can then advantageously takethe periodic solution as �rst approximation� as intermediate orbit� touse Gyld�en�s language�

There is even more� here is a fact which I have not been able to demon�strate rigorously� but which seems very probable to me� nevertheless�

Given equations of the form de�ned in art� �� and any particularsolution of these equations� we can always �nd a periodic solution�whose period� it is true� is very long�� such that the di�erence betweenthe two solutions is as small as we wish� during as long a time aswe wish� In addition� these periodic solutions are so valuable for usbecause they are� so to say� the only breach by which we may attemptto enter an area heretofore deemed inaccessible�

The periodic orbit theme is pursued in Chapter �Shortly thereafter Hadamard �� produced the �rst example of an abstract

system exhibiting chaos�the geodesics on a surface of constant negative curvature��

Hadamard�s example was later generalized by Anosov and is still one of the bestmathematical examples of chaos in its most extreme form�

In America� George David Birkho� continued in the way of Poincar�e� He ex�amined the use of maps� instead of �ows� and began the process of hunting and

�F� Browder� The mathematical heritage of Henri Poincar�e� Vol� �� Proc� Sym�in Pure Math� Vol� �� American Mathematical Society� Providence� RI� ��

�R� MacKay and J� Meiss� eds�� Hamiltonian dynamical systems �Adam Hilger�Philadelphia� ��

�J� Hadamard� Les surfaces 'a curbures oppos�es et leurs lignes g�eod�esiques� Journ�de Math� � � ��

�G� D� Birkho�� Surface transformations and their dynamical applications� ActaMath� ��

�� Appendix H

naming di�erent critters that he called limit sets of the alpha and omega variety�

Still� without the aid of the computer� Birkho� was often fooled by the beast intothinking that nonintegrability implies complete ergodicity� It took the work of Kol�mogorov� Arnold� and Moser in the early ��s to show that the beast is more subtlein its destructive tendencies� and that it prefers to chew on rational frequencies�

Few physicists were concerned about the beast� or even aware of its existence�during the �rst half of the twentieth century� however� this is more than understand�able when you consider that they had their hands full with quantum mechanics andsome less savory inventions� Still� there were some notable exceptions� such as thosediscussed by Brillouin in his book Scienti�c Uncertainty� and Information��

Near the end of the Second World War� Cartwright and Littlewood�� andLevinson�� showed that the beast liked to play the numbers� and could generatesolutions as random as a coin toss� At �rst they did not quite believe their results�however� experiments with a simple circuit �the van der Pol oscillator� forced themto accept the idea that a fully deterministic system can produce random results�what we now think of as the de�nition of chaos�

In �� the young topologist Steven Smale was sitting on the beach in Rioplaying with the beast when he �rst saw that in its heart lay a horseshoe�� Smalefound that he could name the beast� a hyperbolic limit set� even if he had troubleseeing it since it was very� very thin� Moreover� Smale found a simple way tounravel the horseshoe via symbolic dynamics�� Now that the beast was namedand dissected� mathematicians had some interesting mathematical objects to playwith� and Smale and his friends were o� and running�

At about the same time the meteorologist Ed Lorenz� a former student of G� D�Birkho�� discovered that the beast was not only in the heavens� as evidenced in thethree�body problem� but may well be around us all the time in the atmosphere��

Whereas Smale showed us how to name the beast� Lorenz allowed us to see thebeast with computer simulations and judiciously chosen models� Lorenz pointed

G� D� Birkho�� Collected mathematical works� Vol� �� Dover� New York��

J� Moser� Stable and random motions in dynamical systems� Ann� Math� Studies�� Princeton University Press� Princeton� NJ� ��

��L� Brillouin� Scienti�c uncertainty� and information �Academic Press� NewYork� ��

��M� L� Cartwright and N� Littlewood� On nonlinear di�erential equations of thesecond order� I� J� Lond� Math� Soc� ��

��N� Levinson� A second�order di�erential equation with singular solutions� Ann�Math� ��

��S� Smale� The mathematics of time� Essays on dynamical systems� economicprocesses� and related topics �Springer�Verlag� New York� ��

��R� L� Devaney� An introduction to chaotic dynamical systems� second ed��Addison�Wesley� New York� ��

��E� N� Lorenz� Deterministic nonperiodic �ow� J� Atmos� Sci� ��

From Three Bodies to Strange Sets ��

out the role return maps can play in understanding real systems� how ubiquitousthe beast really is� and how we can become more familiar with the beast throughsimple models such as the logistic map��

This two�pronged approach of using abstract topological methods on the onehand and insightful computer experiments on the other is central to the method�ology employed when studying nonlinear dynamical systems� More than anythingelse� it probably best de�nes the �nonlinear dynamical systems method� as it hasdeveloped since Lorenz and Smale� In fact� J� von Neumann helped invent the elec�tronic computer mainly to solve� and provide insight into� nonlinear equations� In�� in an article called �On the principles of large scale computing machines�� hewrote�

Our present analytic methods seem unsuitable for the solution of theimportant problems arising in connection with non�linear partial dif�ferential equations and� in fact� with virtually all types of non�linearproblems in pure mathematics� � � �

� � � really e�cient high�speed computing devices may� in the �eld ofnon�linear partial di�erential equations as well as in many other �eldswhich are now di�cult or entirely denied of access� provide us withthose heuristic hints which are needed in all parts of mathematics forgenuine progress�

Thank heavens for the Martians�

��E� N� Lorenz� The problem of deducing the climate from the governing equa�tions� Tellus ��

�� Appendix I

Appendix I� Projects

In this appendix we provide a brief guide to the literature on some topics thatmight be suitable for an advanced undergraduate research project�

�� Acoustics

�a� Review articleW� Lauterborn and U� Parlitz� Methods of chaos physics and theirapplication to acoustics� J� Acoust� Soc� Am� ��

�b� Oscillations in gas columnsT� Yazaki� S� Takashima� and F� Mizutani� Complex quasiperiodicand chaotic states observed in thermally induced oscillations of gascolumns� Phys� Rev� Lett� ��

�c� WineglassA� French� A study of wineglass acoustics� Am� J� Phys� ��

�� Biology

�a� Review articleL� Olsen and H� Degn� Chaos in biological systems� Quart� Rev� Bio�phys� ��

�b� Brain wavesP� Rapp� T� Bashore� J� Martinerie� A� Albano� I� Zimmerman� and A�Mees� Dynamics of brain electrical activity� Brain Topography � ��$��

�c� Gene structureH� Je�rey� Chaos game representation of gene structure� Nucleic AcidsRes� ��

�� Chemistry

�a� Chemical clocksS� Scott� Clocks and chaos in chemistry� New Scientist �� December�� E� Mielczarek� J� Turner� D� Leiter� and L� Davis� Chem�ical clocks� Experimental and theoretical models of nonlinear behavior�Am� J� Phys� ��

�� Electronics

�a� Analog simulation of laser rate equationsM� James and F� Moss� Analog simulation of a periodically modulatedlaser model� J� Opt� Soc� Am� B � � �� for moredetails about the circuit contact F� Moss� Dept� of Physics� Universityof St� Louis� St� Louis� MO ��

Projects ��

�b� CircuitsJ� Lesurf� Chaos on the circuit board� New Scientist �� June �� A� Rodriguez�Vazquez� J� Huertas� A� Rueda� B� Perez�Verdu�and L� Chua� Chaos from switched�capacitor circuits� Discrete maps�Proc� IEEE ��

�c� Double scrollT� Matsumoto� L� Chua� and M� Komuro� The double scroll bifurca�tions� Circuit theory and applications �� T� Mat�sumoto� L� Chua� and M� Komuro� The double scroll� IEEE Trans�on Circuits and Systems CAS�� L� Chua� M�Komuro� and T� Matsumoto� The double scroll family� IEEE Trans�on Circuits and Systems CAS�� T� Weldon�An inductorless double scroll circuit� Am� J� Phys� ��

� Hydrodynamics

�a� Dripping faucetR� Cahalan� H� Leidecker� and G� Cahalan� Chaotic rhythms of a drip�ping faucet� Computing in Physics� �� Jul�Aug �� R� Shaw�The dripping faucet as a model chaotic system �Aerial Press� SantaCruz� CA �� H� Yepez� N� Nuniez� A� Salas Brito� C� Vargas� andL� Viente� Chaos in a dripping faucet� Eur� J� Phys� ��

�b� Hele�Shaw cellJ� Nye� H� Lean� and A� Wright� Interfaces and falling drops in Hele�Shaw cell� Eur� J� Phys� ��

�c� Lorenz loopS� Dodd� Chaos in a convection loop� Reed College Senior Thesis�� M� Gorman� P� Widmann� and K� Robbins� Nonlinear dynam�ics of a convection loop� Physica ��D� � �� P� Widmann�M� Gorman� and K� Robbins� Nonlinear dynamics of a convection loopII� Physica ��D� � ��

�d� Surface wavesR� Apfel� �Whispering� waves in a wineglass� Am� J� Phys� �� S� Douady and S� Fauve� Pattern selection in Fara�day instability� Europhys� Lett� � �� J� P� Gollub�Spatiotemporal chaos in interfacial waves �To appear in� New per�spectives in turbulence� edited by S� Orszag and L� Sirovich� Springer�Verlag�� J� Miles and D� Henderson� Parametrically forced surfacewaves� Annu� Rev� Fluid Mech� �� V� Nevolin� Para�metric excitation of surface waves� J� Eng� Phys� �USSR� ��

�� Appendix I

�� Mechanics

�a� Artistic mobilesO� Viet� J� Wesfreid� and E� Guyon� Art cin�etique et chaos m�ecanique�Eur� J� Phys� ��

�b� Bicycle stabilityY� H�ena�� Dynamical stability of the bicycle� Eur� J� Phys� �� J� Papadopoulos� Bicycling handling experiments you can do�preprint� Cornell University� �� C� Miller� The physical anatomyof steering stability� Bike Tech �October ��

�c� Compass in a B��eldH� Meissner and G� Schmidt� A simple experiment for studying thetransition from order to chaos� Am� J� Phys� �� V� Croquette and C� Poitou� Cascade of period doubling bifurcationsand large stochasticity in the motions of a compass� J� Physique Lettres�� L� ��L� ��

�d� Compound pendulumN� Pedersen and O� Soerensen� The compound pendulum in interme�diate laboratories and demonstrations� Am� J� Phys� ��

�e� Coupled pendulaK� Nakajima� T� Yamashita� and Y� Onodera� Mechanical analogue ofactive Josephson transmission line� J� Appl� Phys� ��

�f� Elastic pendulumE� Breitenberger and R� Mueller� The elastic pendulum� A nonlinearparadigm� J� Math� Phys� �� M� Olsson� Whydoes a mass on a spring sometimes misbehave� Am� J� Phys� �� H� Lai� On the recurrence phenomenon of a reso�nant spring pendulum� Am� J� Phys� �� L� Falk�Recurrence e�ects in the parametric spring pendulum� Am� J� Phys�� M� Rusbridge� Motion of the sprung pendu�lum� Am� J� Phys� �� T� Cayton� The laboratoryspring�mass oscillator� An example of parametric instability� Am� J�Phys� �� J� Lipham and V� Pollak� Constructinga �misbehaving� spring� Am� J� Phys� ��

�g� Forced beamSee Appendix C of F� Moon� Chaotic vibrations �John Wiley $ Sons�New York� ��

�h� Forced pendulumR� Leven� B� Pompe� C� Wilke� and B� Koch� Experiments on periodicand chaotic motions of a parametrically forced pendulum� Physica ��D��

Projects ��

�i� Impact oscillatorH� Isomaki� J� Von Boehm� and R� Raty� Devil�s attractors and chaosof a driven impact oscillator� Phys� Lett� ��A �� A� Brahic� Numerical study of a simple dynamical system� Astron� $Astrophys� �� C� N� Bapat and C� Bapat� Impact�pairunder periodic excitation� J� Sound Vib� ��

�j� Impact pendulumD� Moore and S� Shaw� The experimental response of an impactingpendulum system� Int� J� Nonlinear Mech� �� S�Shaw and R� Rand� The transition to chaos in a simple mechanicalsystem� Int� J� Nonlinear Mech� ��

�k� Impact printerP� Tung and S� Shaw� The dynamics of an impact print hammer� J�of Vibration� Acoustics� Stress� and Reliability in Design �� April ��

�l� Swinging Atwood�s machineJ� Casasayas� A� Nunes� and N� B� Tu�llaro� Swinging Atwood�s ma�chine� Integrability and dynamics� J� de Physique �� J� Casasayas� N� B� Tu�llaro� and A� Nunes� In�nity manifold of aswinging Atwood�s machine� Eur� J� Phys� � �� N� B� Tu�llaro� A� Nunes� and J� Casasayas� Unbounded orbits of aswinging Atwood�s machine� Am� J� Phys� �� N� B� Tu�llaro� Integrable motion of a swinging Atwood�s machine�Am� J� Phys� �� N� B� Tu�llaro� Motions of aswinging Atwood�s machine� J� de Physique �� N�B� Tu�llaro� Collision orbits of a swinging Atwood�s machine� J� dePhysique �� N� B� Tu�llaro� T� A� Abbott� and D�J� Gri�ths� Swinging Atwood�s machine� Am� J� Phys� �� B� Bruhn� Chaos and order in weakly coupled systems ofnonlinear oscillators� Physica Scripta ��

�m� Swinging trackJ� Long� The nonlinear e�ects of a ball rolling in a swinging quartercircle track� Reed College Senior Thesis �� R� Benenson and B�Marsh� Coupled oscillations of a ball and a curved�track pendulum�Am� J� Phys� ��

�n� WedgeN� Whelan� D� Goodings and J� Cannizzo� Two balls in one dimensionwith gravity� Phys� Rev� A �� H� Lehtihet and B�Miller� Numerical study of a billiard in a gravitational �eld� Physica��D� �� A� Matulich and B� Miller� Gravity in one dimen�sion� Stability of a three particle system� Celest� Mech� �� B� Miller and K� Ravishankar� Stochastic modeling of a billiardin a gravitational �eld� J� Stat� Phys� ��

�� Appendix I

� Optics

�a� Laser instabilitiesN� B� Abraham� A new focus on laser instabilities and chaos� LaserFocus� May ��

�b� Light absorbing �uidG� Indebetouw and T� Zukowski� Nonlinear optical e�ects in absorbing�uids� Some undergraduate experiments� Eur� J� Phys� ��

�c� Maser equationsD� Kleinman� The maser rate equations and spiking� Bell System Tech�J� �July ��

�d� Semiconductor lasersH� Winful� Y� Chen� and J� Liu� Frequency locking� quasiperiodicity�and chaos in modulated self�pulsing semiconductor lasers� Appl� Phys�Lett� �� J� Camparo� The diode laser in atomicphysics� Contemp� Phys� ��

�e� Diode�pumped YAG laserN� B� Abraham� L� Molter� and G� Alman� Experiments with a diode�pumped Nd�YAG laser� private communication� Address� Departmentof Physics� Bryn Mawr College� Bryn Mawr� PA �� USA�

� Spatial�Temporal Chaos

�a� Capillary ripplesN� B� Tu�llaro� Order�disorder transition in capillary ripples� Phys�Rev� Lett� �� H� Riecke� Stable wave�numberkinks in parametrically excited standing waves� Europhys� Lett� �� A� B� Ezerskii� M� I� Rabinovitch� V� P� Reutov� andI� M� Starobinets� Spatiotemporal chaos in the parametric excitationsof a capillary ripple� Sov� Phys� JEPT �� W�Eisenmenger� Dynamic properties of the surface tension of water andaqueous solutions of surface active agents with standing capillary wavesin the frequency range from �� kc�s to �� Mc�s� Acustica ��

�b� Cellular automataJ� P� Reilly and N� B� Tu�llaro� Worlds within worlds�An introductionto cellular automata� The Physics Teacher �February ��

�c� Coupled lattice mapsR� Kapral� Pattern formation in two�dimensional arrays of coupled�discrete�time oscillators� Phys� Rev� A ��

�d� FerrohydrodynamicsE� M� Karp� Pattern selection and pattern competition in a ferrohy�drodynamic system� Reed College Senior Thesis ��

Projects ��

�e� Video feedbackJ� Crutch�eld� Space�time dynamics in video feedback� Physica �D�� V� Golubev� M� Rabinovitch� V� Talanov� V� Shklover�and V� Yakhno� Critical phenomena in inhomogeneous excitable media�JEPT Lett� �� G� Hausler� G� Seckmeyer� and T�Weiss� Chaos and cooperation in nonlinear pictorial feedback systems�Appl� Opt� ��

Index

�k� �� recurrent point� �� limit set� �� pseudo orbit� ��limit set� �� period three implies chaos��

abstract dynamical system� ��alternating binary tree� �� Arnold� V� I�� Artin crossing convention� � �asymptotic behavior� ��asymptotically stable� ��attracting� �

set� �� attractor� ��

identi�cation� ��autonomous� ��

backward limit set� �� backward orbit� ��Baker�s map� �basin of attraction� �� beam� �� Bendixson� Ivar� ��bi�in�nite symbol sequence� ��bifurcation� ��

diagram� �� summary diagram� �

binary expansion� �� binary tree� ��BINGO� �� Birkho�� G� D�� Birman and Williams template theo�

rem� ��bisection method� ��Borromean rings� � �

bouncing ballC program� ��impact map� � � ��impact relation� ��phase map� �� sticking solution� �� strange attractor� ��system� ��velocity map� ��

braid� � �group� � relations� � word� �

braid linking matrix� � branch line� ��branch torsion� ��

C programming language� ��Cantor set� �� Cartwright and Littlewood� �� center� �� center manifold� ��

theorem� ��center space� ��chain recurrent� ��chain recurrent set� ��chain rule� ��chaos� �� chaos vs� noise� �� chaotic� ��

attractor� ��repeller� ��set� ��

circle map� � �close return� ��

��

INDEX ��

closure� � �coe�cient of restitution� �coexisting solutions� ��conservative� �� contraction� ��correlation dimension� ��correlation integral� �cross section� �� crossing convention� � �

damping coe�cient� �delay time� ��derivative

of a composite function� ��of a map� ��

deterministic� ��Devaney� R� L�� devil�s staircase� ��di�eomorphism� ��di�erential� ��di�erential equations

�rst�order system of� ��geometric de�nition� ��numerical solution of� ��

di�erentiator� ��disconnected set� � discrete Fourier transform

C program� �� dissipative� �� divergence� �� domain of attraction� �� Du�ng equation� ��

template� �dynamical quest� �� dynamical systems theory� �

eigenvalues� �� real matrix� ��elastic collision� �embedded time series� embedded variable� ��embedding�

di�erential� ��time�delayed� ��

equation of �rst variation� ��equilibrium

point� ��solution� ��state� ��

eventually �xed� ��eventually periodic point� �evolution operator� ��

linear� �� expansion� ��extended phase space� ��

fast Fourier transform� ��Feigenbaum� �Feigenbaum�s delta� FFT� ��gure�eight knot� � ��nite Fourier series� ��rst return map� �� rst�order system� di�erential equa�

tions� ��xed point� �� ip bifurcation� ��ow� �� forced damped pendulum� �� forward limit set� �� forward orbit� ��Fourier series� ��Fourier transform

C program� �� fractal� �fractal dimension� �� framed braid� � �framing� � �frequency spectrum� ��fully developed chaos� �� functional composition� ��fundamental frequency� ��

geometric braid� � �geometric convergence� geometric series� �� global torsion� �� granny knot� � �graphical iteration of a one�dimensional

map� ��

�� INDEX

Hadamard� J�� half�twist� � �harmonic balance� �� harmonic oscillator� � �� Hausdor�� F�� Heaviside function� �H�enon map� � � �

Jacobian� ��H�enon� Michel� � H�enon�s trick� ��heteroclinic

point� ��tangle� ��

Holmes� P�� homeomorphism� �� homoclinic

intersection� � �point� ��solution� ��tangle� � ��

Hopf bifurcation� � Hopf link� � �horseshoe� � � ��

intertwining matrix� ��linear map� ��template� ��

hyperbolic�xed point� ��point� ��set� ��

hyperbolicity� ��hysteresis� ��

IHJM optical map� �incommensurate� ��inelastic collision� �initial condition� ��inset� �� integral curve� �� integration trick� ��intertwining matrix� ��invariant set� �� invertible map� ��itinerary� ��

Jacobian� ��

Katok�s theorem� �� kneading theory� �� knot� ��

problem� � �types� � �

laser rate equation� �� Levinson� N�� Li and Yorke� ��lift� ��limit cycle� �� linear approximation� ��linear di�erential equation� �linear map� � linear resonance� ��linear stability� ��linear system� �linearization� �� link� � �linking number� � � ��local stability� ��local torsion� ��logistic map� �longitudinal vibrations� ��Lorentz force law� �� Lorenz equations� �

divergence� ��intertwining matrix� ��template� ��

Lorenz� E� N�� Lyapunov exponent� ��

main resonance� ��Mandelbrot� B�� manifold� ��map� � � ��maps to� ��Martians� ��Mathematica� �� matrix of partial derivatives� ��metric� ��mod operator� � modulated laser equations� �

INDEX ��

multifractal� ��

negative cross� � �neutral stability� �nonautonomous� ��noninvertible map� ��nonlinear dynamics� �nonwandering

point� ��set� ��

numerical methodscalculation of a Poincar�e map� ��solution of ordinary di�erential equa�

tions� ��

Ode� ��orbit� � � �� orbit stability diagram� ordering relation� �� ordering tree� �orientation� ��orientation preserving� ��oriented knot� � �outset� ��

parabolic map� ��parametric oscillation� ��partition� �� pedagogical approach� � �� pendulum� ��perfect set� � period� ��period doubling� ��

bifurcation� �route to chaos� ��

period one orbit� ��periodic orbit extraction� ��

C program� ��periodic point� �� periodic solution� ��phase plane� �� phase space� ��

experimental reconstruction� ��phase�locked loop� �� piezoelectric �lm� ��

Pirogon template� ��PLL� �� Poincar�e map� �� Poincar�e section� experimental� ��Poincar�e� Jules Henri� ��

�� Poincar�e�Bendixson theorem� ��point attractor� ��positive cross� � �power spectrum� ��preimage� ��primary resonance� �� principle of superposition� �pruning� �

quadratic map� �� bifurcation diagram� bifurcation diagram program� C program� ��

qualitative analysis� ��qualitative solution� quasiperiodic motion� �� quasiperiodic route to chaos� �� quasiperiodic solution� ��

recurrence� � �� criterion� �� plot� ��

recurrent point� �� Reidemeister moves� � �relative rotation rate� ��

arithmetic� ��C program� ��calculation� ��Mathematica package� ��

repeller� �� repelling� �resonance� �� response curve� ��ribbon graph� � �rotating�wave approximation� ��rotation matrix� ��rubber sheet model� ��

saddle� ��

� INDEX

saddle point� �� saddle�node bifurcation� �SAM� �� Sarkovskii�s theorem� ��secondary resonance� �� self�rotation rate� ��self�similarity� �semiconjugate� ��semideterministic� ��semi�ow� ��sensitive dependence� ��

��separatrix� �� sequence space� ��shadowing� ��shift equivalent� �� shift map� �� sideband� ��Simple Theorem of Fermat� ��sine circle map� �sink� �� slowly varying amplitude� ��Smale horseshoe� ��Smale� S�� Smale�Birkho� homoclinic theorem� ��solid torus� ��solution curve� ��source� �� spectral signature� ��spectrum analyzer� ��square knot� � �stability� ��

diagram� in one�dimensional maps� �of a periodic point� �

stable manifold� ��of a map� ��

stable node� ��stable space� ��standard insertion� ��standard map� ��state� �� static� ��strange

attractor� �� repeller� ��set� � �

strange vs� chaotic attractor� ��stretching and folding� ��stroboscopic map� ��subharmonic� �� suspension� �� swinging Atwood�s machine� �� symbol space� ��symbolic

coordinate� ��dynamics� �� future� ��itinerary� ��

tangent bifurcation� �tangent bundle� ��tangent manifold� ��tangle� ��Taylor expansion� �� template� �� tent map� ��time delay� ��time series� �� time�T map� ��topological chaos� ��topological conjugacy� ��topological entropy� ��topological invariant� � �topology� ��torus attractor� � torus doubling route to chaos� ��trajectory� �� transcritical bifurcation� �transient

behavior� ��precursor� ��solution� �

transverse vibrations� ��trapping region� �� tree� �trefoil� � �triggering circuit� ��

INDEX ��

ultraharmonic� �� ultrasubharmonic� �� unimodal map� ��unit interval� ��universality theory� unknot� � �unstable manifold� ��

of a map� ��unstable node� ��unstable periodic orbit� �unstable space� ��

vector �eld� �� Verhulst� P� F�� von Neumann� J��

wandering point� ��Whitehead link� � �winding number� ��

z� ��

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