Applied Mathematical Sciences Volume 73

Editors F. lohn 1. E. Marsden L. Sirovich

Advisors M. Ghil 1.K. Hale 1. Keller K. Kirchgassner B.l. Matkowsky 1.T. Stuart A. Weinstein

Applied Mathematical Sciences

I. Johll: Partial Differential Equations. 4th ed. 2. SirOl'ich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential Equations. 2nd ed. 4. Percus: Combinatorial Methods. 5. VOII MiseslFriedrichs: Fluid Dynamics. 6. FreibergerlGrenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space.

II. WolOl'ich: Linear Multivariable Systems. 12. Berkol'itz: Optimal Control Theory. 13. BllimalllCole: Similarity Methods for Differential Equations. 14. Yoshi:awa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions. 15. Braull: Differential Equations and Their Applications. 3rd cd. 16. LefscltelZ: Applications of Algebraic Topology. 17. ColIlll:IWellerling: Optimization Problems. IS. Grellllllder: Pattern Synthesis: Lectures in Pattern Theory. Vol I. 20. DrilU: Ordinary and Delay Differential Equations. 21. COliralltlFriedrichs: Supersonic Flow and Shock Waves. 22 . RouchelHabelSlLalov: Stability Theory by Liapunov 's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grellallder: Pattern Analysis: Lectures in Pattern Theory. Vol. II. 25. Dm'ies: Integral Transforms and Their Applications. 2nd ed. 26. Klis/1Iler/C/ark: Stochastic Approximation Methods for Constrained and Unconstrained Systems 27. de Boor: A Practical Guide to Splines. 2S. KeilsOl': Markov Chain Models-Rarity and Exponentiality. 29. de Vellbl'ke: A Course in Elasticity. 30. SIIilllycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturm ian Theory for Ordinary Differential Equations. 32. MeislMarkowitz? Numerical Solution of Partial Differential Equations. 33. Grellallder: Regular Structures: Lectures in Pattern Theory. Vol. III. 34. KemrkianlCole: Perturbation methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory. 36. BenXr.mmIGltillKiillen: Dynamic Meteorology: Data Assimilation Methods. 37. Saperswlle: Semidynamical Systems in Infinite Dimensional Spaces. 3S. LicirtenberglLiebermlll': Regular and Stochastic Motion. 39. PicciniISramp<lcchiaIVidossich: Ordinary Differential Equations in R". 40. Naylor/Sell: Linear Operator Theory in Engineering and Science. 41. Sp<trTOII': The Lorenz Equations: Bifureations. Chaos. and Strange Attractors. 42. GlickenlteimerlHolmes: Nonlinear Oscillations. Dynamical Systems and Bifurcations of Vector Fields. 43. OckellllOl,l7ilvler: Inviscid Fluid Flows. 44. Pa:r: Semigroups of Linear Operators and Applications to Partial Differential Equations. 45. G/m/llllfiGusrafsoll: Linear Optimization and Approximation: An Introduction to the Theoretical Analysis

and Numerical Treatment of Semi-Infinite Programs. 46. Wilcox: Scattering Theory for Diffraction Gratings. 47 . Hal<' er al.: An Introduction to Infinite Dimensional Dynamical Systems-Geometric Theory. 4S. Mlim/l': Asymptotic Analysis. 49. L<Il/r:lrenskaya: The Boundary-Value Problems of Mathematical Physics. 50. Wilcox: Sound Propagation in Stratified Fluids. 51. GolllbitskylSchaeffer: Bifurcation and Groups in Bifurcation Theory. Vol. I. 52. Clripor: Variational Inequalities and Flow in Porous Media. 53. Majda: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. 54. WaSOlI': Linear Turning Point Theory.

(continuedfollowing index)

Stephen Wiggins

Global Bifurcations andChaos Analytical Methods

With 200 IIlustrations

Springer Science+Business Media, LLC

Stephen Wiggins Applied Mechanics 104-44 California Institute of Technology Pasadena, CA 91125 USA

Editors F. John Courant Institute of

Mathematical Sciences New York University New York, NY 10012 USA

J.E. Marsden Department of

Mathematics University of California Berkeley, CA 94720 USA

Mathematics Subject Classification (1980): 34xx, 58xx, 70

Library of Congress Cataloging-in-Publication Data Wiggins, Stephen.

L. Sirovich Division of

Applied Mathematics Brown University Providence, RI 02912 USA

Global bifurcations and chaos : analytical methods / Stephen Wiggins. p. cm.-(Applied mathematical sciences ; v. 73)

Bibliography: p. IncIudes index. ISBN 978-1-4612-1041-2 ISBN 978-1-4612-1042-9 (eBook) DOI 10.1007/978-1-4612-1042-9

3. Differential equations-Numerical solutions. 1. Title. II. Series: Applied mathematical sciences (Springer Science+Business Media,

LLC); v. 73. QAI.A647 voI. 73 [QA372] 51Os-dcl9 [514'.74] 88-19959

© 1988 by Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1988 Softcover reprint of the hardcover 1 st edition 1988 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Camera-ready copy provided by the author.

9 8 7 6 5 4 3 2 1

ISBN 978-1-4612-1041-2

To Meredith

PREFACE

The study of chaotic phenomena in deterministic nonlinear dynamical systems

has attracted much attention over the last fifteen years. For the applied scientist,

this study poses three fundamental questions. First, and most simply, what is meant

by the term "chaos"? Second, by what mechanisms does chaos occur, and third,

how can one predict when it will occur in a specific dynamical system? This book

begins the development of a program that will answer these questions.

I have attempted to make the book as self-contained as possible, and thus have

included some introductory material in Chapter One. The reader will find much

new material in the remaining chapters. In particular, in Chapter Two, the tech­

niques of Conley and Moser (Moser [1973]) and Afraimovich, Bykov, and Silnikov

[1983] for proving that an invertible map has a hyperbolic, chaotic invariant Cantor

set are generalized to arbitrary (finite) dimensions and to subshifts of finite type.

Similar techniques are developed for the nonhyperbolic case. These nonhyperbolic

techniques allow one to demonstrate the existence of a chaotic invariant set having

the structure of the Cartesian product of a Cantor set with a surface or a "Cantor

set of surfaces". In Chapter Three the nonhyperbolic techniques are applied to the

study of the orbit structure near orbits homoclinic to normally hyperbolic invariant

tori.

In Chapter Four,1 develop a class of global perturbation techniques that enable

one to detect orbits homoclinic or heteroclinic to hyperbolic fixed points, hyperbolic

periodic orbits, and normally hyperbolic invariant tori in a large class of systems.

The methods developed in Chapter Four are similar in spirit to a technique origi­

nally developed by Melnikov [1963] for periodically forced, two-dimensional systems;

viii Preface

however, they are much more general in that they are applicable to arbitrary (but

finite) dimensional systems and allow for slowly varying parameters and quasiperi­

odic excitation. This general theory will hopefully be of interest to the applied

scientist, since it allows one to give a criterion for chaotic dynamics in terms of the

system parameters. Moreover, the methods apply in arbitrary dimensions, where

much work remains to be done in chaos and nonlinear dynamics.

In this book I do not deal with the question of the existence of strange attrac­

tors. Indeed, this remains a major outstanding problem in the subject. However,

this book does provide useful techniques for studying strange attractors, in that

the first step in proving that a system possesses a chaotic attracting set is to prove

that it possesses chaotic dynamics and then to show that the dynamics are con­

tained in an attracting set that has no stable "regular" motions. One cannot deny

that chaotic Cantor sets can radically influence the dynamics of a system; however,

the extent and nature of this influence needs to be studied. This will require the

development of new ideas and techniques.

Over the past two years many people have offered much encouragement and

help in this project, and I take great pleasure in thanking them now.

Phil Holmes and Jerry Marsden gave me the initial encouragement to get

started and criticized several early versions of the manuscript.

Steve Schecter provided extremely detailed criticisms of early versions of the

manuscript which prevented many errors.

Steve Shaw read and commented on all of the manuscript.

Pat Sethna listened patiently to my explanations of various parts of the book

and helped me considerably in clarifying my thoughts and presentation style.

John Allen and Roger Samelson called my attention to a crucial error in some

earlier work.

Darryl Holm, Daniel David, and Mike Tratnik listened to several lengthy ex­

planations of material in Chapters Three and Four and pointed out several errors

in the manuscript.

Much of the material in Chapters Three and Four was first tried out in graduate

applied math courses at Caltech. I am grateful to the students in those courses for

enduring many obscure lectures and offering useful suggestions.

During the past two years Donna Gabai and Jan Patterson worked tirelessly

on the layout and typing of this manuscript. They unselfishly gave of their time

Preface ix

(often evenings and weekends) so that various deadlines could be met. Their skill

and help made the completion of this book immensely easier.

I would also like to acknowledge the artists who drew the figures for this book

and pleasantly tolerated my many requests for revisions. The figures for Chapter

One were done by Betty Wood, and those for Chapter Four by Cecilia Lin. Peggy

Firth, Pat Marble, and Bob Turring of the Caltech Graphic Arts Facilities and Joe

Pierro, Haydee Pierro, Melissa Loftis, Gary Hatt, Marcos Prado, Bill Contado, Abe

Won, and Stacy Quinet of Imperial Drafting Inc. drew the figures for Chapters

Two and Three.

Finally, Meredith Allen gave indispensable advice and editorial assistance

throughout this project.

CONTENTS

Preface

CHAPTER 1.

Introduction: Background for Ordinary Differential Equations and

Dynamical Systems

1.1. The Structure of Solutions of Ordinary Differential Equations

1.1a. Existence and Uniqueness of Solutions

LIb. Dependence on Initial Conditions and Parameters

1.1c. Continuation of Solutions

LId. Autonomous Systems

LIe. Nonautonomous Systems

1.1 f. Phase Flows

1.1g. Phase Space

1.1h. Maps

1.1 i. Special Solutions

l.lj. Stability

1.1k. Asymptotic Behavior

1.2. Conjugacies

1.3. Invariant Manifolds

1.4. Transversality, Structural Stability, and Genericity

1.5. Bifurcations

1.6. Poincare Maps

CHAPTER 2.

Chaos: Its Descriptions and Conditions for Existence

2.1. The Smale Horseshoe

2.1a. Definition of the Smale Horseshoe Map

2.1b. Construction of the Invariant Set

VB

1

1

2

3

4

8

9

11

12

14

15

16

20

22

26

56

62

67

75

76

76

79

xii

2.lc. Symbolic Dynamics

2.1d. The Dynamics on the Invariant Set

2.1e. Chaos

Contents

2.2. Symbolic Dynamics

86

90

93

94

96 2.2a. The Structure of the Space of Symbol Sequences

2.2b. The Shift Map

2.2c. The Subshift of Finite Type

2.2d. The Case of N = 00

2.3. Criteria for Chaos: The Hyperbolic Case

2.3a. The Geometry of Chaos

2.3b. The Main Theorem

2.3c. Sector Bundles

2.3d. More Alternate Conditions for Verifying Al and A2

2.3e. Hyperbolic Sets

2.3f. The Case of an Infinite Number of Horizontal Slabs

2.4. Criteria for Chaos: The Nonhyperbolic Case

2.4a. The Geometry of Chaos

2.4b. The Main Theorem

2.4c . Sector Bundles

CHAPTER 3.

Homoclinic and Heteroclinic Motions

3.1. Examples and Definitions

3.2. Orbits Homoclinic to Hyperbolic Fixed Points of Ordinary Differential

100

101

106

108

108

118

128

134

145

149

150

151

159

161

171

171

Equations 182

3.2a. The Technique of Analysis 183

3.2b. Planar Systems 199

3.2c. Third Order Systems 207

i) Orbits Homoclinic to a Saddle Point with Purely Real Eigenvalues 208

ii) Orbits Homoclinic to a Saddle-Focus 227

3.2.d. Fourth Order Systems 258

i) A Complex Conjugate Pair and Two Real Eigenvalues 261

ii) Silnikov's Example in R4 267

Contents xiii

3.2e. Orbits Homoclinic Fixed Points of 4-Dimensional Autonomous

Hamiltonian Systems 275

i) The Saddle-Focus 276

ii) The Saddle with Purely Real Eigenvalues 286

iii) Devaney's Example: Transverse Homoclinic Orbits in an Integrable

Systems 298

3.2f. Higher Dimensional Results 298

3.3. Orbits Heteroclinic to Hyperbolic Fixed Points of Ordinary Differential

Equations

i) A Heteroclinic Cycle in ]R3

ii) A Heteroclinic Cycle in ]R4

3.4. Orbits Homoclinic to Periodic Orbits and Invariant Tori

CHAPTER 4.

Global Perturbation Methods for Detecting Chaotic Dynamics

4.1. The Three Basic Systems and Their Geometrical Structure

4.1a. System I

i) The Geometric Structure of the Unperturbed Phase Space

ii) Homoclinic Coordinates

iii) The Geometric Structure of the Perturbed Phase Space

iv) The Splitting of the Manifolds

4.1b. System II

i) The Geometric Structure of the Unperturbed Phase Space

ii) Homoclinic Coordinates

iii) The Geometric Structure of the Perturbed Phase Space

iv) The Splitting of the Manifolds

4.1c. System III

i) The Geometric Structure of the Unperturbed Phase Space

ii) Homoclinic Coordinates

iii) The Geometric Structure of the Perturbed Phase Space

iv) The Splitting of the Manifolds

v) Horseshoes and Arnold Diffusion

4.1d. Derivation of the Melnikov Vector

i) The Time Dependent Melnikov Vector

300

301

306

313

334

335

339

340

350

352

359

370

370

372

373

375

380

381

383

384

387

394

396

402

xiv

ii) An Ordinary Differential Equation for the Melnikov Vector

iii) Solution of the Ordinary Differential Equation

iv) The Choice of 5;,£ and 5;,£ v) Elimination of to

4.1e. Reduction to a Poincare Map

4.2. Examples

4.2a. Periodically Forced Single Degree of Freedom Systems

i) The Pendulum: Parametrically Forced at O(f) Amplitude,

o (1) Frequency

ii) The Pendulum: Parametrically Forced at 0(1) Amplitude,

O(f) Frequency

4.2.b. Slowly Varying Oscillators

i) The Duffing Oscillator with Weak Feedback Control

ii) The Whirling Pendulum

4.2c. Perturbations of Completely Integrable, Two Degree of Freedom

Hamiltonian System

i) A Coupled Pendulum and Harmonic Oscillator

ii) A Strongly Coupled Two Degree of Freedom System

Contents

404

406

414

416

417

418

418

419

426

429

430

440

452

452

455

4.2d. Perturbation of a Completely Integrable Three Degree of Freedom

System: Arnold Diffusion 458

4.2e. Quasiperiodically Forced Single Degree of Freedom Systems 460

i) The Duffing Oscillator: Forced at O(f) Amplitude with N 0(1)

Frequencies 461

ii) The Pendulum: Parametrically Forced at O(f) Amplitude, 0(1)

Frequency and 0(1) Amplitude, O(f) Frequency

4.3. Final Remarks

i) Heteroclinic Orbits

ii) Additional Applications of Melnikov's Method

iii) Exponentially Small Melnikov Functions

References

Index

468

470

470

470

471

477

489