Nonlinear Physical ScienceSeries Editors: Albert C.J. Luo · Nail H. Ibragimov

Discretization and Implicit Mapping Dynamics

Albert C.J. Luo

Nonlinear Physical Science

Nonlinear Physical Science

Nonlinear Physical Science focuses on recent advances of fundamental theories and principles,analytical and symbolic approaches, as well as computational techniques in nonlinear physicalscience and nonlinear mathematics with engineering applications.

Topics of interest in Nonlinear Physical Science include but are not limited to:

– New findings and discoveries in nonlinear physics and mathematics– Nonlinearity, complexity and mathematical structures in nonlinear physics– Nonlinear phenomena and observations in nature and engineering– Computational methods and theories in complex systems– Lie group analysis, new theories and principles in mathematical modeling– Stability, bifurcation, chaos and fractals in physical science and engineering– Nonlinear chemical and biological physics– Discontinuity, synchronization and natural complexity in the physical sciences

International Advisory Board

Ping Ao, University of Washington, USA; Email: aoping@u.washington.eduJan Awrejcewicz, The Technical University of Lodz, Poland; Email: awrejcew@p.lodz.plEugene Benilov, University of Limerick, Ireland; Email: Eugene.Benilov@ul.ieEshel Ben-Jacob, Tel Aviv University, Israel; Email: eshel@tamar.tau.ac.ilMaurice Courbage, Université Paris 7, France; Email: maurice.courbage@univ-paris-diderot.frMarian Gidea, Northeastern Illinois University, USA; Email: mgidea@neiu.eduJames A. Glazier, Indiana University, USA; Email: glazier@indiana.eduShijun Liao, Shanghai Jiaotong University, China; Email: sjliao@sjtu.edu.cnJose Antonio Tenreiro Machado, ISEP-Institute of Engineering of Porto, Portugal;Email: jtm@dee.isep.ipp.ptNikolai A. Magnitskii, Russian Academy of Sciences, Russia; Email: nmag@isa.ruJosep J. Masdemont, Universitat Politecnica de Catalunya (UPC), Spain;Email: josep@barquins.upc.eduDmitry E. Pelinovsky, McMaster University, Canada; Email: dmpeli@math.mcmaster.caSergey Prants, V.I.Il’ichev Pacific Oceanological Institute of the Russian Academy of Sciences,Russia; Email: prants@poi.dvo.ruVictor I. Shrira, Keele University, UK; Email: v.i.shrira@keele.ac.ukJian Qiao Sun, University of California, USA; Email: jqsun@ucmerced.eduAbdul-Majid Wazwaz, Saint Xavier University, USA; Email: wazwaz@sxu.eduPei Yu, The University of Western Ontario, Canada; Email: pyu@uwo.ca

More information about this series at http://www.springer.com/series/8389

Series editors

Albert C.J. LuoDepartment of Mechanical and IndustrialEngineeringSouthern Illinois University EdwardsvilleEdwardsville, IL 62026-1805, USAe-mail: aluo@siue.edu

Nail H. IbragimovDepartment of Mathematics and ScienceBlekinge Institute of TechnologyS-371 79 Karlskrona, Swedene-mail: nib@bth.se

Albert C.J. Luo

Discretization and ImplicitMapping Dynamics

123Higher EducationPress

Albert C.J. LuoDepartment of Mechanical and IndustrialEngineering

Southern Illinois University EdwardsvilleEdwardsville, ILUSA

ISSN 1867-8440 ISSN 1867-8459 (electronic)Nonlinear Physical ScienceISBN 978-3-662-47274-3 ISBN 978-3-662-47275-0 (eBook)DOI 10.1007/978-3-662-47275-0

Jointly published with Higher Education Press, BeijingISBN: 978-7-04-042835-3 Higher Education Press, Beijing

Library of Congress Control Number: 2015939425

Springer Heidelberg New York Dordrecht London© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2015This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publishers, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publishers nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

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Preface

This book discusses discretization of differential equations of continuous nonlinearsystems and implicit mapping dynamics of periodic flows to chaos. In recent years,approximate analytical solutions for periodic motions to chaos in continuousnonlinear systems were developed by the author through finite Fourier series.However, for many nonlinear dynamical systems, it is difficult to achieve suchapproximate analytical solutions of periodic motions to chaos. With computerextensive applications in numerical computations, one has used the discrete formsof differential equations of nonlinear systems to obtain numerical solutions viarecurrent iterations. The discrete forms in recurrent iterations will cause accumu-lated computational errors for numerical results. Once the iteration numberincreases, numerical results given by the discrete forms cannot approximatelyrepresent true solutions of nonlinear dynamical systems. To improve the compu-tational accuracy, one has tried to adopt implicit maps as discrete forms to achievenumerical results. However, such implicit mapping forms cannot be iterateddirectly, which cause the difficulty to extensive applications of discrete implicitmaps in continuous nonlinear systems. In this book, the author would like tosystematically discuss implicit mapping dynamics of periodic motions to chaos incontinuous dynamical systems, and discrete Fourier series based on the discretenodes of periodic motions will be used to obtain the harmonic responses in fre-quency space, which can be measured from experiments.

This book includes six chapters. In Chap. 1, a brief literature survey iscompleted. Chapter 2 reviewed the nonlinear theory for stability and bifurcation offixed points in discrete nonlinear systems. In Chap. 3, discretization of differentialequations is discussed comprehensively. The explicit and implicit discrete schemesin nonlinear dynamical system are discussed through one-step and multi-stepdiscretization of differential equations, and the corresponding stability and con-vergence of the explicit and implicit discrete maps are discussed. In Chap. 4,implicit mapping dynamics of period-m fixed points in discrete dynamical systemsare discussed with positive and negative discrete maps, and the complete solutionsof Ying-Yang states of period-m fixed points are presented. In Chap. 5, themethodology for the solutions of periodic motions in continuous dynamical systems

vii

with/without time delay is presented through the mapping dynamics of discreteimplicit mappings under specific truncated errors. The discrete Fourier series ofperiodic motions are discussed from discrete nodes of periodic motions, and thecorresponding approximate analytical expression can be obtained. Harmonicamplitude quantity levels can be analyzed for periodic motions in continuousnonlinear systems. Chapter 6 discusses the bifurcation trees of periodic motions tochaos in the Duffing oscillator to demonstrate the implicit mapping dynamics of thediscretized Duffing oscillator. Such semi-analytical results of periodic motions inthe Duffing oscillator are compared with the approximate analytical solutions ofperiodic motions based on the finite Fourier series solutions.

Finally, I would like to appreciate my former student, Dr. Yu Guo, for com-pleting all numerical computations. Herein, I thank my wife (Sherry X. Huang) andmy children (Yanyi Luo, Robin Ruo-Bing Luo, and Robert Zong-Yuan Luo) fortheir understanding and infinite support.

Albert C.J. Luo

viii Preface

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Book Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Nonlinear Discrete Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Fixed Points and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Stability Switching Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Bifurcation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Discretization of Continuous Systems . . . . . . . . . . . . . . . . . . . . . . 513.1 Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Basic Discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.1 Forward Euler’s Method . . . . . . . . . . . . . . . . . . . . . 553.2.2 Backward Euler’s Method . . . . . . . . . . . . . . . . . . . . 593.2.3 Trapezoidal Rule Discretization . . . . . . . . . . . . . . . . 633.2.4 Midpoint Method . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Introduction to Runge–Kutta Methods . . . . . . . . . . . . . . . . . . 723.3.1 Taylor Series Method . . . . . . . . . . . . . . . . . . . . . . . 733.3.2 Runge–Kutta Method of Order 2 . . . . . . . . . . . . . . . 78

3.4 Explicit Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . 813.4.1 Runge–Kutta Method of Order 3 . . . . . . . . . . . . . . . 893.4.2 Runge–Kutta Method of Order 4 . . . . . . . . . . . . . . . 96

3.5 Implicit Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . 1013.5.1 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . 1023.5.2 Implicit Runge–Kutta Methods . . . . . . . . . . . . . . . . . 1053.5.3 Gauss Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.5.4 Radau Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143.5.5 Lobatto Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

ix

3.5.6 Diagonally Implicit RK Methods . . . . . . . . . . . . . . . 1213.5.7 Stability of Implicit Runge–Kutta Methods. . . . . . . . . 124

3.6 Multi-step Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.6.1 Adams–Bashforth Methods . . . . . . . . . . . . . . . . . . . 1283.6.2 Adams–Moulton Methods . . . . . . . . . . . . . . . . . . . . 1373.6.3 Explicit Adams Methods . . . . . . . . . . . . . . . . . . . . . 1443.6.4 Implicit Adams Methods . . . . . . . . . . . . . . . . . . . . . 1463.6.5 General Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.7 Generalized Implicit Multi-step Methods . . . . . . . . . . . . . . . . 152References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4 Implicit Mapping Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.1 Single-Step Implicit Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.2 Discrete Systems with Multiple Maps . . . . . . . . . . . . . . . . . . 1704.3 Complete Dynamics of a Henon Map System. . . . . . . . . . . . . 1744.4 Multi-step Implicit Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 181References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

5 Periodic Flows in Continuous Systems . . . . . . . . . . . . . . . . . . . . . 1995.1 Continuous Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . 1995.2 Continuous Time-Delay Systems . . . . . . . . . . . . . . . . . . . . . 218

5.2.1 Interpolated Time-Delay Nodes . . . . . . . . . . . . . . . . 2185.2.2 Integrated Time-Delay Nodes . . . . . . . . . . . . . . . . . . 242

5.3 Discrete Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

6 Periodic Motions to Chaos in Duffing Oscillator. . . . . . . . . . . . . . 2816.1 Period-1 Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2816.2 Period-m Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2856.3 Bifurcation Trees of Periodic Motions . . . . . . . . . . . . . . . . . . 2876.4 Frequency–Amplitude Characteristics . . . . . . . . . . . . . . . . . . 294

6.4.1 Period-1 Motions to Chaos. . . . . . . . . . . . . . . . . . . . 2966.4.2 Period-3 Motions . . . . . . . . . . . . . . . . . . . . . . . . . . 299

6.5 Numerical Simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

x Contents

Chapter 1Introduction

For solutions of periodic motions in nonlinear dynamical systems, analytical andnumerical techniques have been adopted. The analytical methods include themethod of averaging, perturbation methods, harmonic balance method, and gen-eralized harmonic balance method. Through the analytical methods, one can obtainthe analytical expressions of approximate solutions of periodic motions indynamical systems. The numerical methods are based on discrete maps obtained bydiscretization of differential equations for dynamical systems. The discrete mapsinclude explicit and implicit maps. The explicit maps can be directly used to obtainnumerical solutions of differential equations for dynamical systems, but the com-putational errors for the recurrence iteration of explicit maps will be accumulated innumerical results. Once the recurrence iteration times become large, the numericalresults may not be adequate for numerical solutions of dynamical systems. Herein,implicit maps will be used to develop mapping structures for periodic motions. Theimplicit maps cannot be simply used by the recurrence iteration. For periodic flowsin nonlinear dynamics, mapping structures based on implicit maps can be devel-oped. Of course, an explicit mapping can be expressed by an implicit map as aspecial case. Based on the mapping structures, analytical prediction of periodicflows in nonlinear dynamical systems can be completed. The mapping structuregives a set of nonlinear algebraic equations, which can be solved. Without therecurrence iteration, the solution errors of node points of periodic flows are fixedwithout computational errors caused by iterations. The purpose of this book is todevelop a semi-analytical method for periodic flows to chaos in nonlineardynamical systems with/without time delay through implicit mapping structures.

1.1 A Brief History

To determine periodic flows in nonlinear dynamical systems, existing techniquesfor periodic motions in nonlinear systems are reviewed briefly. The analyticalmethods for periodic motions are discussed first. Lagrange (1788) developed themethod of averaging for periodic motions in the three-body problem as a pertur-bation of the two-body problem. The idea is based on the solutions of linear

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2015A.C.J. Luo, Discretization and Implicit Mapping Dynamics,Nonlinear Physical Science, DOI 10.1007/978-3-662-47275-0_1

1

systems. Such an idea was further extended by Poincare in the end of the nineteenthcentury. Thus, Poincare (1899) developed the perturbation theory for motions ofcelestial bodies. van der Pol (1920) used the method of averaging for the periodicsolutions of oscillation systems in circuits. Such an application caused great interestin the perturbation theory for the approximate analytical solution of periodicmotions in nonlinear systems. Until 1928, the asymptotic validity of the method ofaveraging was not proved. Fatou (1928) gave the proof of the asymptotic validitythrough the solution existence theorems of differential equations. Krylov andBogoliubov (1935) further developed the method of averaging, and the detailedpresentation was given in Bogoliubov and Mitropolsky (1961). Hayashi (1964)presented the perturbation methods including averaging method and principle ofharmonic balance. Barkham and Soudack (1969) extended the Krylov–Bogoliubovmethod for the approximate solutions of nonlinear autonomous second-order dif-ferential equations [also see, Barkham and Soudack (1970)]. Nayfeh (1973)employed the multiple-scale perturbation method to develop approximate solutionsof periodic motions in the Duffing oscillators. Holmes and Rand (1976) discussedthe stability and bifurcation of periodic motions in the Duffing oscillator. Nayfehand Mook (1979) used the perturbation method to investigate nonlinear structuralvibrations, and Holmes (1979) demonstrated chaotic motions in nonlinear oscilla-tors through the Duffing oscillator with a twin-well potential. Ueda (1980)numerically simulated chaos by period-doubling of periodic motions of Duffingoscillators. A generalized harmonic balance approach was used by Garcia-Margalloand Bejarano (1987) to determine approximate solutions of nonlinear oscillationswith strong nonlinearity. Rand and Armbruster (1987) determined the stability ofperiodic solutions by the perturbation method and bifurcation theory. Yuste andBejarano (1989) employed the elliptic functions instead of trigonometric functionsto extend the Krylov–Bogoliubov method. Coppola and Rand (1990) used theaveraging method with elliptic functions for approximation of limit cycle. Wanget al. (1992) used the harmonic balance method and the Floquet theory to inves-tigate the nonlinear behaviors of the Duffing oscillator with a bounded potentialwell [also see, Kuo et al. (1992)]. Luo and Han (1997) determined the stability andbifurcation conditions of periodic motions of the Duffing oscillator. However, onlysymmetric periodic motions of the Duffing oscillators were investigated. Luo andHan (1999) investigated the analytical prediction of chaos in nonlinear rods throughthe Duffing oscillator. Peng et al. (2008) presented the approximate symmetricsolution of period-1 motions in the Duffing oscillator by the harmonic balancemethod with three harmonic terms. Luo (2012a) developed a generalized harmonicbalance method for the approximate analytical solutions of periodic motions andchaos in nonlinear dynamical systems. This method used the finite-term Fourierseries to approximately express periodic motions, and the coefficients are time-varying. With averaging, a dynamical system of coefficients is obtained, and fromsuch a dynamical system, the approximate solutions of periodic motions areachieved and the corresponding stability and bifurcation analysis are completed.Luo and Huang (2012a) used such a generalized harmonic balance method withfinite terms for the analytical solutions of period-1 motions in the Duffing oscillator

2 1 Introduction

with a twin-well potential. Luo and Huang (2012b) also employed a generalizedharmonic balance method to find analytical solutions of period-m motions in such aDuffing oscillator. The analytical bifurcation trees of periodic motions in theDuffing oscillator to chaos were obtained [also see, (Luo and Huang 2012c, d,2013a, b, c, d)]. Such analytical bifurcation trees show the connection from periodicsolution to chaos analytically. For a better understanding of nonlinear behaviors innonlinear dynamical systems, analytical bifurcation trees of period-1 motions tochaos in a periodically forced oscillator with quadratic nonlinearity were presentedin Luo and Yu (2013a, b, 2015), and period-m motions in the periodically forcedvan der Pol equation were presented in Luo and Laken (2013). The analyticalsolutions of periodic oscillations in the van der Pol oscillator can be used to verifythe conclusions in Cartwright and Littlewood (1947) and Levinson (1948). Theresults for the parametric quadratic nonlinear oscillator in Luo and Yu (2014)analytically show the complicated period-1 motions and the corresponding bifur-cation structures. The detailed presentation for analytical methods for periodic flowsin nonlinear dynamical systems can be found in Luo (2014a, b).

In recent years, time-delayed systems are of great interest since such systemsextensively exist in engineering (e.g., Tlusty 2000; Hu and Wang 2002). Theinfinite dimensional state space causes the significant difficulty to solve such time-delayed problems. Thus, one used numerical methods for the corresponding com-plicated behaviors. On the other hand, one is interested in the stability and bifur-cation of equilibriums of the time-delayed systems (e.g., Stepan 1989; Sun 2009;Insperger and Stepan 2011). In addition, one is also interested in analytical solu-tions of periodic motions in time-delayed dynamical systems. Perturbation methodshave been used for such periodic motions in delayed dynamical systems. Forinstance, the approximate solutions of the time-delayed nonlinear oscillator wereinvestigated by the method of multiple scales (e.g., Hu et al. 1998; Wang and Hu2006). The harmonic balance method was also used to determine approximatesolutions of periodic motions in delayed nonlinear oscillators [e.g., MacDonald(1995); Liu and Kalmar-Nagy (2010); Lueng and Guo (2014)]. However, suchapproximate solutions of periodic motions in the time-delayed oscillators are basedon one or two harmonic terms, which are not accurate enough. In addition, thecorresponding stability and bifurcation analysis of such approximate solutions ofperiodic motions may not be adequate. Luo (2013) presented an alternative way forthe accurate analytical solutions of periodic flows in time-delayed dynamical sys-tems (see also, Luo 2014c). This method is without any small-parameter require-ment. In addition, this approach can also be applied to the coefficient varying withtime. Luo and Jin (2014a) analytically presented the bifurcation tree of period-1motions to chaos in a periodically forced, time-delayed quadratic nonlinear oscil-lator. Luo and Jin (2014b, c, d) discussed the bifurcation trees of period-m motionsto chaos in the periodically forced Duffing oscillator with a linear time-delayeddisplacement.

From the literature survey, for some simple nonlinear systems, the approximateanalytical solutions of periodic motions can be obtained. However, for most of thenonlinear dynamical systems, it is very difficult to obtain analytical solutions of

1.1 A Brief History 3

periodic motions. Thus, numerical results of periodic motions in complicatednonlinear dynamical systems become very significant in engineering. In fact,human being has a long history as old as human civilization to use numericalalgorithms to get approximate numerical results instead of exact results. Forinstance, the Rhind Papyrus of ancient Egypt describes a root-finding method forsolving a simple equation in about 1650 BC, and Archimedes of Syracuse (287–212BC) used numerical algorithm to approximately compute lengths, areas, and vol-umes of geometric figures. Based on the ideas and spirits of numerical approxi-mations, Isaac Newton and Gottfried Leibnitz developed the calculus byinfinitesimal elements to linear approximation and infinitesimal summarization tointegration. Because of calculus development, one can describe more complicatedmathematical models for real physical problems, but it is very difficult to solve suchaccurate mathematical models explicitly. This is an important impetus for one todevelop numerical methods to get approximate solutions of the accurate mathe-matical models. Thus, Newton developed several numerical methods to findapproximate solutions. For instance, numerical methods for root-finding andpolynomial interpolation were developed by Newton. Since then, Euler (1707–1783), Lagrange (1736–1813), and Gauss (1777–1855) further developed numer-ical methods for approximate results, such as Euler method for differential equa-tions, Lagrange interpolation method, and Gauss interpolation. The more detailedinformation about numerical methods can be found in Goldestine (1977).

This book will focus on numerical methods for nonlinear dynamical systems. Forthis issue, Euler developed an explicit method to achieve approximate solutionsnumerically. Such Euler method is a one-step discrete method. This method is stillused in numerical computation, but its computational accuracy is very low, andnumerical solutions are not accurate. Bashforth and Adams (1883) presented a multi-step discrete method for numerical solutions of differential equations.Moulton (1926)extended such a method to the Adams–Moulton method. The Adams–Bashforthmethod is the explicit method as a predictor, and the Adams–Moulton method is theimplicit method as a corrector. In addition, the Adams–Bashforth method can beextended for the practical application of the Taylor series method as presented inNordsieck (1962). Milne (1949) used the entire interval for integration based onNewton–Cotes quadrature formulas. The recent theory of linear multi-step methodwas systematically discussed by Dahlquist (1956, 1959). The general formulas werepresented, and the corresponding consistency, stability, and convergence were dis-cussed by the linear stability theory. Runge (1895) started modern one-step methodswith the order of two and three for numerical solutions of differential equations. Heun(1900) raised the order of themethod from two and three to four. Kutta (1901) gave theformulation of the method with the order conditions. Nystrom (1925) made the cor-rection of the fifth-order method of Kutta and showed how to apply the Runge–Kuttamethod to the second-order differential equations. Butcher (1963) discussed thecoefficients of Runge–Kutta method, and the implicit Runge–Kutta methods werepresented in Butcher (1964, 1975).

With extensive applications of computers, numerical computations become verypopular to obtain numerical results for differential equations through discretization.

4 1 Introduction

Once the discrete maps are obtained for dynamical systems, discrete dynamicalsystems can be used to investigate nonlinear dynamics of dynamical systems. Basedon nonlinear maps, one discovered the existence of chaotic motions in nonlineardynamical systems through iteration of discrete maps.

In 2005, Luo (2005a, b) presented a mapping dynamics of discrete dynamicalsystems which is a more generalized symbolic dynamics. The systematicaldescription of mapping dynamics in discontinuous dynamical systems was pre-sented in Luo (2009). The discrete maps can be any implicit and/or explicit func-tions rather than explicit maps in numerical iterative methods only. From discretemapping structures, periodic motions in discrete dynamical systems can be pre-dicted analytically, and the stability and bifurcation analysis of periodic motions innonlinear dynamical systems can be completed. Such an idea was applied to dis-continuous dynamical systems in Luo (2009, 2012b, c).

1.2 Book Layout

The main body in this book will discuss discretization of differential equations ofnonlinear continuous dynamical systems to obtain implicit maps for periodic flows.The mapping structures will be employed to analytically predict the periodic flowsin nonlinear continuous systems, and the corresponding stability and bifurcationcan be discussed.

In Chap. 2, a theory for nonlinear discrete systems is reviewed. The local andglobal theories of stability and bifurcation for nonlinear discrete systems are dis-cussed. The stability switching and bifurcation on specific eigenvectors of thelinearized system at fixed points under a specific period are presented. The higherorder singularity and stability for nonlinear discrete systems on the specificeigenvectors are discussed.

In Chap. 3, the discretization of continuous systems is presented. The explicitand implicit discrete maps are discussed for numerical predictions of continuoussystems. Basic discrete schemes are presented which include forward and backwardEuler methods, and midpoint and trapezoidal rule methods. An introduction toRunge–Kutta methods is presented, and the Taylor series method and second-orderRunge–Kutta method are introduced. The explicit Runge–Kutta methods for thirdand fourth order are systematically presented. The implicit Runge–Kutta methodsare discussed based on the polynomial interpolation, which include a generalizedimplicit Runge–Kutta method, Guass method, Radau method, and Lotta methods.In addition to one-step methods, implicit and explicit multi-step methods are dis-cussed, including Adams–Bashforth method, Adams–Moulton methods, andexplicit and implicit Adams methods.

In Chap. 4 presented is a Ying–Yang theory for implicit, discrete, nonlinearsystems with consideration of positive and negative iterations of discrete iterativemaps. In existing analysis, the solutions relative to “Yang” in nonlinear dynamicalsystems are extensively investigated. However, the solutions pertaining to “Ying”

1.1 A Brief History 5

in nonlinear dynamical systems are not discussed too much. A set of concepts on“Ying” and “Yang” in implicit, nonlinear, discrete dynamical systems are intro-duced. Based on the Ying–Yang theory, the complete dynamics of implicit discretesystems can be discussed. A discrete dynamical system with the Henon map isinvestigated as an example. Period-m solutions, stability, and bifurcations for multi-step, implicit discrete systems are discussed.

In Chap. 5, periodic flows in continuous nonlinear systems are discussed throughdiscrete implicit mappings. The period-1 flows in nonlinear systems are discussedby the one-step discrete maps, and then, the period-m flows in nonlinear dynamicalsystems are also discussed through the one-step discrete maps. Multi-step, implicitdiscrete maps are employed to discuss the period-1 and period-m motions innonlinear dynamical systems. Periodic flows in nonlinear time-delayed dynamicalsystems are discussed with time-delay discrete nodes interpolated by two non-delaydiscrete nodes. In addition, periodic flows in time-delayed nonlinear dynamicalsystems are also discussed through the delay nodes determined by integration.Through the discrete nodes in periodic flows, the periodic flows are approximatedby the discrete Fourier series and the frequency space of the periodic flows can bedetermined through amplitude spectrums.

In Chap. 6, periodic motions in the Duffing oscillator are discussed through themapping structures of discrete implicit maps. The discrete implicit maps areobtained from the differential equation of the Duffing oscillator. From mappingstructures, bifurcation trees of periodic motions are predicted analytically throughnonlinear algebraic equations of implicit maps, and the corresponding stability andbifurcation analysis of periodic motions in the bifurcation trees are presented. Thebifurcation trees of periodic motions are also presented through the harmonicamplitudes of the discrete Fourier series. Finally, from the analytical prediction,numerical simulation results of periodic motions are performed to verify the ana-lytical prediction. The harmonic amplitude spectrums are also presented, and thecorresponding analytical expression of periodic motions can be obtainedapproximately.

References

Barkham, P. G. D., & Soudack, A. C. (1969). An extension to the method of Krylov andBogoliubov. International Journal of Control, 10, 377–392.

Barkham, P. G. D., & Soudack, A. C. (1970). Approximate solutions of nonlinear, non-autonomoussecond-order differential equations. International Journal of Control, 11, 763–767.

Bashforth, F., & Adams, J. C. (1883). Theories of capillary action. Cambridge University Press:London.

Bogolyubov, N. N., & Mitropolsky, Y. A. (1961). Asymptotic methods in the theory of non-linearoscillations. New York: Gordon and Breach.

Butcher, J. C. (1963). Coefficients for the study of Rung-Kutta integration processes. Journal ofAustralian Mathematical Society, 3, 185–201.

Butcher, J. C. (1964). Implicit Runge-Kutta processes. Mathematics of Computation, 19, 408–417.

6 1 Introduction

Butcher, J. C. (1975). A stability property of implicit Runge-Kutta methods. BIT NumericalMathematics, 15, 358–361.

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Krylov, N. M., & Bogolyubov, N. N. (1935). Methodes approchees de la mecanique non-lineairedans leurs application a l’Aeetude de la perturbation des mouvements periodiques de diversphenomenes de resonance s’y rapportant, Academie des Sciences d’Ukraine:Kiev (in French).

Kutta, W. (1901). Beitrag zur naherungsweisen intergration totaler differentialgleichungen. Z MathPhys, 46, 435–453.

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oscillators. Journal of Vibration and Control, 20, 501–517.Levinson, N. (1948). A simple second order differential equation with singular motions. In:

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8 1 Introduction

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References 9

Chapter 2Nonlinear Discrete Systems

In this chapter, a theory for nonlinear discrete systems is reviewed. The local andglobal theory of stability and bifurcation for nonlinear discrete systems is presented.The stability switching and bifurcation on specific eigenvectors of the linearizedsystem at fixed points under a specific period are discussed. The higher-ordersingularity and stability for nonlinear discrete systems on the specific eigenvectorsare also presented.

2.1 Definitions

Definition 2.1 For Xa � Rn and K � Rm with a 2 Z, consider a vector functionfa : Xa � K ! Xa which is Cr (r ≥ 1)-continuous, and there is a discrete (or dif-ference) equation in a form of

xkþ1 ¼ faðxk; paÞ for xk; xkþ1 2 Xa; k 2 Z and pa 2 K ð2:1Þ

with an initial condition of xk ¼ x0, the solution of Eq. (2.1) is given by

xk ¼ faðfað. . .ðfa|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}k

ðx0; paÞÞÞÞ

for xk 2 Xa; k 2 Z and p 2 K:

ð2:2Þ

(i) The difference equation with the initial condition is called a discretedynamical system.

(ii) The vector function faðxk;paÞ is called a discrete vector field on Xa.(iii) The solution xk for each k 2 Z is called a flow of discrete system.(iv) The solution xk for all k 2 Z on domain Xa is called the trajectory, phase

curve, or orbit of the discrete dynamical system, which is defined as

C ¼ xkjxkþ1 ¼ faðxk; paÞ for k 2 Z and pa 2 Kf g � [aXa: ð2:3Þ

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2015A.C.J. Luo, Discretization and Implicit Mapping Dynamics,Nonlinear Physical Science, DOI 10.1007/978-3-662-47275-0_2

11

(v) The discrete dynamical system is called a uniform discrete system if

xkþ1 ¼ faðxk; paÞ ¼ fðxk; pÞ for k 2 Z and xk 2 Xa: ð2:4ÞOtherwise, this discrete dynamical system is called a non-uniform discrete

system.

Definition 2.2 For the discrete dynamical system in Eq. (2.1), the relation betweenstate xk and state xkþ1 (k 2 Z) is called a discrete map if

Pa : xk ����!fa xkþ1 and xkþ1 ¼ Paxk ð2:5Þ

with the following properties:

Pðk;lÞ : xk ������!fa1 ;fa2 ;...;fal xkþl and xkþl ¼ Pal � Pal�1 � � � � � Pa1xk ð2:6Þ

where

Pðk;lÞ ¼ Pal � Pal�1 � � � � � Pa1 : ð2:7Þ

If Pal ¼ Pal�1 ¼ � � � ¼ Pa1 ¼ Pa, then

Pða;lÞ � PðlÞa ¼ Pa � Pa � � � � � Pa ð2:8Þ

with

PðnÞa ¼ Pa � Pðn�1Þ

a and Pð0Þa ¼ I: ð2:9Þ

The total map with l-different sub-maps is shown in Fig. 2.1. The map Pak withthe relation function fak (ak 2 Z) is given by Eq. (2.5). The total map Pðk;lÞ is givenin Eq. (2.7). The domains Xak (ak 2 Z) can fully overlap each other or can becompletely separated without any intersection.

Definition 2.3 For a vector function in fa 2 Rn; fa : Rn ! Rn. The operator normof fa is defined by

fak k ¼Xni¼1

maxxkk k� 1;pa

jfaðiÞðxk; paÞj: ð2:10Þ

For an n� n matrix faðxk; paÞ ¼ Aaxk and Aa ¼ ðaijÞn�n, the corresponding normis defined by

Aak k ¼Xni;j¼1

aij�� ��: ð2:11Þ

12 2 Nonlinear Discrete Systems

Definition 2.4 For Xa � Rn and K � Rm with a 2 Z, the vector functionfaðxk; paÞ with fa : Xa � K ! Rn is differentiable at xk 2 Xa if

@faðxk; paÞ@xk

����ðxk ;pÞ

¼ limDxk!0

faðxk þ Dxk; paÞ � faðxk; paÞDxk

: ð2:12Þ

@fa=@xk is called the spatial derivative of faðxk; paÞ at xk, and the derivative isgiven by the Jacobian matrix

@faðxk; paÞ@xk

¼ @faðiÞ@xkðjÞ

� �n�n

: ð2:13Þ

Definition 2.5 For Xa � Rn and K � Rm, consider a vector function fðxk; pÞ withf : Xa � K ! Rn where xk 2 Xa and p 2 K with k 2 Z. The vector functionfðxk; pÞ is said to satisfy the Lipschitz condition if

fðyk; pÞ � fðxk; pÞk � Lk kyk � xkk ð2:14Þ

with xk; yk 2 Xa and L a constant. The constant L is called the Lipschitz constant.

2.2 Fixed Points and Stability

Definition 2.6 Consider a discrete, dynamical system xkþ1 ¼ faðxk; paÞ inEq. (2.4).

(i) A point xk 2 Xa is called a fixed point or period-1 solution of a discretenonlinear system xkþ1 ¼ faðxk; paÞ under a map Pa if for xkþ1 ¼ xk ¼ xk

xk ¼ faðxk ; pÞ: ð2:15Þ

The linearized system of the nonlinear discrete system xkþ1 ¼ faðxk; paÞ inEq. (2.4) at the fixed point xk is given by

1kα +Ω

kαΩ

kx 1k+xkαf

1αΩ2αΩ

1αf

lαΩ1Pα

kPα

Fig. 2.1 Maps and vector functions on each sub-domain for discrete dynamical system

2.1 Definitions 13

ykþ1 ¼ DPaðxk ; pÞyk ¼ Dfaðxk ; pÞyk ð2:16Þ

where

yk ¼ xk � xk and ykþ1 ¼ xkþ1 � xkþ1: ð2:17Þ

(ii) A set of points xj 2 Xaj ðaj 2 ZÞ is called the fixed point set or period-1 pointset of the total map Pðk;lÞ with l-different sub-maps in nonlinear discrete systemof Eq. (2.5) if

xkþjþ1 ¼ faj0 ðxkþj; paj0 Þ for j 2 Zþ and j0 ¼ modðj; lÞ þ 1;

xkþmodðj;lÞ ¼ xk :ð2:18Þ

The linearized equation of the total map Pðk;lÞ gives

ykþjþ1 ¼ DPaj0 ðxkþj; paj0 Þykþj ¼ Dfaj0 ðxkþj; paj0 Þykþj with

ykþjþ1 ¼ xkþjþ1 � xkþjþ1 and ykþj ¼ xkþj � xkþj for

j 2 Zþ and j0 ¼ modðj; lÞ þ 1:

ð2:19Þ

The resultant equation for each individual map is

ykþjþ1 ¼ DPðk;lÞðxk ; pÞykþj for j 2 Zþ ð2:20Þ

where

DPðk;nÞðxk ; pÞ ¼Y1

j¼lDPajðxkþj�1; pÞ

¼ DPalðxkþl�1; panÞ � � � � � DPa2ðxkþ1; pa2Þ � DPa1ðxk ; pa1Þ¼ DfðalÞðxkþl�1; panÞ � � � � � Dfða2Þðxkþ1; pa2Þ � Dfða1Þðxk ; pa1Þ:

ð2:21Þ

The fixed point xk lies in the intersected set of two domains Xk and Xkþ1, asshown in Fig. 2.2. In the vicinity of the fixed point xk , the incremental relations inthe two domains Xk and Xkþ1 are different. In other words, setting yk ¼ xk � xk andykþ1 ¼ xkþ1 � xkþ1, the corresponding linearization is generated as in Eq. (2.16).Similarly, the fixed point of the total map with n-different sub-maps requires theintersection set of two domains Xk and Xkþn, and there are a set of equations toobtain the fixed points from Eq. (2.18). The other values of fixed points lie indifferent domains, i.e., xj 2 Xj ðj ¼ k þ 1; k þ 2; . . .; k þ n� 1Þ, as shown inFig. 2.3.

The corresponding linearized equations are given in Eq. (2.19). From Eq. (2.20),the local characteristics of the total map can be discussed as a single map. Thus, the

14 2 Nonlinear Discrete Systems

dynamical characteristics for the fixed point of the single map will be discussedcomprehensively, and the fixed points for resultant map are applicable. The resultscan be extended to any period-m flows with PðmÞ.

Definition 2.7 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) with a fixed point xk . The linearized system of the discrete nonlinearsystem in the neighborhood of xk is ykþ1 ¼ Dfðxk ; pÞyk (yl ¼ xl � xk andl ¼ k; k þ 1) in Eq. (2.16). The matrix Dfðxk ; pÞ possesses n1 real eigenvaluesjkjj\1 (j 2 N1), n2 real eigenvalues jkjj[ 1 (j 2 N2), n3 real eigenvalues kj ¼ 1(j 2 N3), and n4 real eigenvalues kj ¼ �1 (j 2 N4). N ¼ f1; 2; . . .; ng and Ni ¼fi1; i2; . . .; inig [Ø ði ¼ 1; 2; 3; 4Þ with im 2 N (m ¼ 1; 2; . . .; ni) and R4

i¼1ni ¼ n.Ni � N [ Ø; [4

i¼1Ni ¼ N; Ni \ Np ¼ Ø ðp 6¼ iÞ. Ni ¼ Ø if ni = 0. The corre-sponding eigenvectors for contraction, expansion, invariance, and flip oscillationare fvjg (j 2 Ni) (i ¼ 1; 2; 3; 4), respectively. The stable, unstable, invariant, andflip subspaces of ykþ1 ¼ Dfðxk ; pÞyk in Eq. (2.16) are linear subspace spanned byfvjg (j 2 Ni) (i ¼ 1; 2; 3; 4), respectively, i.e.,

kx 1k+x

αf

k∗x

αΩ

Fig. 2.2 A fixed point between domains Xk and Xkþ1 for a discrete dynamical system

1jα +Ω

jαΩ

k j+x 1k j+ +xjαf

1αΩ2αΩ

1αf

lαΩ

1Pα

jPα

lαf

lPα

k l+x

kx

Fig. 2.3 Fixed points with l-maps for discrete dynamical system

2.2 Fixed Points and Stability 15

Es ¼ span vjðDfðxk ; pÞ � kjIÞvj ¼ 0;

jkjj\1; j 2 N1 � N [Ø

����� �

;

Eu ¼ span vjðDfðxk ; pÞ � kjIÞvj ¼ 0;

kj�� ��[ 1; j 2 N2 � N [Ø

����( )

;

Ei ¼ span vjðDfðxk ; pÞ � kjIÞvj ¼ 0;

kj ¼ 1; j 2 N3 � N [Ø

����� �

;

Ef ¼ span vjðDfðxk ; pÞ � kjIÞvj ¼ 0;

kj ¼ �1; j 2 N4 � N [Ø

����� �

ð2:22Þ

where

Es ¼ Esm [ Es

o [ Esz with

Esm ¼ span vj

ðDfðxk ; pÞ � kjIÞvj ¼ 0;

0\kj\1; j 2 Nm1 � N [Ø

����� �

;

Eso ¼ span vj

ðDfðxk ; pÞ � kjIÞvj ¼ 0;

�1\kj\0; j 2 No1 � N [Ø

����� �

;

Esz ¼ span vj

ðDfðxk ; pÞ � kjIÞvj ¼ 0;

kj ¼ 0; j 2 Nz1 � N [Ø

����� �

;

ð2:23Þ

Eu ¼ Eum [ Eu

o with

Eum ¼ span vj

ðDfðxk ; pÞ � kjIÞvj ¼ 0;

kj [ 1; j 2 Nm2 � N [Ø

����� �

;

Euo ¼ span vj

ðDfðxk ; pÞ � kjIÞvj ¼ 0;

�1[ kj; j 2 No2 � N [Ø

����� �

:

ð2:24Þ

Herein, subscripts “m” and “o” represent the monotonic and oscillatory evolutions.

Definition 2.8 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) with a fixed point xk . The linearized system of the discrete nonlinearsystem in the neighborhood of xk is ykþ1 ¼ Dfðxk ; pÞyk (yl ¼ xl � xk andl ¼ k; k þ 1) in Eq. (2.16). The matrix Dfðxk ; pÞ has complex eigenvalues aj ibjwith eigenvectors uj ivj (j 2 f1; 2; . . .; ng), and the base of vector is

B ¼ u1; v1; . . .; uj; vj; . . .; un; vn

: ð2:25Þ

The stable, unstable, center subspaces of ykþ1 ¼ Dfkðxk ; pÞyk in Eq. (2.16) are linearsubspaces spanned by fuj; vjg(j 2 Ni, i ¼ 1; 2; 3), respectively. Set N ¼f1; 2; . . .; ng plus Ni ¼ fi1; i2; . . .; inig [Ø � N [Ø with im 2 N (m ¼ 1; 2; . . .; ni)and R4

i¼1ni ¼ n. [4i¼1Ni ¼ N with Ni \ Np ¼ Ø(p 6¼ i). Ni ¼ Ø if ni ¼ 0: The stable,

unstable, center subspaces of ykþ1 ¼ Dfðxk ; pÞyk in Eq. (2.16) are defined by

16 2 Nonlinear Discrete Systems

Es ¼ span ðuj; vjÞrj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2j þ b2j

q\1;

ðDfðxk ; pÞ � ðaj ibjÞIÞðuj ivjÞ ¼ 0;

j 2 N1 � f1; 2; . . .; ng [Ø

��������

8>><>>:

9>>=>>;;

Eu ¼ span ðuj; vjÞrj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2j þ b2j

q[ 1;

ðDfðxk ; pÞ � ðaj ibjÞIÞðuj ivjÞ ¼ 0;

j 2 N2 � f1; 2; . . .; ng [Ø

��������

8>><>>:

9>>=>>;;

Ec ¼ span ðuj; vjÞrj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2j þ b2j

q¼ 1;

ðDfðxk ; pÞ � ðaj ibjÞIÞðuj ivjÞ ¼ 0;

j 2 N3 � f1; 2; . . .; ng [Ø

��������

8>><>>:

9>>=>>;:

ð2:26Þ

Definition 2.9 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) with a fixed point xk . The linearized system of the discrete nonlinearsystem in the neighborhood of xk is ykþ1 ¼ Dfðxk ; pÞyk (yl ¼ xl � xk andl ¼ k; k þ 1) in Eq. (2.16). The fixed point or period-1 point is hyperbolic if no anyeigenvalues of Dfðxk ; pÞ are on the unit circle (i.e., jkij 6¼ 1 for i ¼ 1; 2; . . .; n).

Theorem 2.1 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) with a fixed point xk . The linearized system of the discrete nonlinearsystem in the neighborhood of xk is ykþ1 ¼ Dfðxk ; pÞyk (yj ¼ xj � xk andj ¼ k; k þ 1) in Eq. (2.16). The eigenspace of Dfðxk ; pÞ (i.e., E � Rn) in the lin-earized dynamical system is expressed by direct sum of three subspaces

E ¼ Es � Eu � Ec ð2:27Þ

where Es;Eu and Ec are the stable, unstable, and center subspaces, respectively.

Proof The proof can be referred to Luo (2011). h

Definition 2.10 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) with a fixed point xk . Suppose there is a neighborhood of the equilibriumxk as UkðxkÞ � Xk , and in the neighborhood,

limjjyk jj!0

jjfðxk þ yk; pÞ � Dfðxk ; pÞykjjjjykjj

¼ 0; ð2:28Þ

and

ykþ1 ¼ Dfðxk ; pÞyk: ð2:29Þ

2.2 Fixed Points and Stability 17

(i) A Cr invariant manifold

Slocðxk; xkÞ ¼ fxk 2 UkðxkÞj limj!þ1

xkþj ¼ xk and

xkþj 2 UkðxkÞ with j 2 Zþgð2:30Þ

is called the local stable manifold of xk , and the corresponding global stablemanifold is defined as

Sðxk; xkÞ ¼ [j2Z� fðSlocðxkþj; xkþjÞÞ ¼ [j2Z� fðjÞðSlocðxk; xkÞÞ: ð2:31Þ

(ii) A Cr invariant manifold Ulocðxk; xkÞ

Ulocðxk; xkÞ ¼ fxk 2 UkðxkÞj limj!�1

xkþj ¼ xk and

xkþj 2 UkðxkÞ with j 2 Z�gð2:32Þ

is called the local unstable manifold of x, and the corresponding globalunstable manifold is defined as

Uðxk; xkÞ ¼ [j2Zþ fðUlocðxkþj; xkþjÞÞ ¼ [j2Zþ fðjÞðUlocðxk; xkÞÞ: ð2:33Þ

(iii) A Cr�1 invariant manifold Clocðx; xÞ is called the center manifold of x ifClocðx; xÞ possesses the same dimension of Ec for x 2 Clocðx; xÞ, and thetangential space of Clocðx; xÞ is identical to Ec.

As in continuous dynamical systems, the stable and unstable manifolds areunique, but the center manifold is not unique. If the nonlinear vector field f isC1-continuous, then a Cr center manifold can be found for any r\1.

Theorem 2.2 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) with a hyperbolic fixed point xk . The corresponding solution is xkþj ¼fðxkþj�1; pÞ with j 2 Z. Suppose there is a neighborhood of the hyperbolic fixedpoint xk (i.e., UkðxkÞ � Xa), and fðxk; pÞ is Cr (r 1)-continuous in UkðxkÞ. Thelinearized system is ykþjþ1 ¼ Dfðxk ; pÞykþj (ykþj ¼ xkþj � xk ) in UkðxkÞ. If thehomeomorphism between the local invariant subspace EðxkÞ � UkðxkÞ andthe eigenspace E of the linearized system exists with the condition in Eq. (2.28), thelocal invariant subspace is decomposed by

Eðxk; xkÞ ¼ Slocðxk; xkÞ �Ulocðxk; xkÞ: ð2:34Þ(a) The local stable invariant manifold Slocðx; xÞ possesses the following

properties:

(i) for xk 2 Slocðxk; xkÞ; Slocðxk; xkÞ possesses the same dimension of Es

and the tangential space of Slocðxk; xkÞ is identical to Es;

18 2 Nonlinear Discrete Systems

(ii) for xk 2 Slocðxk; xkÞ; xkþj 2 Slocðxk; xkÞ and limj!1

xkþj ¼ xk for all j 2 Zþ;

(iii) For xk 62 Slocðxk; xkÞ; jjxkþj � xk jj d for d[ 0 with j; j1 2 Zþ andj j1 0:

(b) The local unstable invariant manifold Ulocðxk; xkÞ possesses the followingproperties:

(i) for xk 2 Ulocðxk; xkÞ; Ulocðxk; xkÞ possesses the same dimension of Eu

and the tangential space of Ulocðxk; xkÞ is identical to Eu;(ii) for xk 2 Ulocðxk; xkÞ; xkþj 2 Ulocðxk; xkÞ and lim

j!�1xkþj ¼ xk for all

j 2 Z�;(iii) for xk 62 Ulocðxk; xkÞ; jjxkþj � xk jj d for d[ 0 with j1; j 2 Z� and

j� j1 � 0:

Proof See Nitecki (1971). h

Theorem 2.3 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) with a fixed point xk . The corresponding solution is xkþj ¼ fðxkþj�1; pÞwith j 2 Z. Suppose there is a neighborhood of the fixed point xk (i.e.,UkðxkÞ � Xa), and fðxk; pÞ is Cr (r 1)-continuous in UkðxkÞ. The linearizedsystem is ykþjþ1 ¼ Dfðxk ; pÞykþj (ykþj ¼ xkþj � xk) in UkðxkÞ. If the homeomor-phism between the local invariant subspace EðxkÞ � UkðxkÞ and the eigenspace Eof the linearized system exists with the condition in Eq. (2.28), in addition to thelocal stable and unstable invariant manifolds, there is a Cr�1 center manifoldClocðxk; xkÞ. The center manifold possesses the same dimension of Ec forx 2 Clocðxk; xkÞ, and the tangential space of Clocðx; xÞ is identical to Ec. Thus,the local invariant subspace is decomposed by

Eðxk; xkÞ ¼ Slocðxk; xkÞ �Ulocðxk; xkÞ � Clocðxk; xkÞ: ð2:35ÞProof See Guckenhiemer and Holmes (1990). h

Definition 2.11 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) on domain Xa 2 Rn. Suppose there is a metric space ðXa; qÞ, then the mapP under the vector function fðxk; pÞ is called the contraction map if

qðxð1Þkþ1; xð2Þkþ1Þ ¼ qðfðxð1Þk ; pÞ; fðxð2Þk ; pÞÞ� kqðxð1Þk ; xð2Þk Þ ð2:36Þ

for k 2 ð0; 1Þ and xð1Þk ; xð2Þk 2 Xa with qðxð1Þk ; xð2Þk Þ ¼ jjxð1Þk � xð2Þk jj:Theorem 2.4 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) on domain Xa 2 Rn. Suppose there is a metric space ðXa; qÞ, if the map Punder the vector function fðxk; pÞ is the contraction map, then there is a uniquefixed point xk which is globally stable.

Proof The proof can be referred to Luo (2011). h

2.2 Fixed Points and Stability 19

Definition 2.12 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) with a fixed point xk . The corresponding solution is given by xkþj ¼fðxkþj�1; pÞ with j 2 Z. Suppose there is a neighborhood of the fixed point xk (i.e.,UkðxkÞ � Xa), and fðxk; pÞ is Cr (r 1)-continuous in UkðxkÞ. The linearizedsystem is ykþjþ1 ¼ Dfðxk ; pÞykþj (ykþj ¼ xkþj � xk ) in UkðxkÞ. Consider a realeigenvalue ki of matrix Dfðxk ; pÞ (i 2 N ¼ f1; 2; . . .; ng) and there is a corre-

sponding eigenvector vi. On the invariant eigenvector vðiÞk ¼ vi, consider yðiÞk ¼cðiÞk vi and yðiÞkþ1 ¼ cðiÞkþ1vi ¼ kic

ðiÞk vi, and thus, cðiÞkþ1 ¼ kic

ðiÞk .

(i) xðiÞk on the direction vi is stable if

limk!1

jcðiÞk j ¼ limk!1

jðkiÞkj � jcðiÞ0 j ¼ 0 for jkij\1: ð2:37Þ

(ii) xðiÞk on the direction vi is unstable if

limk!1

jcðiÞk j ¼ limk!1

jðkiÞkj � jcðiÞ0 j ¼ 1 for jkij[ 1: ð2:38Þ

(iii) xðiÞk on the direction vi is invariant if

limk!1

cðiÞk ¼ limk!1

ðkiÞkcðiÞ0 ¼ cðiÞ0 for ki ¼ 1: ð2:39Þ

(iv) xðiÞk on the direction vi is flipped if

lim2k!1

cðiÞk ¼ lim2k!1

ðkiÞ2k � cðiÞ0 ¼ cðiÞ0

lim2kþ1!1

cðiÞk ¼ lim2kþ1!1

ðkiÞ2kþ1 � cðiÞ0 ¼ �cðiÞ0

9=;for ki ¼ �1: ð2:40Þ

(v) xðiÞk on the direction vi is degenerate if

cðiÞk ¼ ðkiÞkcðiÞ0 ¼ 0 for ki ¼ 0: ð2:41Þ

Definition 2.13 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) with a fixed point xk . The corresponding solution is given by xkþj ¼fðxkþj�1; pÞ with j 2 Z. Suppose there is a neighborhood of the fixed point xk (i.e.,UkðxkÞ � Xa), and fðxk; pÞ is Cr (r 1)-continuous in UkðxkÞ: Consider a pair ofcomplex eigenvalues ai ibi of matrix Dfðxk ; pÞ (i 2 N ¼ f1; 2; . . .; ng, i ¼ ffiffiffiffiffiffiffi�1

p)

and there is a corresponding eigenvector ui ivi. On the invariant plane of

ðuðiÞk ; vðiÞk Þ ¼ ðui; viÞ, consider xðiÞk ¼ xðiÞkþ þ xðiÞk� with

20 2 Nonlinear Discrete Systems

xðiÞk ¼ cðiÞk ui þ dðiÞk vi; xðiÞkþ1 ¼ cðiÞkþ1ui þ dðiÞkþ1vi: ð2:42Þ

Thus, cðiÞk ¼ ðcðiÞk ; dðiÞk ÞT with

cðiÞkþ1 ¼ EicðiÞk ¼ riRic

ðiÞk ð2:43Þ

where

Ei ¼ai bi�bi ai

� �and Ri ¼

cos hi sin hi� sin hi cos hi

� �;

ri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2i þ b2i

q; cos hi ¼ ai=ri and sin hi ¼ bi=ri;

ð2:44Þ

and

Eki ¼

ai bi�bi ai

� �kand Rk

i ¼cos khi sin khi� sin khi cos khi

� �: ð2:45Þ

(i) xðiÞk on the plane of ðui; viÞ is spirally stable if

limk!1

jjcðiÞk jj ¼ limk!1

rki jjRki jj � jjcðiÞ0 jj ¼ 0 for ri ¼ jkij\1: ð2:46Þ

(ii) xðiÞk on the plane of ðui; viÞ is spirally unstable if

limk!1

jjcðiÞk jj ¼ limk!1

rki jjRki jj � jjcðiÞ0 jj ¼ 1 for ri ¼ jkij[ 1: ð2:47Þ

(iii) xðiÞk on the plane of ðui; viÞ is on the invariant circles if

jjcðiÞk jj ¼ rki jjRki jj � jjcðiÞ0 jj ¼ jjcðiÞ0 jj for ri ¼ jkij ¼ 1: ð2:48Þ

(iv) xðiÞk on the plane of ðui; viÞ is degenerate in the direction of ui if bi ¼ 0:

Definition 2.14 Consider a discrete, nonlinear dynamical system xkþ1 ¼ fðxk; pÞ inEq. (2.4) with a fixed point xk . The corresponding solution is given by xkþj ¼fðxkþj�1; pÞ with j 2 Z. Suppose there is a neighborhood of the fixed point xk (i.e.,UkðxkÞ � Xa), and fðxk; pÞ is Cr (r 1)-continuous in UkðxkÞ with Eq. (2.28). Thelinearized system is ykþjþ1 ¼ Dfðxk ; pÞykþj (ykþj ¼ xkþj � xk) in UkðxkÞ. Thematrix Dfðxk ; pÞ possesses n eigenvalues ki (i ¼ 1; 2; . . .; n).

(i) The fixed point xk is called a hyperbolic point if jkij 6¼ 1 (i ¼ 1; 2; . . .; n).(ii) The fixed point xk is called a sink if jkij\1 (i ¼ 1; 2; . . .; n).

2.2 Fixed Points and Stability 21