Korsch . Jodl Chaos

Springer-Verlag Berlin Heidelberg GmbH

H. J. Korsch H.-J. JodI

CHAOSA Program Collection for the PC

Second EditionWith 250 Figures,Many Numerical Experiments,and CD-ROM for Windows 95 and NT

Springer

Professor Dr. H. J. Korsch Professor Dr. H.-J. Jodl Fachbereich Physik, Universität Kaiserslautern Erwin-Schrödinger-Strasse 0-67663 Kaiserslautern, Germany e-mail: korsch@physik.uni-kl.de

jodl@physik.uni-kl.de

The cover picture shows a Lorenz attractor generated with the program ODE, ordinary differential equations, included on the CD-ROM

Additional material to this book can be downloaded from http://extras.springer.com

ISBN 978-3-662-03868-0 ISBN 978-3-662-03866-6 (eBook) DOI 10.1007/978-3-662-03866-6

Library of Congress Cataloging-in-Publication Data applied for.

Die Deutsche Bibliothek - CIP-Einheitsaufnahme

CHAOS: a program collection for the PC; with many numerical experimentsl H. J. Korsch; H.-J. Jod!. - Berlin; Heidelberg; New York; London; Paris; Tokyo; Hong Kong; Barcelona: Springer Literaturangaben

Buch. - 2. ed. - 1998 CD-ROM. -2. ed. - 1998

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are hable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1994, 1999 Originally published by Springer-Verlag Berlin Heidelberg New York in 1999.

Softcover reprint of the hardcover 2nd edition 1999

Tbe use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and therefore free for general use.

Please note: Before using the programs in this book, please consult the technical manuals provided by the manufacturer of the computer - and of any additional plug-in boards - to be used. Tbe authors and the publisher accept no legal responsibility for any damage caused by improper use of the instruc­tions and programs contained herein. Although these programs have been tested with extreme care, we can offer no formal guarantee that they will function correctly. Tbe programs on the enclosed CD-ROM are under copyright protection and may not be reproduced without written permission by Springer­Verlag. One copy of the programs may be made as a back-up, but all further copies violate copyright law.

Cover Design: E. Kirchner, Heidelberg Typesetting: Camera-ready copies from the authors using a Springer LATEX macro package SPIN: 10661573 56/3144 - 5 4 3 2 I 0 - Printed on acid-free paper

Preface to the Second Edition

The st ill growing interest in chaotic dynamics in physics and the friendly receiptgiven to the first edit ion encouraged us to prepare a second edit ion of this book.

During the last years, we observed an increasing introduction to chaotic(or nonlinear) dyn amics already in basic courses in physics. Here the computeris often used as an ideal to ol for the demonstration of chaot ic phenomena incomputer "experiments" during lectures. More and more student s realize thatthey can benefit from the simultaneous interaction with computer programs andreading of text s, provided that specialized and easy to use programs with manysuggest ions for such "experiments" exist. The resonance from many st udentsand colleagues gave us the impression that our collection of programs helps thest udents to explore the highly non-tri vial behavior of dynamical syst ems.

We have taken the oppor tunity to correct some minor errors and to clarifya few points in the text of the book, but most of the program codes of thefirst edit ion remained unchanged. However , the rapid development of computeroperat ing syste ms made it necessary to modify some of the comput er codesand to change the installat ion routin e. The programs in this collect ion wereoriginally written to ru n within the operating system DOS. Now t hey havebeen tes ted to run under Windows 95 and NT as well. Because of the increaseof speed, the computing t imes are not iceably reduced and much more elaboratenumerical experiments may be performed in acceptable times.

In addit ion to the st udents and research associates who cont ributed consid­erably to the first edit ion, we would like to thank Dr. Martin Menzel, Dr . LeoSchoendorff and Bernd Schellhaaf3 for their assistance in preparing this secondedit ion.

Kaiserslautern ,October 1998

H. J . Korsch and H.-J. JodI

Preface to the First Edition

The problem, expressed in its general form, is an old one and appears undermany guises. Why are the clouds the way they are? Is the solar system stable?What determines the structure of turbulence in liquids, the noise in electroniccircuits, the stability of a plasma? What is new, that is to say with respect toNewton 's Principia published three hundred years ago, and which has emergedover the last few decades, is the heuristic use of computers to enhance our un­derstanding of the mathematics of nonlinear dynamical processes and to explorethe complex behavior that even simple systems often exhibit.

It is the purpose of this book to teach chaos through a simultaneousreading of the text and interaction with our selected computer programs.

The use of computers is not only essential for studying nonlinear phenom­ena, but also enables the intuitive geometric and heuristic approach to be devel­oped, taught to students, and integrated into the scientist's skills, techniques,and methods.

• Computers allow us to penetrate unexplored regions of mathematics anddiscover foreseen links between ideas.

• Numerical solutions of complex nonlinear problems - displayed by graphsor videos - as opposed to analytical solutions, which are often limited dueto the approximations made, become possible.

• The visualization of mathematics will also be a focus of this book: onegood graph or simulation video that highlights the evolution of a coherentcomplex pattern can be worth more than a hundred equations.

• Nonlinear problems are almost always difficult, often having unexpectedsolutions.

• In attempting to understand the details of the computer solution onemay uncover a new kind of problem, or a new aspect that leads to deeperunderstanding.

• Appropriate graphical displays, especially those that are constructed andcomposed on the screen as one interacts with the computer, will improvethe ability to choose from among promising paths. This procedure natu­rally complements the usual approaches of experiment, theoretical formu­lation and asymptotic approximation.

VIII Preface to the First Edition

• The benefits of the computational approach clearly depend on t he avail­ability of various graphical displays: small effects that may signal new phe­nomena (zoom); proper mapping of dat a in place of a search for st ruc turesin voluminous printouts of columns of numb ers; pictures which clearly pro­du ce an insight into the physics; color or real-time videos which enhanceperception, enabling one to correlate old and new results and recognizeunexpected phenomena.

• Interactive software in general, or educat ional software in physics, mustprovide the ability to display in one, two or three dimensions, to showspatial and temporal correlations, to rotate, displace or st retch objects,to acquire diagnost ic variables and essent ial summaries or to opt imizecompar isons.

Are we providing thi s kind of training in our universities, so that our stu­dent s may learn thi s method of working at the 'nonlinear' fronti er? We have tofind new methods to teach students to experiment with computers in the waythat we now teach them to experiment with lasers, or deepen their knowledgeof theory. Therefore, the philosophy of our approach in thi s book is to practicethe use of a computer in computational physics directed at a convincing topic,i.e. nonlinear physics and chaos. Our programs are written in such a style thatphysical problems can quickly be tackled, and t ime is not waste d to programdet ails such as the use of algorithms to solve an equation, or the input and out­put of data. Of course, the student will event ually need to master the elementsof programming himself as he comes closer to independent resear ch. Therefore,this book is aimed at those who have completed a course of study in physicsand are on the threshold of research. Another advantage is that 'mini-research 'can be carried out , thanks to the nature of the topic, chaos, and to the too l,the compute r. These allow one to discuss physical problems which are onlyment ioned in textbooks nowadays as a potenti al topic of study, e.g. the dou blependulum.

From the point of view of the complexity of the mathematics and physics ,this book is designed mainly for students in the third or four th year in ascience or engineering faculty. In a limited way, it might also be useful to thoseworking at the fronti ers of nonlinear physics, since this topic is relatively newand far from having well-established solut ions or wide applicat ions.

This book is organized in the following way: in Chap. 1 'Overview andBasic Concepts ' , we attempt to int roduce typical features of chao t ic behaviorand to point out the broad applicability of chaos in science as well as to makethe reader familiar with the terminology and theoret ical concepts. In Chap. 2'Nonlinear Dynamics and Deterministic Chaos', we will develop the necessarybasis, which will be deepened and app lied in subsequent chapters. Chapter 3'Billiard Systems ' and Chap. 4 'Gravitational Billiards ' will treat two of the'classical examples' of simple conservative mechanical syste ms. In Chap. 5, theclass of different pendula, such as kicked, inverted, coupled, oscillatory or ro­tat ing, is representatively discussed through the double pendulum. Phenomena

Preface to the First Edition IX

appearing in chaotic scattering systems are represented by the three disk scat­tering in Chap. 6. The subsequent chapters treat systems explicitly dependenton time: namely, in Chap. 7 a periodically kicked particle in a box 'Fermi Ac­celeration', and in Chap. 8 a driven anharmonic oscillator 'Duffing Oscillator'.The celebrated one-dimensional iterated maps are the topic of Chap.9, andthe observed period-doubling scenario can be studied via the physical exampleof nonlinear electronic circuits in Chap . 10. Numerical experiments with two­dimensional maps are considered in Chap. 11 'Mandelbrot and Julia Sets ', whileChap.12 'Ordinary Differential equations' provides a quite general platformfrom which to investigate systems governed by coupled first order differentialequations. Finally, further technical questions of hardware requirements, pro­gram installation, and the use of the programs are addressed in the appendices.Most chapters follow the same substructure:

- Theoretical Background

- Numerical Techniques

- Interaction with the Program

- Computer Experiments

- Real Experiments and Empirical Evidence

- References

Many books and articles have been written on chaotic experiments, andsome of the 'classical' experiments are mentioned in Chap . 1. Therefore, thelast subsection in each chapter is intended to give the reader confidence toprogress from his studies on the computer to real experiments and empiricalevidence; e.g. comparing the trajectory of a double pendulum in reality andon the screen. Of course, some aspects of the system are better studied inthe computer experiment, others in the real one; in addition, they complementeach other, e.g. looking for bifurcations in a nonlinear electronic circuit on theoscilloscope and on the screen (Chap. 10). One is at first impressed by theapparently chaotic motion of a real double pendulum, but deeper insight intothe structure of this chaotic behavior is gained by looking at Poincare maps inphase space . The experiments chosen here and briefly reported (for details seethe cited literature) are mainly for educational purposes, to be reconstructedand used in student laboratories or in lectures. Therefore, they are not meantto represent those experiments investigated in current nonlinear research.

The most effective way of using this book may be to read a chapter whileworking simultaneously on the computer using the appropriate program. As al­ready mentioned, the reader should use the programs - rather than programmajor parts himself - in much the same way as he would use standard ser­vice software in combination with commercial research apparatus. On the otherhand, the use of our programs should not be a simple push button procedure,but involve serious interaction with the software . For example, some parameters,

X Preface to the First Edition

initial values, boundary conditions are already pre-set to execute numerical ex­periments discussed in the book, while other numerical experiments described indetail require changes in the pre-set parameters. Further studies are suggestedand the reader may proceed independently, guided by some hints and the citedliterature. The programs are flexible and organized in such a way that he canset up his own computer experiments, e.g. define his own boundary in a billiardproblem or explore his favorite system of nonlinear differential equations.

Reading this book and working with the programs requires a knowledge ofclassical mechanics and a basic understanding of chaotic phenomena. The shortoverview on chaotic dynamics and chaos theory in Chaps . 1 and 2 cannot serveas a substitute for a textbook. Within the last decade a number of such bookshave been published reflecting the rapid growth of the field. Some focus onexperiments, some concentrate on theory, some deal with basic concepts, whileothers are simply a selection of original articles. The reader should consult someof these texts while exploring the nonlinear world by means of the computerprograms in this book .

We hope that the selected examples of chaotic systems will help the readerto gain a basic understanding of nonlinear dynamics, and will also demonstratethe usefulness of computers for teaching physics on the PC .

Most of the programs, at least in their preliminary version, were developedby students during a seminar 'Computer Assisted Physics' (1985-1990) . Withthe aid of two grants (PPP 1987-1989, PPPP 1991-1994), we were able to im­prove, test , standardize and update those programs. Chapter 1 contains a listof all coauthors for every program. We wish to acknowledge funding from theBundesministerium fiir Bildung und Wissenschaft (BMBW), from the Univer­sity of Kaiserslautern via the Kultusministerium of Rheinland Pfalz and theDeutsche Forschungsgemeinschaft (DFG) for the hardware. We are indebted tothe Volkswagenstiftung (VW) for supporting one of us (H.-J. J.) to finish thisbook during a sabbatical.

Finally, we wish to recognize the contribution of undergraduate and gradu­ate students and research associates who worked so enthusiastically with us onproblems associated with chaos and on the use of computers in physics teach­ing. We would particularly like to thank the graduate students Bjorn Baser andAndreas Schuch for their considerable assistance in developing and debuggingthe computer codes, the instructions for interacting with the programs as wellas the large number of computer experiments. Finally, we would like to thankFrank Bensch and Bruno Mirbach, who read parts of the manuscript and mademany useful suggestions.

Kaiserslautern,May 1994

H. J . Korsch and H.-J . JodI

Table of Contents

1 Overview and Basic Concepts1.1 Introduction .1.2 The Programs .1.3 Literature on Chaotic Dynamics .

2 Nonlinear Dynamics and Deterministic Chaos2.1 Deterministic Chaos .2.2 Hamiltonian Systems .

2.2.1 Integrable and Ergodic Systems2.2.2 Poincare Sections . .2.2.3 The KAM Theorem ...2.2.4 Homoclinic Points . . . .

2.3 Dissipative Dynamical Systems2.3.1 Attractors . .. .2.3.2 Rout es to Chaos . ...

2.4 Special Topics . . . . . . . . . .2.4.1 The Poincare-Birkhoff Theorem2.4.2 Continued Fractions .2.4.3 The Lyapunov Exponent . . . .2.4.4 Fixed Points of One-Dimensional Maps2.4.5 Fixed Points of Two-Dimensional Maps .2.4.6 BifurcationsReferences . .

3 Billiard Systems3.1 Deformations of a Circle Billiard .3.2 Numerical Techniques . . . .3.3 Interacting with the Program . .3.4 Computer Experiments . . . . . .

3.4.1 From Regularity to Chaos3.4.2 Zooming In .3.4.3 Sensitivity and Determinism3.4.4 Suggestions for Additional Experiments .

Stability of Two-Bounce Orbits .. .. .

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XII Table of Contents

Bifurcations of Periodic Orbits .A New Integrable Billiard? .Non-Convex Billiards .

3.5 Suggest ions for Further Studies .. . .3.6 Real Exp eriments and Empirical Evidence

References . . . . . . . . . . . . . .

4 Gravitational Billiards: The Wedge4.1 The Poincare Mapping . . . .4.2 Int eracting with the Program .4.3 Computer Experiment s .

4.3.1 Periodic Motion and Ph ase Space Organizati on4.3.2 Bifurcation Phenomena. . . . . . . . . .4.3.3 'P lane Filling' Wedge Billiards .4.3.4 Suggestions for Additional Experiments .

Mixed A - B Orbits . .Pure B Dynamics . . .The Stochast ic RegionBreathing Chaos .. .

4.4 Suggestions for Further Studies4.5 Real Exp erim ents and Empirical Evidence

References . . . . . .

5 The Double Pendulum5.1 Equations of Motion5.2 Numerica l Algorithms5.3 Interacting with the Program5.4 Computer Experiments . . . .

5.4.1 Different Types of Motion5.4.2 Dynamics of the Double Pendulum5.4.3 Destruction of Invariant Curves . .5.4.4 Suggestions for Addi tional Exp eriments .

Testing the Numerical Int egration .Zooming In .Different Pendulum Parameters . .

5.5 Real Exp eriments and Empirical EvidenceReferences . . . .

6 Chaotic Scattering6.1 Scat tering off Three Disks6.2 Numerica l Techniques ..6.3 Interacting with the Program6.4 Computer Experiments . . . .

6.4.1 Scattering Functio ns and Two-Disk Collisions6.4.2 Tree Organizat ion of Three-Disk Collisions6.4.3 Unstable Periodic Orb its .

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Table of Contents XIII

6.4.4 Fractal Singularity Structure . . . . . . .6.4.5 Suggestions for Additional Exp eriments .

Long-Lived Traj ectories . .. .Incomplete Symbolic DynamicsMultiscale Fractals . . . . . . .

6.5 Suggest ions for Further Studies . . . .6.6 Real Exp eriments and Empirical Evidence

References . . . .

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7 Fermi Acceleration 1377.1 Fermi Mapping . 1387.2 Int eracting with the Program 1397.3 Computer Exp eriments . . . . 141

7.3.1 Exploring Phase Space for Different Wall Oscillations 1417.3.2 KAM Curves and Stochastic Acceleration 1447.3.3 Fixed Points and Linear Stability . . . . 1477.3.4 Absolute Barri ers . . . . . . . . . . . . . 1487.3.5 Suggestions for Addi tion al Exp eriments . 150

Higher Order Fixed Points 150Standard Mapping . . . . . . . . . . 151Bifurcation Ph enomena . . . . . . . . 152Influence of Different Wall Velocities 153

7.4 Suggest ions for Furth er Studies . . . . . . 1547.5 Real Experiments and Empirical Evidence 155

References . . . . . . 155

8 The Dulling Oscillator 1578.1 The Duffing Equat ion. 1578.2 Numerical Techniques 1618.3 Interacting with the Program 1618.4 Computer Exp eriments. . . . 165

8.4.1 Chaoti c and Regular Oscillations 1658.4.2 The Free Duffing Oscillator ... 1668.4.3 Anh arm onic Vibrations: Resonances and Bist ability 1668.4.4 Coexist ing Limit Cycles and Strange Attractors 1708.4.5 Suggest ions for Additional Experiments. 173

Harmonic Oscillator .. . 173Gravit ational Pendulum . . . 174Exact Harm onic Response . . 174Period-Doubling Bifurcations 175Strange At tractors . . . . . . 177

8.5 Suggestions for Further Studies . . . 1778.6 Real Experiments and Empirical Evidence 178

References . . . . . . . . . . . . . . . . . . 179

XIV Table of Contents

9 Feigenbaum Scenario . . . . . .9.1 One-Dimensional Maps .. . .9.2 Interacting with the Program9.3 Comput er Exp eriments .. ..

9.3.1 Period-Doubling Bifurcations9.3.2 The Chaotic Regime9.3.3 Lyapunov Exponents . . . . .9.3.4 The Tent Map . . . . . . . . .9.3.5 Suggest ions for Additional Exp eriments .

Different Mapping Function s .Periodic Orbit Theory . .Exploring the Circle Map ..

9.4 Suggestions for Further Studies . . .9.5 Real Experiments and Empirical Evidence

References . . . . . . . . . .

10 Nonlinear Electronic Circuits10.1 A Chaos Generat or .10.2 Numerical Techniques .10.3 Interacting with the Program10.4 Computer Experiments .

10.4.1 Hopf Bifurcat ion10.4.2 Period Doubling .10.4.3 Return Map ...10.4.4 Suggestions for Additional Exp eriments.

Comparison with an Electronic CircuitDeviations from the Logistic Mapping.Boundary Crisis . . . . . . . . . . .

10.5 Real Experiments and Empirical EvidenceReferences . . . . . . . . .

11 Mandelbrot and Julia Sets11.1 Two-Dimensional Iterated Maps .11.2 Numerical and Coloring Algorithms11.3 Int eracting with th e Program . .11.4 Computer Exp eriments .

11.4.1 Mand elbrot and Juli a Sets .11.4.2 Zooming into the Mandelbrot Set11.4.3 General Two-Dimensional Quadratic Mappin gs

11.5 Suggest ion for Addi tional Experiments ..Components of the Mandelbrot SetDistort ed Mandelbrot Maps . . . .Furth er Experiments . . . . . . . .

11.6 Real Exp eriments and Empirical EvidenceReferences . . . . . . . . . . . . . . . . . .

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Table of Contents XV

12 Ordinary Differential Equations 24912.1 Numerical Techniques . . .. 25012.2 Interacting with the Program 25012.3 Computer Experiments . . . . 255

12.3.1 The Pendulum . . . . 25512.3.2 A Simple Hopf Bifurcation . 25712.3.3 The Duffing Oscillator Revisited . 26012.3.4 Hill's Equation . . . . 26312.3.5 The Lorenz Attr actor . . . 27012.3.6 The Rossler Attractor .. 27412.3.7 The Henon-Heiles System 27512.3.8 Suggestions for Additional Exp eriments . 279

Lorenz System: Limit Cycles and Intermittency 279The Restri cted Three Body Problem 280

12.4 Suggestions for Furth er Studies 284References . . . . . . . . . . . . . . . . . . . 289

A System Requirements and Program Installation 291A.l Syst em Requirements . . 291A.2 Installing the Programs. 291A.3 Programs and Files . . . 292

B General Remarks on Using the Programs 295B.l Mask Menus . . . . . . . . . . . . . 295B.2 The File-Select Box . . . . . . . . . . . . 296B.3 Input of Mat hematical Expressions . . . 297B.4 Selection of Points or Areas in Graph ics 298

Glossary 299

Index 307