Table Of ContentChaotic
Dynamics of
Nonlinear Systems
Chaotic
Dynamics of
Nonlinear Systems
S. NEIL RASBAND
Department of Physics and Astronomy
Brigham Young University
Provo, Utah
•
WILEY
A Wiley-Interscience Publication
JOHN WILEY & SONS
New York Chichester Brisbane Toronto Singapore
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Copyright © 1990 by John Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of any part of this work
beyond that permitted by Section 107 or 108 of the
1976 United States Copyright Act without the permission
of the copyright owner is unlawful. Requests for
permission or further information should be addressed to
the Permissions Department, John Wiley & Sons, Inc.
Library of Congress Cataloging in Publication Data:
Rasband, S. Neil.
Chaotic dynamics of nonlinear systems/S. Neil Rasband.
p. em.
"A Wiley-Interscience publication."
Bibliography: p.
Includes index.
ISBN 0-471-63418-2
1. Chaotic behavior in systems. 2. Nonlinear theories.
I. Title.
Q172.5.C45R37 1989
003'.75-dc20 89-32903
CIP
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
To Judi
PREFACE
In recent years the scientific community has witnessed the birth and initial
development of a new paradigm for understanding complicated and seemingly
unpredictable behavior. This new paradigm goes by the name of Chaos, re
ferring to a scientific philosophy, an approach, and a set of methods to deal
'vith manifestations of chaos in the physical sciences. To a large extent the
enthusiasm that has developed for Chaos is the result of the breadth of its
applications. These applications of Chaos for understanding complex and
unpredictable behavior range across the spectrum of scientific disciplines. In
deed, Chaos has n1uch the same flavor as classical thermodynamics in that the
fundamental ideas and results seem applicable to a wide variety of different
physical systems. There is probably no physical system exhibiting unpre
dictable behavior that is not presently being scrutinized through the lens of
Chaos by someone.
Despite widespread interest and broad application, Chaos is a young sci
ence and as a consequence, the traditional examples are fewer and the stan
dardization of methods is less well developed. With its roots in many areas
of scientific inquiry, only in recent years have the examples and methods been
welded into a new structure. In this textbook I have tried to introduce Chaos
by presenting those topics and examples that seem to have risen to the top
and become standard fare. However, for reasons of length choices must be
made and the topics selected certainly reflect my own preferences. But I have
tried to represent what most people in the community seem to feel are the im
portant topics. Naturally, not all would agree on every point, and I would not
expect anyone to agree with all of the selections or with the depth to which
I 'have discussed them. Nevertheless, I believe that most of the "classical"
topics in Chaos are represented.
To eli1ninate from the beginning any false expectations, I mention some
topics that are not discussed: quantum chaos, noisy chaos, symbolic dynam
ics, and many, many examples of chaos occurring in specific physical systems.
First and foremost, the book is intended to be useful as a textbook in a
one-sen1ester course taught in a physics department for seniors or first-year
graduate students, and I have used this material for such a course. The audi
ence, however, has not consisted only, or even primarily, of physics students.
Certain topics and chapters require decidedly more background than others.
vii
viii PREFACE
Chapter 8 on conservative dynamics expects the reader to be familiar with
Hamiltonian dynamics. Sections 5.5 and 9.2 use some basic mathematical
tools from differential geometry. However, the vast majority of the presenta
tion depends only on some familiarity with differential equations and linear
vector spaces. Even the reader with a limited knowledge of Hamiltonian dy
namics or certain mathematical tools should be able to follow the presentation
with only rarely a feeling of unfamiliarity.
The absolutely essential prerequiste the author expects the reader to
bring to a study of this book is a willingness to do considerable numerical
experimentation. The programming and numerical skills required for most of
the examples are minimal, but a great deal of insight comes from personally
performing numerical experiments on some of the classical problems in Chaos.
Personal computers are adequate for doing everything in this book but, of
course, may not suffice for tackling research problems in Chaos.
I wish to express my personal gratitude to colleagues who have fueled my
interest by giving me copies of articles from diverse places and by generally en
couraging me in this writing project. Particularly, I thank G. Mason, G. Hart,
R. Shirts and E. lliiuchle. I thank H. Stokes for frequent suggestions on 'lEX
formatting and T. Knudsen for help in preparation of the manuscript. I espe
cially thank my colleague and friend Ross Spencer for reading the manuscript
and making literally hundreds of suggestions. The book is significantly better
than it would otherwise be as a result of his help.
l\1y deepest thanks go to my family since a considerable portion of the
time necessary to complete this work has been taken from hours that rightfully
belonged to them. The completion of this project would not have been possible
without the love and support of my wife and children.
S. Neil Rasband
Provo, Utah
July 1989
CONTENTS
1 Introduction 1
1.1 Chaos and Nonlinearity 1
1.2 The Kicked Harmonic Oscillator 3
1.3 Examples 9
Exercises 10
2 One-Dimensional Maps 13
2.1 The Tent Map 14
2. 2 The Lyapunov Exponent in One Dimension 18
2.3 The Logistic Map 19
2.4 Asymptotic Sets and Bifurcations 25
Exercises 31
3 Universality Theory 33
3.1 Period Doubling and the Composite Functions 33
3.2 Scaling and Self-Similarity in the Logistic Map 47
3.3 Subharmonic Scaling 53
3.4 Experimental Comparisons 58
3.5 Intermittency 60
3.6 Summary 68
Exercises 69
4 Fractal Dimension 71
4.1 Definitions of Fractal Dimension 72
4.2 Classical Examples of Fractal Sets 75
4.3 At tractor for the Universal Quadratic Function 78
4.4 Summary 81
Exercises 82
5 Differential Dynamics 85
5.1 Linearization 86
5.2 Invariant Manifolds 89
5.3 The Poincare Map 92
5.4 Center Manifolds 96
5.5 Normal Forms 103
5.6 Bifurcations 108
5.7 Summary 110
Exercises 110
ix
CONTENTS
X
6 Nonlinear Examples with Chaos 111
6.1 The Disk Dynamo 112
6.2 Attracting Sets and Trapping Regions 117
6.3 The Lorenz System 119
6.4 The Damped, Driven Pendulum 122
6.5 The Circle Map and the Devil's Staircase 128
6.6 Summary 132
Exercises 132
7 Two-Dimensional Maps 135
7.1 Fixed Points 135
7.2 Normal Forms 138
7.3 Hopf Bifurcation and Arnold Tongues 142
7.4 The Henon Map 148
7.5 Renormalization of an Area-Preserving Map 153
7.6 Universality for Area-Preserving Maps 156
7.7 Summary 159
Exercises 160
8 Conservative Dynamics 161
8.1 Hamiltonian Dynamics and Transformation Theory 162
8.2 Two Examples in Action-Angle Coordinates 165
8.3 Nonintegrable Hamiltonians 168
8.4 Canonical Perturbation Theory and Resonances 171
8.5 The Standard Map 178
8.6 Arnold Diffusion 179
8.7 Summary 181
Exercises 181
9 Measures of Chaos 183
9.1 Power Spectrum Analysis 183
9.2 Lyapunov Characteristic Exponents Revisited 187
9.3 Information and K-Entropy 196
9.4 Generalized Information, K-Entropy, and Fractal Dimension 199
9.5 Chaos Measures from a Time Series 200
9.6 Summary 202
Exercises 203
10 Complexity and Chaos 205
10.1 Algorithmic Complexity 206
10.2 The LZ Complexity Measure 207
10.3 Complexity and Simple Maps 211
10.4 Summary 212
Exercises 213
Reprise 215
Glossary 217
References 221
Index 227
