A SURVEY OF NONLINEAR DYNAMICS ("Chaos Theory")

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1 A Survey of Nonlinear Dynamics (Chaos Theory) Downloaded from A SURVEY OF NONLINEAR DYNAMICS ("Chaos Theory")

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3 A Survey of Nonlinear Dynamics (Chaos Theory) Downloaded from A SURVEY OF NONLINEAR DYNAMICS ("Chaos Theory") R. L. Ingraham Department of Physics New Mexico State University USA World Scientific Singapore New Jersey London Hong Kong

4 Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ UK office: 73 Lynton Mead, Totteridge, London N20 8DH A Survey of Nonlinear Dynamics (Chaos Theory) Downloaded from Library of Congress Cataloging-in-Publication Data Ingraham, R. L. (Richard Lee), A survey of nonlinear dynamics : "chaos theory" / R.L. Ingraham. p. cm. Includes bibliographical references and index. 1. Chaotic behavior in systems. I. Title. Q172.5.C \7~dc CIP Copyright 1992 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,electronic or mechanical, including photocopying,recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISBN Printed in Singapore by JBW Printers & Binders Pte. Ltd.

5 A Survey of Nonlinear Dynamics (Chaos Theory) Downloaded from PREFACE This report is intended to give a survey of the whole field of nonlinear dynamics (or "chaos theory," as it is popularly called) in a compressed form. It is slightly expanded from a series of lectures given over the space of a single month in This young and rapidly growing field is already very extensive, so that this survey cannot be deep or detailed. In particular, no pretense of mathematical rigor is made. But I do insist on stating key definitions or theorems carefully so that the reader need not settle for just a qualitative, intuitive understanding. My intention is to touch on the main ideas so that the reader can see if his or her special discipline fits in anywhere and if so, can get an approximate notion of what new ideas or possibilities nonlinear dynamics brings to that field. The cited literature then allows the reader to proceed further if he or she desires. I thank Harry Auvermann for suggesting that I give these lectures in the first place, and David Bandelier for a beautiful job of turning my manuscript into print. Thanks to David Tofsted for invaluable aid in reformatting the report into book form. I acknowledge support from the Atmospheric Sciences Laboratory (representative Harry Auvermann) under the auspices of the Scientific Services Program administered by Battelle for the Army Research Office (D.O. 2008, Contract DAAL03-86-D-0001).

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7 vii CONTENTS PREFACE v 1. INTRODUCTORY REMARKS 1 A Survey of Nonlinear Dynamics (Chaos Theory) Downloaded from Linear Versus Nonlinear The Goals of Nonlinear Dynamics "Chaos" 3 2. FUNDAMENTALS OF CONTINUOUS TIME SYSTEMS Flows Linear Stability Analysis Case of Linear Flows Case of Nonlinear Flows Stability Types of Fixed Points Connection of Stability and Linear Stability Analysis Topological Equivalence of Flows Stable and Unstable Manifolds of a Fixed Point Bifurcations Local Bifurcations Near Fixed Points Bifurcation, General Definition Dissipative Flows and Attractors DISCRETE TIME SYSTEMS: ITERATION OF MAPS Iteration of Maps Linear Stability Analysis Linear Maps Nonlinear Maps Stability Types of a Fixed Point 23

8 viii 3.4 Connection of Stability and Linear Stability Analysis Topological Equivalence of Map Dynamics Stable and Unstable Manifolds of a Fixed Point Periodic Orbits 25 A Survey of Nonlinear Dynamics (Chaos Theory) Downloaded from Bifurcation of Maps One-Dimensional Maps The Logistic Map Period-Doubling Graphical Treatment of Period-Doubling Tangent Bifurcations Lyapunov Exponent for ID Maps Poincare Maps HAMILTONIAN SYSTEMS Generalities Integrable Systems Action-Angle Variables N-Tori Nonintegrable Systems KAM Theorem Poincare-Birkhoff Theorem Resonance Overlap Examples of Nonintegrable Systems MEASURES OF CHAOS Introduction Sensitive Dependence Lyapunov Stability 58

9 ix Maximal Lyapunov Exponent Sensitive Dependence and Chaos in Quantum Mechanics Discrete Fourier Transform and Power Spectrum Algorithmic Complexity, Kolmogorov Entropy, Etc Algorithmic Complexity 63 A Survey of Nonlinear Dynamics (Chaos Theory) Downloaded from Kolmogorov and Other Entropies RENORMALIZATION GROUP Introduction RG for Ising Lattices ID Ising Lattice by the RG D Ising Lattice by the RG RG Applied to Period-Doubling RG Applied to the Tangent Bifurcation RG Applied to Percolation PARTIAL DIFFERENTIAL EQUATIONS General Remarks Convection Equations The Rayleigh Problem The Galerkin Method Rayleigh Problem by the Galerkin Method Cautionary Remarks Patterns of Convection EXPERIMENTAL REALIZATIONS OF NONLINEAR DYNAMICS Introduction Some Specific Experiments PDE Systems 91

10 X A Survey of Nonlinear Dynamics (Chaos Theory) Downloaded from Other Systems 92 REFERENCES 95 APPENDIX A. GENERAL LINEAR STABILITY ANALYSIS 97 APPENDIX B. PROOF OF NO SD IN THE QUANTUM MECHANICAL SELF-CORRELATION 99 GLOSSARY OF PRINCIPAL TERMS 101 INDEX 105