Table Of Content®T HE D\TNAMilCS OF
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®T HE DYNAMICS OF
PATTERNS
MI Rabinovich
University of California, San Diego
A B Ezersky
Russian Academy of Sciences
P D Weidman
University of Colorado
World Scientific
Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
THE DYNAMICS OF PATTERNS
Copyright © 2000 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
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ISBN 981-02-4055-4
ISBN 981-02-4056-2 (pbk)
Printed in Singapore by FuIsland Offset Printing
To our parents,
Dora and Israel
ovetlana ana Doris
Ora ana Merle
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Preface
The creative process of writing this book was less a result of that which
arises out of ones own head, hand, pen and paper than a result of the in
teractive dialogue between authors with a similar vision. Our desire is to
present a clear and concise presentation of principal phenomena, basic con
cepts, and novel ideas in pattern formation using the language of nonlinear
dynamics supplemented by photographs of actual patterns observed in the
laboratory. We believe, perhaps a little presumptuously, that the principle
we adhere to, namely "one chapter - one topic" together with a careful se
lection of examples, has enabled us to write a book that is comprehensible
to graduates, postgraduates, and those researchers who have just started
investigating this extremely fascinating field of science. As is well known,
the elegance of simplicity tempts one to simplify real phenomena by fitting
them to conventional and universal models. We hope that we succeeded in
avoiding this temptation unduly, since every chapter is focused on specific
experiments.
The text is supplemented with two appendices. Appendix A — A Short
Guide to Nonlinear Dynamics — is provided as a review of the theoretical
underpinnings of nonlinear systems and Appendix B — Key Experiments
in Pattern Formation — is included for readers curious about experimental
setups designed to observe pattern phenomena in the laboratory.
For the most part, the book is based on the lectures read by MIR to
students of the University of Nizhny Novgorod, the University of Chicago,
and the University of California at San Diego. Some sections are the result
of laboratory experiments spearheaded by ABE and PDW.
We greatly appreciate numerous discussions on the topics presented
vii
Vlll Preface
herein with H. Abarbanel, V. Afraimovich, I. Aranson, A. Gaponov-Grekhov,
L. Kadanov, J. Lebowitz, A. Libchaber, Ya. Sinai, L. Tsimring and P.
Varona. The authors are indebted to F. Hjguera for a careful reading of
Chapter 10 and to J. Meiss and B. Polotovsky for critical comments concer-
ing Appendix A. In addition, L. Tsimring wrote Chapter 11 upon request
of the authors.
We are ever indebted to Nadezhda Krivatkina and Leonid Rubchinsky
who helped us prepare the manuscript, to Michael Sprague for suggestions
to improve the layout, and to Kendall Hunter for rescuing files sent to
Boulder from Nizhny Novgord. MIR acknowledges support from the En
gineering Research Program of the Office of Basic Energy Science at the
Department of Energy and personally Robert Price who is deeply interested
in the subjects discussed in this book. We also appreciate the support of
International Center for Advanced Studies in Nizhny Novgorod.
M. I. Rabinovich
A. B. Ezersky
P. D. Weidman
Boulder, Colorado
August 2000
Contents
Preface vii
Chapter 1 Patterns: Prelude to a Dynamical Description 1
Chapter 2 Linear Stage of Pattern Formation 15
Chapter 3 Model Equations 27
3.1 Swift-Hohenberg equation 29
3.2 Newell-Whitehead-Segel equation 35
3.3 Coupled amplitude equations 38
3.4 Phase equations 42
Chapter 4 The Ginzburg-Landau Equation 45
4.1 The dissipative Ginzburg-Landau equation 46
4.2 Nerve membrane excitation and the CGL equation 48
4.3 Optical dynamics and the CGL equation 50
4.4 Simple patterns in the CGL equation 52
4.5 Phase equations revisited 55
4.6 Gallery of phenomena 57
Chapter 5 'Crystal' Formation 63
Chapter 6 Quasicrystals 75
6.1 Octagons, decagons, and dodecagons 78
6.2 A generalized Swift-Hohenberg model 81
6.3 The 'turbulent' crystal 82
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x Contents
Chapter 7 Breaking of Order 87
7.1 A simple model for domain walls 89
7.2 Topological defects 92
7.3 The birth of penta-hepta defects 96
7.4 Dislocations and domain walls in Faraday ripples 102
Chapter 8 Localized Patterns 107
8.1 Bistable media 107
8.2 Dynamical disorder of structures 114
8.3 Particle interaction 115
8.4 Chaotic scattering 118
Chapter 9 Spirals 129
9.1 Active spirals 133
9.1.1 Spirals in the complex Ginzburg-Landau equation . . . 133
9.1.2 Spirals in the FitzHugh-Nagumo model 136
9.2 Passive spirals 142
9.2.1 Spirals in the Faraday experiment 143
9.2.2 Spirals in Rayleigh-Benard convection 145
Chapter 10 Patterns in Oscillating Soap Films 151
10.1 Introduction 151
10.2 Observations 152
10.3 Models for vorticity generation 158
10.3.1 Marangoni wave model 160
10.3.2 The role of air 162
Chapter 11 Patterns in Colonies of Microorganisms 173
11.1 Dictyostelium discoideum 174
11.2 Esherichia coli 178
11.3 Bacillus subtilis 184
Chapter 12 Spatial Disorder 189
12.1 Introductory remarks 189
12.2 Characteristics of space series 193
12.3 The Grassberger-Procaccia algorithm 197
12.4 Qualitative description of developing disorder 199
12.5 Dynamical dimension of defect-mediated turbulence 202
