Nonlinear Dynamics and Chaotic Phenomena: An Introduction

414 Pages·1997·11.82 MB·English

by  Bhimsen K. Shivamoggi (auth.)

NONLINEAR DYNAMICS AND CHAOTIC PHENOMENA
FLUID MECHANICS AND ITS APPLICATIONS
Volume 42
Series Editor: R. MOREAU
MADYIAM
Ecole Nationale Superieure d' Hydraulique de Grenoble
Boite Postale 95
38402 Saint Martin d' Heres Cedex, France
Aims and Scope of the Series
The purpose of this series is to focus on subjects in which fluid mechanics plays a
fundamental role.
As well as the more traditional applications of aeronautics, hydraulics, heat and
mass transfer etc., books will be published dealing with topics which are currently
in a state of rapid development, such as turbulence, suspensions and multiphase
fluids, super and hypersonic flows and numerical modelling techniques.
It is a widely held view that it is the interdisciplinary subjects that will receive
intense scientific attention, bringing them to the forefront of technological advance
ment. Fluids have the ability to transport matter and its properties as well as
transmit force, therefore fluid mechanics is a subject that is particulary open to
cross fertilisation with other sciences and disciplines of engineering. The subject of
fluid mechanics will be highly relevant in domains such as chemical, metallurgical,
biological and ecological engineering. This series is particularly open to such new
multidisciplinary domains.
The median level of presentation is the first year graduate student. Some texts are
monographs defining the current state of a field; others are accessible to final year
undergraduates; but essentially the emphasis is on readability and clarity.
For a list o/related mechanics titles, see final pages.
Nonlinear Dynamics
and Chaotic Phenomena
An Introduction
by
BHIMSEN K. SHIVAMOGGI
Departments 0/ Mathematics and Physics,
University o/Central Florida,
Orlando, Florida, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress
ISBN 978-90-481-4926-1 ISBN 978-94-017-2442-5 (eBook)
DOI 10.1007/978-94-017-2442-5
Printed on acid-free paper
AH Rights Reserved
@1997 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997
Softcover reprint of the hardcover 1s t edition 1997
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
v
Salutations to that guru who opened the eye of the one blind due to the (cover) darkness
of ignorance with the needle (coated) with ointment of knowledge.
- Skanda Purana
To Jayashree, Vasudha and Rohini
TABLE OF CONTENTS
Introduction to Chaotic Behavior in Nonlinear Dynamics 1
Phase-Space Dynamics 3
Conservative Dynamical Systems 3
Dissipative Dynamical Systems 4
Routes to Chaos 8
Turbulence in Fluids 11
1. Nonlinear Differential Equations 13
1.1. Deterministic Problems 14
1.2. Equilibrium Points and Stability 15
(i) Liapunov- and Asymptotic Stability 21
(ii) The Center Manifold Theorem 25
1.3. Phase-plane Analysis 32
1.4. Fully Nonlinear Evolution 49
1.5. Non-autonomous Systems 51
2. Bifurcation Theory 61
2.l. Stability and Bifurcation 62
2.2. Saddle-Node, Transcritical and Pitchfork Bifurcations 65
2.3. Hopf Bifurcation 78
2.4. Bifurcation Theory of One-Dimensional Maps 85
2.5. Appendix 88
3. Hamiltonian Dynamics 93
3.1. Hamilton's Equations 94
3.2. Phase Space 98
3.3. Canonical Transformations 105
3.4. The Hamilton-Jacobi Equation 112
3.5. Action-Angle Variables 118
3.6. Infinitesimal Canonical Transformations 121
3.7. Poisson's Brackets 122
4. Integrable Systems 127
4.l. Separable Hamiltonian Systems 127
4.2. Integrable Systems 129
4.3. Dynamics on the Tori 137
4.4. Canonical Perturbation Theory 140
4.5. Kolmogorov-Amol'd-Moser Theory 149
4.6. Breakdown of Integrability and Criteria for Transition to Chaos 160
(i) Local Criteria 160
(ii) Local Stability vs. Global Stability 163
viii TABLE OF CONTENTS
(iii) Global Criteria 166
4.7. Magnetic Island Overlap and Stochasticity in Magnetic
Confinement Systems 171
4.8 Appendix: The Problem of Internal Resonance in
Nonlinearly-Coupled Systems 177
5. Chaos in Conservative Systems 197
5.1. Phase-Space Dynamics of Conservative Systems 197
5.2. Poincare's Surface of Section 198
5.3. Area-preserving Mappings 205
5.4. Twist Maps 207
5.5. Tangent Maps 211
5.6. Poincare-Birkhoff Fixed-Point Theorem 214
5.7. Homoc1inic and Heteroc1inic Points 217
5.8. Quantitative Measures of Chaos 224
(i) Liapunov Exponents 225
(ii) Kolmogorov Entropy 229
(iii) Autocorrelation Function 231
(iv) Power Spectra 232
5.9. Ergodicity and Mixing 233
(i) Ergodicity 233
(ii) Mixing 236
(iii) Baker's Transformation 239
6. Chaos in Dissipative Systems 247
6.1. Phase-Space Dynamics of Dissipative Systems 247
6.2. Strange Attractors 251
6.3. Fractals 254
6.4. Multi-fractals 263
6.5. Analysis of Time Series Data 274
6.6. The Lorenz Attractor 276
(i) Equilibrium Solutions and Their Stability 277
(ii) Slightly Supercritical Case 280
(iii) Existence of an Attractor 284
(iv) Chaotic Behavior of the Nonlinear Solutions 284
6.6. Period-Doubling Bifurcations 286
(i) Difference Equations 286
(ii) The Logistic Map 289
Appendix 6.1: The Hausdorff-Besicovitch Dimension 310
Appendix 6.2: The Derivation of Lorenz's Equations 312
Appendix 6.3: The Derivation of Universality for One-Dimensional Maps 314
TABLE OF CONTENTS ix
7. Fractals and Multi-Fractals in Turbulence 319
7.1. Scale Invariance of the Navier-Stokes Equations and the
Kolmogorov (1941) Theory 328
7.2. The f3 -model for Turbulence 332
7.3. The Multi-fractal Models 336
7.4. The Random-f3 Model 340
7.5. The Intermediate Dissipation Range 348
8. Singularity Analysis and the Painleve Property of Dynamical Systems 353
8.1. The Painleve Property 353
8.2. Singularity Analysis 357
8.3. The Painleve Property for Partial Differential Equations 367
Exercises 375
References 385
Index 399
Preface
FolJowing the formulation of the laws of mechanics by Newton, Lagrange
sought to clarify and emphasize their geometrical character. Poincare and Liapunov
successfuIJy developed analytical mechanics further along these lines. In this approach,
one represents the evolution of all possible states (positions and momenta) by the flow in
phase space, or more efficiently, by mappings on manifolds with a symplectic geometry,
and tries to understand qualitative features of this problem, rather than solving it
explicitly.
One important outcome of this line of inquiry is the discovery that vastly
different physical systems can actually be abstracted to a few universal forms, like
Mandelbrot's fractal and Smale's horse-shoe map, even though the underlying processes
are not completely understood. This, of course, implies that much of the observed
diversity is only apparent and arises from different ways of looking at the same system.
Thus, modern nonlinear dynamics 1 is very much akin to classical thermodynamics in
that the ideas and results appear to be applicable to vastly different physical systems.
Chaos theory, which occupies a central place in modem nonlinear dynamics,
refers to a deterministic development with chaotic outcome. Computers have contributed
considerably to progress in chaos theory via impressive complex graphics. However, this
approach lacks organization and therefore does not afford complete insight into the
underlying complex dynamical behavior. This dynamical behavior mandates concepts
and methods from such areas of mathematics and physics as nonlinear differential
equations, bifurcation theory, Hamiltonian dynamics, number theory, topology, fractals,
and others.
This book has grown out of my lecture notes for an interdisciplinary graduate
level course on nonlinear dynamics which I have taught for the past several years to a
mix of students in applied mathematics, physics, and engineering. I have endeavored to
describe the basic concepts, language and results of nonlinear dynamical systems. This
book is accessible, therefore, to first-year graduate students in applied mathematics,
physics, and engineering, and is useful, of course, to any theoretically inclined
researcher in the physical sciences and engineering. In order to aIJow an interdisciplinary
readership, I have adopted an informal style, kept the mathematical
I In view of the ubiquity of nonlinear phenomena, it is rather awkward 10 call the subject nonlinear dynamics.
As Stanislaus Ulam pUI it (Gleick (1987», "Calling the subject nonlinear dynamics is like calling zoology
'non·elephant sludies'".
xi
xii PREFACE
formalism to a minimum, and omitted much detail. Thus, I have stated a few theorems
for which the proofs are only sketched or are replaced by illustrative examples which
make the general principles plausible.
The book starts with a discussion of nonlinear ordinary differential equations,
bifurcation theory, and Hamiltonian dynamics. It then embarks on a systematic
discussion of the traditional topics of modern nonlinear dynamics - integrable systems,
Poincare maps, chaos, fractals and strange attractors. The Baker's transformation, the
logistic map and the Lorenz system are discussed in detail in view of their central place
in the subject. Then, there is a detailed discussion of the application of fractals and
multi-fractals to fully-developed turbulence - a problem whose understanding has been
considerably enriched by the application of the concepts and methods of modern
nonlinear dynamics. Finally, there is a discussion of the Painleve property of nonlinear
differential equations which seems to provide a test of integrability. On the application
side, one may discern a special emphasis on some aspects of fluid dynamics and plasma
physics reflecting my continued involvement in these areas of physics. I have provided a
few exercises that range from simple applications to occasional considerable extension
of the theory. Finally, the list of references given at the end of the book contains
primarily books and papers I have used in developing my lecture material (and this
book). This list is not to be construed as complete (experts may notice omission of many
important papers).
Orlando Bhimsen K. Shivamoggi
1997