This concise and comprehensive overview of nonlinear processes addresses all those interested in natural sciences and mathematics. It also contains a beautifully illustrated color insert easily accessible to the interested layperson. Thus it is suitable for leisure reading and also for an introductory (under)graduate course in nonlinear physics. Both well-established and more recent new results are discussed, outlining the relation between classical aspects of nonlinear physics and important current problems like the birth of chaos in simple deterministic systems and the emergence of order out of disorder and turbulence. Keywords: Chaos, fractals, strange attractors, turbulence.
Author(s): Andrei V. Gaponov-Grekhov, Mikhail I. Rabinovich, E.F. Hefter, N. Krivatkina
Edition: 1
Publisher: Springer
Year: 1993
Language: English
Pages: 201
Title page......Page 1
Date-line......Page 2
Foreword......Page 3
Preface......Page 5
Contents......Page 7
1. Introduction......Page 11
2.1.1 A Marble in the Chute......Page 21
2.1.2 Spring Pendulum and Nonlinear Optics......Page 24
2.1.3 Nonlinear Landau Damping and Amplification......Page 28
2.2.1 The Fermi-Pasta-Ulam Paradox......Page 32
2.2.2 Solitons as Particles......Page 35
2.2.3 Solitons and Shock Waves......Page 37
2.3 Self-Excited Oscillations......Page 41
2.3.1 Examples and Definitions......Page 42
2.3.2 Competition and Synchronization......Page 46
2.3.3 Self-Excited Oscillations in Chains and Continuous Systems......Page 48
2.4.1 Acquisition of a New Quality......Page 50
2.4.2 Bifurcations of Equilibrium States......Page 53
2.4.4 Bifurcations - Changes of Stability in Periodic Motion......Page 55
2.5.1 The Role of Small Parameters......Page 56
2.5.2 Running Mandelstam Lattices. Modulation of Waves by Waves......Page 58
2.5.3 Generation of Modulation......Page 61
2.5.4 Self-Modulation......Page 62
2.5.5 Recurrence......Page 64
2.5.6 Modulation Solitons......Page 66
3.1 Historical Remarks......Page 69
3.2 Marble in an Oscillating Chute......Page 70
3.3 Stochastic Self-Excited Oscillations......Page 74
3.3.1 The Lorenz Attractor......Page 75
3.3.2 Synchronization - Beats - Chaos......Page 78
3.3.3 Autonomous Noise Generator......Page 79
3.3.4 Scenarios for the Birth of Strange Attractors......Page 82
3.4.1 Dimension and Entropy......Page 85
3.4.2 The Cantor Structure of a Strange Attractor......Page 86
3.4.3 Dimension and Lyapunov Exponent......Page 88
3.4.4 Deterministically Generated and Random Signals......Page 91
4.1 Order and Disorder - Examples......Page 95
4.2.1 Examples of Equations......Page 100
4.2.2 Multistability. Defects......Page 102
4.3.1 Convective Self-Structures......Page 108
4.3.2 Localization Mechanisms......Page 110
4.3.3 Self-Structures in Three-Dimensional Media......Page 111
4.3.4 Interaction of "Elementary Particles"......Page 113
4.3.5 Birth and Interaction of Spiral Waves......Page 115
4.4.1 How to Remember......Page 117
4.4.2 "Camera + TV + Feedback" Analogue......Page 119
4.4.3 Critical Phenomena >......Page 122
4.4.4 Structures in Neuron-Like Media......Page 123
5.1 Prehistory......Page 127
5.2 Basic Models of Dynamic Theory......Page 130
5.3.1 Experiments......Page 132
5.3.2 Development of Turbulence and Multi-Dimensional Attractors......Page 135
5.4.1 Flow Dimension......Page 137
5.4.2 Spatial Bifurcations......Page 139
5.5 Discussion......Page 140
6.1 The Where and the How......Page 143
6.2 Randomness Born out of Nonrandomness......Page 144
6.3 An Unstable Path and Steady Motion. Are They Incompatible?......Page 146
6.4 Does Chance Rule the World?......Page 147
6.5 What is the Character of Nature? Integer or Fractal?......Page 149
6.6 Fractal Fingers......Page 151
6.7 Self-Organizing Structures......Page 153
6.9 The New Life of an Old Problem......Page 155
6.10 Spatial Evolution of Disorder......Page 156
6.11 What Does Your Camera See When It is Watching TV?......Page 157
6.12 Multistability and Memory......Page 158
6.13 Nonlinear Dynamics in Society......Page 159
Color Plates......Page 161
Literature......Page 187
Acknowledgements of the Figures......Page 195
Subject Index......Page 197
