Nonlinear Dynamics: Non-Integrable Systems and Chaotic Dynamics

by Alexander B. Borisov, Vladimir V. Zverev

Overview

The book provides a concise and rigor introduction to the fundamentals of methods for solving the principal problems of modern non-linear dynamics. This monograph covers the basic issues of the theory of integrable systems and the theory of dynamical chaos both in nonintegrable conservative and in dissipative systems. A distinguishing feature of the material exposition is to add some comments, historical information, brief biographies and portraits of the researchers who made the most significant contribution to science. This allows one to present the material as accessible and attractive to students to acquire indepth scientific knowledge of nonlinear mechanics, feel the atmosphere where those or other important discoveries were made. The book can be used as a textbook for advanced undergraduate and graduate students majoring in high-tech industries and high technology (the science based on high technology) to help them to develop lateral thinking in early stages of training.

Contents:
Nonlinear Oscillations
Integrable Systems
Stability of Motion and Structural Stability
Chaos in Conservative Systems
Chaos and Fractal Attractors in Dissipative Systems
Conclusion
References
Index

Product Details

ISBN-13:	9783110430677
Publisher:	De Gruyter
Publication date:	11/21/2016
Series:	De Gruyter Studies in Mathematical Physics , #36
Sold by:	Barnes & Noble
Format:	eBook
Pages:	299
File size:	35 MB
Note:	This product may take a few minutes to download.
Age Range:	18 Years

About the Author

A. B. Borisov, Institute of Metal Physics, Ekaterinburg, Russia,
V. V. Zverev, Ural Federal University, Ekaterinburg, Russia.

Table of Contents

1 Nonlinear Oscillations 1

1.1 Nonlinear Oscillations of a Conservative Single-Degree-of-Freedom System 3

1.1.1 Qualitative Description of Motion by the Phase Plane Method 5

1.2 Oscillations of a Mathematical Pendulum. Elliptic Functions 9

1.3 Small-Amplitude Oscillations of a Conservative Single-Degree-of-Freedom System 14

1.3.1 Straightforward Expansion 15

1.3.2 The Method of Multiple Scales 18

1.3.3 The Method of Averaging: The Van der Pol Equation 22

1.3.4 The Generalized Method of Averaging. The Krylov-Bogolyubov Approach 24

1.4 Forced Oscillations of an Anharmonic Oscillator 27

1.4.1 Straightforward Expansion 28

1.4.2 A Secondary Resonance at ω ≈ ±3 29

1.4.3 A Primary Resonance: Amplitude-Frequency Response 31

1.5 Self-Oscillations: Limit Cycles 37

1.5.1 An Analytical Solution of the Van der Pol Equation for Small Nonlinearity Parameter Values 39

1.5.2 An approximate solution of the Van der Pol equation for large nonlinearity parameter values 42

1.6 External Synchronization of Self-Oscillating Systems 45

1.7 Parametric Resonance 55

1.7.1 The Floquet Theory 56

1.7.2 An Analytical Solution of the Mathieu Equation for Small Nonlinearity Parameter Values 60

2 Integrable Systems 65

2.1 Equations of Motion for a Rigid Body 65

2.1.1 Euler's Angles 68

2.1.2 Euler's Kinematic Equations 71

2.1.3 Moment of Inertia of a Rigid Body 73

2.1.4 Euler's Dynamic Equations 76

2.1.5 S.V. Kovalevskaya's Algorithm for Integrating Equations of Motion for a Rigid Body about a Fixed Point 79

2.2 The Painlevé Property for Differential Equations 85

2.2.1 A Brief Overview of the Analytic Theory of Differential Equations 85

2.2.2 A Modern Algorithm of Analysis of Integrable Systems 88

2.2.3 Integrability of the Generalized Henon-Heiles Model 94

2.2.4 The Linearization Method for Constructing Particular Solutions of a Nonlinear Model 98

2.3 Dynamics of Particles in the Toda Lattice: Integration by the Method of the Inverse Scattering Problem 100

2.3.1 Lax's Representation 104

2.3.2 The Direct Scattering Problem 108

2.3.3 The inverse scattering transform 116

2.3.4 N-Soliton Solutions 120

2.3.5 The Inverse Scattering Problem and the Riemann Problem 128

2.3.6 Solitons as Elementary Excitations of Nonlinear Integrable Systems 133

2.3.7 The Darboux-Backlund Transformations 135

2.3.8 Multiplication of Integrable Equations: The modified Toda Lattice 139

3 Stability of Motion and Structural Stability 145

3.1 Stability of Motion 145

3.1.1 Stability of Fixed Points and Trajectories 145

3.1.2 Succession Mapping or the Poincare Map 149

3.1.3 Theorem about the Volume of a Phase Drop 151

3.1.4 Poincare-Bendixson Theorem and Topology of the Phase Plane 153

3.1.5 The Lyapunov Exponents 155

3.2 Structural Stability 162

3.2.1 Topological Reconstruction of the Phase Portrait 162

3.2.2 Coarse Systems 165

3.2.3 Cusp Catastrophe 167

3.2.4 Catastrophe Theory 169

4 Chaos in Conservative Systems 174

4.1 Determinism and Irreversibility 174

4.2 Simple Models with Unstable Dynamics 180

4.2.1 Homoclinic Structure 180

4.2.2 The Anosov Map 182

4.2.3 The Tent Map 183

4.2.4 The Bernoulli Shift 187

4.3 Dynamics of Hamiltonian Systems Close to Integrable 189

4.3.1 Perturbed Motion and Nonlinear Resonance 189

4.3.2 The Zaslavsky-Chirikov Map 193

4.3.3 Chaos and Kolmogorov-Arnold-Moser Theory 195

5 Chaos and Fractal Attractors in Dissipative Systems 200

5.1 On the Nature of Turbulence 200

5.2 Dynamics of the Lorenz Model 203

5.2.1 Dissipativity of the Lorenz Model 205

5.2.2 Boundedness of the Region of Stationary Motion 205

5.2.3 Stationary Points 206

5.2.4 The Lorenz Model's Dynamic Regimes as a Result of Bifurcations 207

5.2.5 Motion on a Strange Attractor 208

5.2.6 Hypothesis About the Structure of a Strange Attractor 209

5.2.7 The Lorenz Model and the Tent Map 211

5.2.8 Lyapunov Exponents 212

5.3 Elements of Cantor Set Theory 213

5.3.1 Potential and Actual Infinity 213

5.3.2 Cantor's Theorem and Cardinal Numbers 217

5.3.3 Cantor sets 222

5.4 Cantor Structure of Attractors in Two-Dimensional Mappings 225

5.4.1 The Henon Map 225

5.4.2 The lkeda Map 227

5.4.3 An Analytical Theory of the Cantor Structure of Attractors 228

5.5 Mathematical Models of Fractal Structures 230

5.5.1 Massive Cantor Set 231

5.5.2 A binomial multiplicative process 232

5.5.3 The Spectrum of Fractal Dimensions 237

5.5.4 The Lyapunov Dimension 240

5.5.5 A Relationship Between the Mass Exponent and the Spectral Function 241

5.5.6 The Mass Exponent of the Multiplicative Binomial Process 243

5.5.7 A Multiplicative Binomial Process on a Fractal Carrier 244

5.5.8 A Temporal Data Sequence as a Source of Information About an Attractor 245

5.6 Universality and Scaling in the Dynamics of One-Dimensional Maps 249

5.6.1 General Regularities of a Period-Doubling Process 250

5.6.2 The Feigenbaum-Cvitanovic Equation 258

5.6.3 A Universal Regularity in the Arrangement of Cycles: A Universal Power Spectrum 262

5.7 Synchronization of Chaotic Oscillations 269

5.7.1 Synchronization in a System of Two Coupled Maps 270

5.7.2 Types and Criteria of Synchronization 272

Conclusion 274

References 277

Index 281

Editorial Reviews

"The technical calculations are done in detail. The book is a helpful addition to the topics considered in nonlinear dynamics, and for a student starting out with nonlinear dynamics this could be the ideal book."
W.-H. Steeb in: Mathematical Reviews Clippings 7/2017, S. 3

From the Publisher