Chaotic Transitions in Deterministic and Stochastic Dynamical Systems

by Simiu, Emil

• ISBN13: 9780691050942
• ISBN10: 0691050945
• Format: Hardcover
• Copyright: 2002-04-01
• Publisher: Princeton Univ Pr

Summary

The classical Melnikov method provides information on the behavior of deterministic planar systems that may exhibit transitions, i.e. escapes from and captures into preferred regions of phase space. This book develops a unified treatment of deterministic and stochastic systems that extends the applicability of the Melnikov method to physically realizable stochastic planar systems with additive, state-dependent, white, colored, or dichotomous noise. The extended Melnikov method yields the novel result that motions with transitions are chaotic regardless of whether the excitation is deterministic or stochastic. It explains the role in the occurrence of transitions of the characteristics of the system and its deterministic or stochastic excitation, and is a powerful modeling and identification tool. The book is designed primarily for readers interested in applications. The level of preparation required corresponds to the equivalent of a first-year graduate course in applied mathematics. No previous exposure to dynamical systems theory or the theory of stochastic processes is required. The theoretical prerequisites and developments are presented in the first part of the book. The second part of the book is devoted to applications, ranging from physics to mechanical engineering, naval architecture, oceanography, nonlinear control, stochastic resonance, and neurophysiology.

Author Biography

Emil Simiu is a NIST Fellow, National Institute of Standards and Technology, and Research Professor, Whiting School of Engineering, The Johns Hopkins University. A specialist in flow-structure interaction, he is the coauthor of Wind Effects on Structures and was the 1984 recipient of the Federal Engineer of the Year award

Table of Contents

Preface	p. xi
Introduction	p. 1
Fundamentals	p. 9
Transitions in Deterministic Systems and the Melnikov Function	p. 11
Flows and Fixed Points. Integrable Systems. Maps: Fixed and Periodic Points	p. 13
Homoclinic and Heteroclinic Orbits. Stable and Unstable Manifolds	p. 20
Stable and Unstable Manifolds in the Three-Dimensional Phase Space {x[subscript 1], x[subscript 2], t}	p. 23
The Melnikov Function	p. 27
Melnikov Functions for Special Types of Perturbation. Melnikov Scale Factor	p. 29
Condition for the Intersection of Stable and Unstable Manifolds. Interpretation from a System Energy Viewpoint	p. 36
Poincare Maps, Phase Space Slices, and Phase Space Flux	p. 38
Slowly Varying Systems	p. 45
Chaos in Deterministic Systems and the Melnikov Function	p. 51
Sensitivity to Initial Conditions and Lyapounov Exponents. Attractors and Basins of Attraction	p. 52
Cantor Sets. Fractal Dimensions	p. 57
The Samle Horseshoe Map and the Shift Map	p. 59
Symbolic Dynamics. Properties of the Space [Sigma subscript 2]. Sensitivity to Initial Conditions of the Smale Horseshoe Map. Mathematical Definition of Chaos	p. 65
Smale-Birkhoff Theorem. Melnikov Necessary Condition for Chaos. Transient and Steady-State Chaos	p. 67
Chaotic Dynamics in Planar Systems with a Slowly Varying Parameter	p. 70
Chaos in an Experimental System: The Stoker Column	p. 71
Stochastic Processes	p. 76
Spectral Density, Autocovariance, Cross-Covariance	p. 76
Approximate Representations of Stochastic Processes	p. 87
Spectral Density of the Output of a Linear Filter with Stochastic Input	p. 94
Chaotic Transitions in Stochastic Dynamical Systems and the Melnikov Process	p. 98
Behavior of a Fluidelastic Oscillator with Escapes: Experimental and Numerical Results	p. 100
Systems with Additive and Multiplicative Gaussian Noise: Melnikov Processes and Chaotic Behavior	p. 102
Phase Space Flux	p. 106
Condition Guaranteeing Nonoccurrence of Escapes in Systems Excited by Finite-Tailed Stochastic Processes. Example: Dichotomous Noise	p. 109
Melnikov-Based Lower Bounds for Mean Escape Time and for Probability of Nonoccurrence of Escapes during a Specified Time Interval	p. 112
Effective Melnikov Frequencies and Mean Escape Time	p. 119
Slowly Varying Planar Systems	p. 122
Spectrum of a Stochastically Forced Oscillator: Comparison between Fokker-Planck and Melnikov-Based Approaches	p. 122
Applications	p. 127
Vessel Capsizing	p. 129
Model for Vessel Roll Dynamics in Random Seas	p. 129
Numerical Example	p. 132
Open-Loop Control of Escapes in Stochastically Excited Systems	p. 134
Open-Loop Control Based on the Shape of the Melnikov Scale Factor	p. 134
Phase Space Flux Approach to Control of Escapes Induced by Stochastic Excitation	p. 140
Stochastic Resonance	p. 144
Definition and Underlying Physical Mechanism of Stochastic Resonance. Application of the Melnikov Approach	p. 145
Dynamical Systems and Melnikov Necessary Condition for Chaos	p. 146
Signal-to-Noise Ratio Enhancement for a Bistable Deterministic System	p. 147
Noise Spectrum Effect on Signal-to-Noise Ratio for Classical Stochastic Resonance	p. 149
System with Harmonic Signal and Noise: Signal-to-Noise Ratio Enhancement through the Additional of a Harmonic Excitation	p. 152
Nonlinear Transducing Device for Enhancing Signal-to-Noise Ratio	p. 153
Concluding Remarks	p. 155
Cutoff Frequency of Experimentally Generated Noise for a First-Order Dynamical System	p. 156
Introduction	p. 156
Transformed Equation Excited by White Noise	p. 157
Snap-Through of Transversely Excited Buckled Column	p. 159
Equation of Motion	p. 160
Harmonic Forcing	p. 161
Stochastic Forcing. Nonresonance Conditions. Melnikov Processes for Gaussian and Dichotomous Noise	p. 163
Numerical Example	p. 164
Wind-Induced Along-Shore Currents over a Corrugated Ocean Floor	p. 167
Offshore Flow Model	p. 168
Wind Velocity Fluctuations and Wind Stresses	p. 170
Dynamics of Unperturbed System	p. 172
Dynamics of Perturbed System	p. 173
Numerical Example	p. 174
The Auditory Nerve Fiber as a Chaotic Dynamical System	p. 178
Experimental Neurophysiological Results	p. 179
Results of Simulations Based on the Fitzhugh-Nagumo Model. Comparison with Experimental Results	p. 182
Asymmetric Bistable Model of Auditory Nerve Fiber Response	p. 183
Numerical Simulations	p. 186
Concluding Remarks	p. 190
Derivation of Expression for the Melnikov Function	p. 191
Construction of Phase Space Slice through Stable and Unstable Manifolds	p. 193
Topological Conjugacy	p. 199
Properties of Space [Sigma subscript 2]	p. 201
Elements of Probability Theory	p. 203
Mean Upcrossing Rate [tau superscript -1 subscript u] for Gaussian Processes	p. 211
Mean Escape Rate [tau superscript -1 subscript [epsilon] for Systems Excited by White Noise	p. 213
References	p. 215
Index	p. 221
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