Chaotic Transitions in Deterministic and Stochastic Dynamical Systems: Applications of Melnikov Processes in Engineering, Physics, and Neuroscience

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.

Table of Contents:
Preface xi Chapter 1. Introduction 1 PART 1.FUNDAMENTALS 9 Chapter 2. Transitions in Deterministic Systems and the Melnikov Function 11 2.1 Flows and Fixed Points.Integrable Systems.Maps: Fixed and Periodic Points 12 2.2 Homoclinic and Heteroclinic Orbits.Stable and Unstable Manifolds 20 2.3 Stable and Unstable Manifolds in the Three-Dimensional Phase Space 23 2.4 The Melnikov Function 27 2.5 Melnikov Functions for Special Types of Perturbation.Melnikov Scale Factor 29 2.6 Condition for the Intersection of Stable and Unstable Manifolds. Interpretation from a System Energy Viewpoint 36 2.7 Poincare Maps,Phase Space Slices,and Phase Space Flux 38 2.8 Slowly Varying Systems 45 Chapter 3. Chaos in Deterministic Systems and the Melnikov Function 51 3.1 Sensitivity to Initial Conditions and Lyapounov Exponents. Attractors and Basins of Attraction 52 3.2 Cantor Sets.Fractal Dimensions 57 3.3 The Smale Horseshoe Map and the Shift Map 59 3.4 Symbolic Dynamics. Properties of the Space Z2. Sensitivity to Initial Conditions of the Smale Horseshoe Map. Mathematical Definition of Chaos 65 3.5 Smale-Birkhoff Theorem. Melnikov Necessary Condition for Chaos. Transient and Steady-State Chaos 67 3.6 Chaotic Dynamics in Planar Systems with a Slowly Varying Parameter 70 3.7 Chaos in an Experimental System: The Stoker Column 72 Chapter 4. Stochastic Processes 76 4.1 Spectral Density, Autocovariance, Cross-Covariance 76 4.2 Approximate Representations of Stochastic Processes 87 4.3 Spectral Density of the Output of a Linear Filter with Stochastic Input 94 Chapter 5. Chaotic Transitions in Stochastic Dynamical Systems and the Melnikov Process 98 5.1 Behavior of a Fluidelastic Oscillator with Escapes: Experimental and Numerical Results 100 5.2 Systems with Additive and Multiplicative Gaussian Noise: Melnikov Processes and Chaotic Behavior 102 5.3 Phase Space Flux 106 5.4 Condition Guaranteeing Nonoccurrence of Escapes in Systems Excited by Finite-Tailed Stochastic Processes. Example: Dichotomous Noise 109 5.5 Melnikov-Based Lower Bounds for Mean Escape Time and for Probability of Nonoccurrence of Escapes during a Specified Time Interval 112 5.6 Effective Melnikov Frequencies and Mean Escape Time 119 5.7 Slowly Varying Planar Systems 122 5.8 Spectrum of a Stochastically Forced Oscillator: Comparison between Fokker-Planck and Melnikov-Based Approaches 122 PART 2. APPLICATIONS 127 Chapter 6. Vessel Capsizing 129 6.1 Model for Vessel Roll Dynamics in Random Seas 129 6.2 Numerical Example 132 Chapter 7. Open-Loop Control of Escapes in Stochastically Excited Systems 134 7.1 Open-Loop Control Based on the Shape of the Melnikov Scale Factor 134 7.2 Phase Space Flux Approach to Control of Escapes Induced by Stochastic Excitation 140 Chapter 8. Stochastic Resonance 144 8.1 Definition and Underlying Physical Mechanism of Stochastic Resonance. Application of the Melnikov Approach 145 8.2 Dynamical Systems and Melnikov Necessary Condition for Chaos 146 8.3 Signal-to-Noise Ratio Enhancement for a Bistable Deterministic System 147 8.4 Noise Spectrum Effect on Signal-to-Noise Ratio for Classical Stochastic Resonance 148 8.5 System with Harmonic Signal and Noise: Signal-to-Noise Ratio Enhancement through the Addition of a Harmonic Excitation 152 8.6 Nonlinear Transducing Device for Enhancing Signal-to-Noise Ratio 153 8.7 Concluding Remarks 154 Chapter 9. Cutoff Frequency of Experimentally Generated Noise for a First-Order Dynamical System 156 9.1 Introduction 156 9.2 Transformed Equation Excited by White Noise 157 Chapter 10. Snap-Through of Transversely Excited Buckled Column 159 10.1 Equation of Motion 160 10.2 Harmonic Forcing 161 10.3 Stochastic Forcing. Nonresonance Conditions. Melnikov Processes for Gaussian and Dichotomous Noise 163 10.4 Numerical Example 164 Chapter 11. Wind-Induced Along-Shor Currents over a Corrugated Ocean Floor 167 11.1 Offshore Flow Model 168 11.2 Wind Velocity Fluctuations and Wind Stresses 170 11.3 Dynamics of Unperturbed System 172 11.4 Dynamics of Perturbed System 173 11.5 Numerical Example 174 Chapter 12. The Auditory Nerve Fiber as a Chaotic Dynamical System 178 12.1 Experimental Neurophysiological Results 179 12.2 Results of Simulations Based on the Fitzhugh-Nagumo Model. Comparison with Experimental Results 182 12.3 Asymmetric Bistable Model of Auditory Nerve Fiber Response 183 12.4 Numerical Simulations 186 12.5 Concluding Remarks 190 Appendix A1 Derivation of Expression for the Melnikov Function 191 Appendix A2 Construction of Phase Space Slice through Stable and Unstable Manifolds 193 Appendix A3 Topological Conjugacy 199 Appendix A4 Properties of Space Z2 201 Appendix A5 Elements of Probability Theory 203 Appendix A6 Mean Upcrossing Rate for Gaussian Processes 211 Appendix A7 Mean Escape Rate for Systems Excited by White Noise 213 References 215 Index 221