Chaotic Transitions in Deterministic and Stochastic Dynamical Systems

Preface xiChapter 1. Introduction 1PART 1.FUNDAMENTALS 9Chapter 2. Transitions in Deterministic Systems and the Melnikov Function 112.1 Flows and Fixed Points.Integrable Systems.Maps: Fixed and Periodic Points 122.2 Homoclinic and Heteroclinic Orbits.Stable and Unstable Manifolds 202.3 Stable and Unstable Manifolds in the Three-Dimensional PhaseSpace 232.4 The Melnikov Function 272.5 Melnikov Functions for Special Types of Perturbation.MelnikovScale Factor 292.6 Condition for the Intersection of Stable and Unstable Manifolds. Interpretation from a System Energy Viewpoint 362.7 Poincare Maps,Phase Space Slices,and Phase Space Flux 382.8 Slowly Varying Systems 45Chapter 3. Chaos in Deterministic Systems and the Melnikov Function 513.1 Sensitivity to Initial Conditions and Lyapounov Exponents. Attractors and Basins of Attraction 523.2 Cantor Sets.Fractal Dimensions 573.3 The Smale Horseshoe Map and the Shift Map 593.4 Symbolic Dynamics. Properties of the Space Z2. Sensitivity to Initial Conditions of the Smale Horseshoe Map. Mathematical Definition of Chaos 653.5 Smale-Birkhoff Theorem. Melnikov Necessary Condition for Chaos. Transient and Steady-State Chaos 673.6 Chaotic Dynamics in Planar Systems with a Slowly Varying Parameter 703.7 Chaos in an Experimental System: The Stoker Column 72Chapter 4. Stochastic Processes 764.1 Spectral Density, Autocovariance, Cross-Covariance 764.2 Approximate Representations of Stochastic Processes 874.3 Spectral Density of the Output of a Linear Filter with Stochastic Input 94Chapter 5. Chaotic Transitions in Stochastic Dynamical Systems and the Melnikov Process 985.1 Behavior of a Fluidelastic Oscillator with Escapes: Experimental and Numerical Results 1005.2 Systems with Additive and Multiplicative Gaussian Noise: Melnikov Processes and Chaotic Behavior 1025.3 Phase Space Flux 1065.4 Condition Guaranteeing Nonoccurrence of Escapes in Systems Excited by Finite-Tailed Stochastic Processes. Example: Dichotomous Noise 1095.5 Melnikov-Based Lower Bounds for Mean Escape Time and for Probability of Nonoccurrence of Escapes during a Specified Time Interval 1125.6 Effective Melnikov Frequencies and Mean Escape Time 1195.7 Slowly Varying Planar Systems 1225.8 Spectrum of a Stochastically Forced Oscillator: Comparison between Fokker-Planck and Melnikov-Based Approaches 122PART 2. APPLICATIONS 127Chapter 6. Vessel Capsizing 1296.1 Model for Vessel Roll Dynamics in Random Seas 1296.2 Numerical Example 132Chapter 7. Open-Loop Control of Escapes in Stochastically Excited Systems 1347.1 Open-Loop Control Based on the Shape of the Melnikov Scale Factor 1347.2 Phase Space Flux Approach to Control of Escapes Induced by Stochastic Excitation 140Chapter 8. Stochastic Resonance 1448.1 Definition and Underlying Physical Mechanism of Stochastic Resonance. Application of the Melnikov Approach 1458.2 Dynamical Systems and Melnikov Necessary Condition for Chaos 1468.3 Signal-to-Noise Ratio Enhancement for a Bistable Deterministic System 1478.4 Noise Spectrum Effect on Signal-to-Noise Ratio for Classical Stochastic Resonance 1488.5 System with Harmonic Signal and Noise: Signal-to-Noise Ratio Enhancement through the Addition of a Harmonic Excitation 1528.6 Nonlinear Transducing Device for Enhancing Signal-to-Noise Ratio 1538.7 Concluding Remarks 154Chapter 9. Cutoff Frequency of Experimentally Generated Noise for a First-Order Dynamical System 1569.1 Introduction 1569.2 Transformed Equation Excited by White Noise 157Chapter 10. Snap-Through of Transversely Excited Buckled Column 15910.1 Equation of Motion 16010.2 Harmonic Forcing 16110.3 Stochastic Forcing. Nonresonance Conditions. Melnikov Processes for Gaussian and Dichotomous Noise 16310.4 Numerical Example 164Chapter 11. Wind-Induced Along-Shor Currents over a Corrugated Ocean Floor 16711.1 Offshore Flow Model 16811.2 Wind Velocity Fluctuations and Wind Stresses 17011.3 Dynamics of Unperturbed System 17211.4 Dynamics of Perturbed System 17311.5 Numerical Example 174Chapter 12. The Auditory Nerve Fiber as a Chaotic Dynamical System 17812.1 Experimental Neurophysiological Results 17912.2 Results of Simulations Based on the Fitzhugh-Nagumo Model. Comparison with Experimental Results 18212.3 Asymmetric Bistable Model of Auditory Nerve Fiber Response 18312.4 Numerical Simulations 18612.5 Concluding Remarks 190Appendix A1 Derivation of Expression for the Melnikov Function 191Appendix A2 Construction of Phase Space Slice through Stable and Unstable Manifolds 193Appendix A3 Topological Conjugacy 199Appendix A4 Properties of Space Z2 201Appendix A5 Elements of Probability Theory 203Appendix A6 Mean Upcrossing Rate for Gaussian Processes 211Appendix A7 Mean Escape Rate for Systems Excited by White Noise 213References 215Index 221