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

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 Poincaré 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