Chapter 3. Chaos in Deterministic Systems and the Melnikov Function

Chapters in this book

1. Frontmatter i
2. Contents vii
3. Preface xi
4. Chapter 1. Introduction 1
5. PART 1. FUNDAMENTALS
6. Chapter 2. Transitions in Deterministic Systems and the Melnikov Function 11
7. Chapter 3. Chaos in Deterministic Systems and the Melnikov Function 51
8. Chapter 4. Stochastic Processes 76
9. Chapter 5. Chaotic Transitions in Stochastic Dynamical Systems and the Melnikov Process 98
10. PART 2. APPLICATIONS
11. Chapter 6. Vessel Capsizing 129
12. Chapter 7. Open-Loop Control of Escapes in Stochastically Excited Systems 134
13. Chapter 8. Stochastic Resonance 144
14. Chapter 9. Cutoff Frequency of Experimentally Generated Noise for a First-Order Dynamical System 156
15. Chapter 10. Snap-Through of Transversely Excited Buckled Column 159
16. Chapter 11. Wind-Induced Along-Shore Currents over a Corrugated Ocean Floor 167
17. Chapter 12. The Auditory Nerve Fiber as a Chaotic Dynamical System 178
18. Appendix A1 Derivation of Expression for the Melnikov Function 191
19. Appendix A2 Construction of Phase Space Slice through Stable and Unstable Manifolds 193
20. Appendix A3 Topological Conjugacy 199
21. Appendix A4 Properties of Space ∑₂ 201
22. Appendix A5 Elements of Probability Theory 203
23. Appendix A6 Mean Upcrossing Rate τ_u^-1 for Gaussian Processes 211
24. Appendix A7 Mean Escape Rate τ_∊^-1 for Systems Excited by White Noise 213
25. References 215
26. Index 221