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

Cover -- Title -- Copyright -- Contents -- Preface -- Chapter 1. Introduction -- PART 1. FUNDAMENTALS -- Chapter 2. Transitions in Deterministic Systems and the Melnikov Function -- 2.1 Flows and Fixed Points. Integrable Systems. Maps: Fixed and Periodic Points -- 2.2 Homoclinic and Heteroclinic Orbits. Stable and Unstable Manifolds -- 2.3 Stable and Unstable Manifolds in the Three-Dimensional Phase Space {x1, x2, t} -- 2.4 The Melnikov Function -- 2.5 Melnikov Functions for Special Types of Perturbation. Melnikov Scale Factor -- 2.6 Condition for the Intersection of Stable and Unstable Manifolds. Interpretation from a System Energy Viewpoint -- 2.7 Poincaré Maps, Phase Space Slices, and Phase Space Flux -- 2.8 Slowly Varying Systems -- Chapter 3. Chaos in Deterministic Systems and the Melnikov Function -- 3.1 Sensitivity to Initial Conditions and Lyapounov Exponents. Attractors and Basins of Attraction -- 3.2 Cantor Sets. Fractal Dimensions -- 3.3 The Smale Horseshoe Map and the Shift Map -- 3.4 Symbolic Dynamics. Properties of the Space ∑2. Sensitivity to Initial Conditions of the Smale Horseshoe Map. Mathematical Definition of Chaos -- 3.5 Smale-Birkhoff Theorem. Melnikov Necessary Condition for Chaos. Transient and Steady-State Chaos -- 3.6 Chaotic Dynamics in Planar Systems with a Slowly Varying Parameter -- 3.7 Chaos in an Experimental System: The Stoker Column -- Chapter 4. Stochastic Processes -- 4.1 Spectral Density, Autocovariance, Cross-Covariance -- 4.2 Approximate Representations of Stochastic Processes -- 4.3 Spectral Density of the Output of a Linear Filter with Stochastic Input -- Chapter 5. Chaotic Transitions in Stochastic Dynamical Systems and the Melnikov Process -- 5.1 Behavior of a Fluidelastic Oscillator with Escapes: Experimental and Numerical Results.

5.2 Systems with Additive and Multiplicative Gaussian Noise: Melnikov Processes and Chaotic Behavior -- 5.3 Phase Space Flux -- 5.4 Condition Guaranteeing Nonoccurrence of Escapes in Systems Excited by Finite-Tailed Stochastic Processes. Example: Dichotomous Noise -- 5.5 Melnikov-Based Lower Bounds for Mean Escape Time and for Probability of Nonoccurrence of Escapes during a Specified Time Interval -- 5.6 Effective Melnikov Frequencies and Mean Escape Time -- 5.7 Slowly Varying Planar Systems -- 5.8 Spectrum of a Stochastically Forced Oscillator: Comparison between Fokker-Planck and Melnikov-Based Approaches -- PART 2. APPLICATIONS -- Chapter 6. Vessel Capsizing -- 6.1 Model for Vessel Roll Dynamics in Random Seas -- 6.2 Numerical Example -- Chapter 7. Open-Loop Control of Escapes in Stochastically Excited Systems -- 7.1 Open-Loop Control Based on the Shape of the Melnikov Scale Factor -- 7.2 Phase Space Flux Approach to Control of Escapes Induced by Stochastic Excitation -- Chapter 8. Stochastic Resonance -- 8.1 Definition and Underlying Physical Mechanism of Stochastic Resonance. Application of the Melnikov Approach -- 8.2 Dynamical Systems and Melnikov Necessary Condition for Chaos -- 8.3 Signal-to-Noise Ratio Enhancement for a Bistable Deterministic System -- 8.4 Noise Spectrum Effect on Signal-to-Noise Ratio for Classical Stochastic Resonance -- 8.5 System with Harmonic Signal and Noise: Signal-to-Noise Ratio Enhancement through the Addition of a Harmonic Excitation -- 8.6 Nonlinear Transducing Device for Enhancing Signal-to-Noise Ratio -- 8.7 Concluding Remarks -- Chapter 9. Cutoff Frequency of Experimentally Generated Noise for a First-Order Dynamical System -- 9.1 Introduction -- 9.2 Transformed Equation Excited by White Noise -- Chapter 10. Snap-Through of Transversely Excited Buckled Column -- 10.1 Equation of Motion.

10.2 Harmonic Forcing -- 10.3 Stochastic Forcing. Nonresonance Conditions. Melnikov Processes for Gaussian and Dichotomous Noise -- 10.4 Numerical Example -- Chapter 11. Wind-Induced Along-Shore Currents over a Corrugated Ocean Floor -- 11.1 Offshore Flow Model -- 11.2 Wind Velocity Fluctuations and Wind Stresses -- 11.3 Dynamics of Unperturbed System -- 11.4 Dynamics of Perturbed System -- 11.5 Numerical Example -- Chapter 12. The Auditory Nerve Fiber as a Chaotic Dynamical System -- 12.1 Experimental Neurophysiological Results -- 12.2 Results of Simulations Based on the Fitzhugh-Nagumo Model. Comparison with Experimental Results -- 12.3 Asymmetric Bistable Model of Auditory Nerve Fiber Response -- 12.4 Numerical Simulations -- 12.5 Concluding Remarks -- Appendix A1 Derivation of Expression for the Melnikov Function -- Appendix A2 Construction of Phase Space Slice through Stable and Unstable Manifolds -- Appendix A3 Topological Conjugacy -- Appendix A4 Properties of Space ∑2 -- Appendix A5 Elements of Probability Theory -- Appendix A6 Mean Upcrossing Rate Τ^−1u for Gaussian Processes -- Appendix A7 Mean Escape Rate Τ^−1∊ for Systems Excited by White Noise -- References -- Index.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.