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 appli...
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Princeton
Princeton University Press
2002
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| Edition: | 1 |
| Series: | Princeton Series in Applied Mathematics |
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| ISBN: | 0691144346, 9780691144344, 9780691050942, 0691050945 |
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| Abstract | 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. |
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| AbstractList | No detailed description available for "Chaotic Transitions in Deterministic and Stochastic Dynamical Systems". 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. |
| Author | Simiu, Emil |
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| Snippet | No detailed description available for "Chaotic Transitions in Deterministic and Stochastic Dynamical Systems". The classical Melnikov method provides information on the behavior of deterministic planar systems that may exhibit transitions, i.e. escapes from and captures... |
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| SubjectTerms | Chaotic behavior in systems Differentiable dynamical systems Stochastic systems |
| Subtitle | Applications of Melnikov Processes in Engineering, Physics, and Neuroscience |
| TableOfContents | 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 |
| Title | Chaotic Transitions in Deterministic and Stochastic Dynamical Systems |
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