Title Chaotic transitions in deterministic and stochastic dynamical systems : applications of Melnikov processes in engineering, physics, and neuroscience / Emil Simiu

Published Princeton, New Jersey : Princeton University Press, 2002 ©2002

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Description 1 online resource (241 pages) : illustrations Series Princeton Series in Applied Mathematics Princeton series in applied mathematics. Contents Transitions in Deterministic Systems and the Melnikov Function -- Flows and Fixed Points. Integrable Systems. Maps: Fixed and Periodic Points -- Homoclinic and Heteroclinic Orbits. Stable and Unstable Manifolds -- Stable and Unstable Manifolds in the Three-Dimensional Phase Space {x[subscript 1], x[subscript 2], t} -- The Melnikov Function -- Melnikov Functions for Special Types of Perturbation. Melnikov Scale Factor -- Condition for the Intersection of Stable and Unstable Manifolds. Interpretation from a System Energy Viewpoint -- Poincare Maps, Phase Space Slices, and Phase Space Flux -- Slowly Varying Systems -- Chaos in Deterministic Systems and the Melnikov Function -- Sensitivity to Initial Conditions and Lyapounov Exponents. Attractors and Basins of Attraction -- Cantor Sets. Fractal Dimensions -- The Samle Horseshoe Map and the Shift Map -- Symbolic Dynamics. Properties of the Space [Sigma subscript 2]. Sensitivity to Initial Conditions of the Smale Horseshoe Map. Mathematical Definition of Chaos -- Smale-Birkhoff Theorem. Melnikov Necessary Condition for Chaos. Transient and Steady-State Chaos -- Chaotic Dynamics in Planar Systems with a Slowly Varying Parameter -- Chaos in an Experimental System: The Stoker Column -- Stochastic Processes -- Spectral Density, Autocovariance, Cross-Covariance -- Approximate Representations of Stochastic Processes -- Spectral Density of the Output of a Linear Filter with Stochastic Input -- Chaotic Transitions in Stochastic Dynamical Systems and the Melnikov Process Frontmatter -- Contents -- Preface -- Chapter 1. Introduction -- Chapter 2. Transitions in Deterministic Systems and the Melnikov Function -- Chapter 3. Chaos in Deterministic Systems and the Melnikov Function -- Chapter 4. Stochastic Processes -- Chapter 5. Chaotic Transitions in Stochastic Dynamical Systems and the Melnikov Process -- Chapter 6. Vessel Capsizing -- Chapter 7. Open-Loop Control of Escapes in Stochastically Excited Systems -- Chapter 8. Stochastic Resonance -- Chapter 9. Cutoff Frequency of Experimentally Generated Noise for a First-Order Dynamical System -- Chapter 10. Snap-Through of Transversely Excited Buckled Column -- Chapter 11. Wind-Induced Along-Shore Currents over a Corrugated Ocean Floor -- Chapter 12. The Auditory Nerve Fiber as a Chaotic Dynamical System -- 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 ∑ -- Appendix A5 Elements of Probability Theory -- Appendix A6 Mean Upcrossing Rate x -- Appendix A7 Mean Escape Rate x -- References -- Index Summary 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 Analysis Affine transformation Amplitude Arbitrarily large Attractor Autocovariance Big O notation Central limit theorem Change of variables Chaos theory Coefficient of variation Compound Probability Computational problem Control theory Convolution Coriolis force Correlation coefficient Covariance function Cross-covariance Cumulative distribution function Cutoff frequency Deformation (mechanics) Derivative Deterministic system Diagram (category theory) Diffeomorphism Differential equation Dirac delta function Discriminant Dissipation Dissipative system Dynamical system Eigenvalues and eigenvectors Equations of motion Even and odd functions Excitation (magnetic) Exponential decay Extreme value theory Flow velocity Fluid dynamics Forcing (recursion theory) Fourier series Fourier transform Fractal dimension Frequency domain Gaussian noise Gaussian process Harmonic analysis Harmonic function Heteroclinic orbit Homeomorphism Homoclinic orbit Hyperbolic point Inference Initial condition Instability Integrable system Invariant manifold Iteration Joint probability distribution LTI system theory Limit cycle Linear differential equation Logistic map Marginal distribution Moduli (physics) Multiplicative noise Noise (electronics) Nonlinear control Nonlinear system Ornstein-Uhlenbeck process Oscillation Parameter space Parameter Partial differential equation Perturbation function Phase plane Phase space Poisson distribution Probability density function Probability distribution Probability theory Probability Production-possibility frontier Relative velocity Scale factor Shear stress Spectral density Spectral gap Standard deviation Stochastic process Stochastic resonance Stochastic Stream function Surface stress Symbolic dynamics The Signal and the Noise Topological conjugacy Transfer function Variance Vorticity Bibliography Includes bibliographical references and index Notes In English Print version record Subject Differentiable dynamical systems. Chaotic behavior in systems. Stochastic systems. MATHEMATICS -- Calculus. MATHEMATICS -- Mathematical Analysis. MATHEMATICS -- Applied. Chaotic behavior in systems Differentiable dynamical systems Stochastic systems Form Electronic book ISBN 9781400832507 1400832500 9780691050942 0691050945

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