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.

ISBN 0691050945   Pages 246 ISBN13 9780691050942 (What's this?)   Volumes 1 Publisher The University Press Group Ltd   Weight (grammes) 485 Imprint Princeton University Press   Published in New Jersey Format Hardback   Series title Princeton Series in Applied Mathematics Publication date 01 Apr 2002   Height (mm) 229 Library of Congress 2001059163   Width (mm) 152 DEWEY 003.76   Spine width (mm) 21 DEWEY edition DC21   Academic level Professional / Scholarly, Tertiary education

		Preface
Ch. 1		Introduction		1
Ch. 2		Transitions in Deterministic Systems and the Melnikov Function		11
2.1		Flows and Fixed Points. Integrable Systems. Maps: Fixed and Periodic Points		13
2.2		Homoclinic and Heteroclinic Orbits, Stable and Unstable Manifolds		20
2.3		Stable and Unstable Manifolds in the Three-Dimensional Phase Space [X[subscript 1], X[subscript 2], t]		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		Poincare Maps, Phase Space Slices, and Phase Space Flux		38
2.8		Slowly Varying Systems		45
Ch. 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 [Sigma][subscript 2], 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		71
Ch. 4		Stochastic Processes		76
4.1		Spectral Density, Autocovariance, Cross-Covariance		76
4.2		Approximate Representations of Stochastic Processes		87
		More...

Highly readable, elegant, and concise... Emil Simiu has succeeded in putting together a highly stimulating book that proposes a promising, unifying approach to various aspects of chaos theory. While encompassing a wide swath of topics, traditionally found only on scattered sources, the book is succinctly written, exhibiting a quality reserved to the best of review works. -- Daniel ben-Avraham Journal of Statistical Physics

 Be the first to write a customer review