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. "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 "The author has chosen an excellent subject, which will probably become a main direction of research in the field of stochastic differential equations. This book is addressed to a wide readership: specialists in dynamical systems and stochastic processes, mathematicians, engineers, physicists, and neuroscientists. The author succeeds in making the material interesting to all these groups of researchers." ―Florin Diacu, Pacific Institute for the Mathematical Sciences, University of Victoria "The author has chosen an excellent subject, which will probably become a main direction of research in the field of stochastic differential equations. This book is addressed to a wide readership: specialists in dynamical systems and stochastic processes, mathematicians, engineers, physicists, and neuroscientists. The author succeeds in making the material interesting to all these groups of researchers." --Florin Diacu, Pacific Institute for the Mathematical Sciences, University of Victoria Emil Simiu is a NIST Fellow, National Institute of Standards and Technology, and Research Professor, Whiting School of Engineering, The Johns Hopkins University. A specialist in flow-structure interaction, he is the coauthor of Wind Effects on Structures and was the 1984 recipient of the Federal Engineer of the Year award. Chaotic Transitions in Deterministic and Stochastic Dynamical Systems Applications of Melnikov Processes in Engineering, Physics, and Neuroscience By Emil Simiu PRINCETON UNIVERSITY PRESS Copyright © 2002 Princeton University Press All rights reserved. ISBN: 978-0-691-14434-4 Contents Preface, xi, Chapter 1. Introduction, 1, PART 1. FUNDAMENTALS, 9, Chapter 2. Transitions in Deterministic Systems and the Melnikov Function, 11, Chapter 3. Chaos in Deterministic Systems and the Melnikov Function, 51, Chapter 4. Stochastic Processes, 76, Chapter 5. Chaotic Transitions in Stochastic Dynamical Systems and the Melnikov Process, 98, PART 2. APPLICATIONS, 127, Chapter 6. Vessel Capsizing, 129, Chapter 7. Open-Loop Control of Escapes in Stochastically Excited Systems, 134, Chapter 8. Stochastic Resonance, 144, Chapter 9. Cutoff Frequency of Experimentally Generated Noise for a First-Order Dynamical System, 156, Chapter 10. Snap-Through of Transversely Excited Buckled Column, 159, Chapter 11. Wind-Induced Along-Shore Currents over a Corrugated Ocean Floor, 167, Chapter 12. The Auditory Nerve Fiber as a Chaotic Dynamical System, 178, Appendix A1 Derivation of Expression for the Melnikov Function, 191, Appendix A2 Construction of Phase Space Slice through Stable and Unstable Manifolds, 193, Appendix A3 Topological Conjugacy, 199, Appendix A4 Properties of Space σ2, 201, Appendix A5 Elements of Probability Theory, 203, Appendix A6 Mean Upcrossing Rate τu-1 for Gaussian Processes, 211, Appendix A7 Mean Escape Rate τε-1 for Systems Excited by White Noise, 213, References, 215, Index, 221, CHAPTER 1 Introduction This work has two main objectives: (1) to present the Melnikov method as a unified theoretical framework for the study of transitions and chaos in a wide class of determini