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

Format:

Book

Author/Creator:

Simiu, Emil, author.

Series:

Princeton series in applied mathematics.

Princeton Series in Applied Mathematics

Language:

English

Subjects (All):

Differentiable dynamical systems.

Chaotic behavior in systems.

Stochastic systems.

Physical Description:

1 online resource (244 p.)

Edition:

1st ed.

Place of Publication:

Princeton, New Jersey : Princeton University Press, 2002.

Language Note:

English

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.

Contents:

Front matter

Contents

Preface

Chapter 1. Introduction

PART 1. FUNDAMENTALS

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

PART 2. APPLICATIONS

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 ∑2

Appendix A5 Elements of Probability Theory

Appendix A6 Mean Upcrossing Rate τu-1 for Gaussian Processes

Appendix A7 Mean Escape Rate τ∊-1 for Systems Excited by White Noise

References

Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9780691050942

0691050945

9781400832507

1400832500

OCLC:

888348995