Chaotic Transitions in Deterministic and Stochastic Dynamical Systems: Applications of Melnikov Processes in Engineering, Physics, and Neuroscience (Princeton Series in Applied Mathematics) - Softcover

Simiu, Emil

Softcover

ISBN 10: 0691144346 ISBN 13: 9780691144344

Verlag: Princeton University Press, 2009

Inhaltsangabe

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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Emil Simiu

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"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

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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 deterministic and stochastic nonlinear planar systems, and (2) to demonstrate the method's usefulness in applications, particularly for stochastic systems. Our interest in the Melnikov method is motivated by its capability to provide criteria and information on the occurrence of transitions and chaotic behavior in a wide variety of systems in engineering, physics, and the life sciences.

To illustrate the type of problem to which the Melnikov method is applicable we consider a celebrated experiment on a system known as the magnetoelastic beam. The experiment demonstrates the remarkable type of dynamic behavior called deterministic chaos (Moon and Holmes, 1979). The system consists of (a) a rigid frame fixed onto a shaking table that may undergo periodic horizontal motions, (b) a beam with a vertical undeformed axis, an upper end fixed onto the frame, and a free lower end, and (c) two identical magnets equidistant from the undeformed position of the beam (Fig. 1.1). The beam experiences nonlinear displacement-dependent forces induced by the magnets, linear restoring forces due to its elasticity, dissipative forces due to its internal friction, the viscosity of the surrounding air, and magnetic damping, and periodic excitation forces due to the horizontal motion of the shaking table. Neither the system properties nor the forces acting on the beam vary randomly with time: the system is fully deterministic.

In the absence of excitation, and depending upon the initial conditions, the beam settles on one of two possible stable equilibria, that is, with the beam's tip closer to the right magnet or closer to the left magnet. The beam also has an unstable equilibrium position—its vertical undeformed axis.

If the excitation is periodic, three distinct types of steady-state dynamic behavior can occur:

1. For sufficiently small excitations, depending again upon the initial conditions, the beam moves periodically about one of its two stable equilibria. The periodic motion is confined to a half-plane bounded by the beam's unstable equilibrium position (Fig. 1.2(a)); in this type of motion there can be no escape from that half-plane.

2. For sufficiently large excitations the motion is periodic about—and crosses periodically—the unstable equilibrium position (Fig. 1.2(b)).

3. For intermediate excitation amplitudes, and for restricted sets of initial conditions and excitation frequencies, the steady-state motion is irregular, even though the system is fully deterministic; hence the term deterministic chaos. The motion evolves about one of the three equilibria, then it undergoes successive transitions, that is, it changes successively to motion about one of the other two equilibria (Fig. 1.2(c)). Transitions in such irregular, deterministic motion are referred to as chaotic. A transition away from motion in a half-plane bounded by the beam's unstable equilibrium position is called an escape. A transition to motion occurring within such a half-plane is called a capture. A succession of escapes and captures is referred to as hopping.

The system just described may be modeled as a dynamical system—a system that evolves in time in accordance with a specified mathematical expression. In this book we are primarily concerned with dynamical systems capable of exhibiting all three types of behavior illustrated in Fig. 1.2. One basic feature of such systems is that they are multistable, meaning that their unforced counterparts have at least two stable equilibria (the term applied to the case of two stable equilibria is bistable). In the particular case of mechanical systems, the dynamic behavior is modeled by nonlinear differential equations expressing a relationship among terms that represent

• inertial forces

• dissipative forces

• potential forces, that is, forces derived from a potential function and dependent solely upon displacements; for Fig. 1.1 these forces are due to the magnets and the elasticity of the beam

• excitation forces dependent explicitly on time

Similar terms occur in equations modeling other types of dynamical system, for example, electrical, thermal, or chemical systems.

For a large number of systems arising in engineering or physics safe operation requires that steady-state motions occur within a restricted region, called a safe region (in Fig. 1.2(a), the displacement coordinates in the restricted regions are bounded by the vertical line that coincides with the axis of the undeformed beam); transitions to motions visiting another region are undesirable. However, for some systems (e.g., systems that enhance heat transfer, and neurological systems whose activity entails escapes or, in neurological terminology, firings) the occurrence of such transitions is a functional requirement.

Although we will also examine systems with a slowly varying third variable, our main focus will be on planar systems, that is, continuous systems with two time-dependent variables, for example, displacement and velocity. For planar systems subjected to periodic excitation an analytical condition that guarantees the nonoccurrence of transitions was derived in a seminal paper by Melnikov (1963). That condition involves a function—the Melnikov function—consisting of a sum of terms related, through the system's potential, to the system dissipation and excitation. The Melnikov condition for nonoccurrence of transitions states that if the Melnikov function has no zeros or at most a double zero, then transitions cannot occur....

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2009
• Sprache: Englisch
• ISBN 10: 0691144346
• ISBN 13: 9780691144344
• Einband: Taschenbuch
• Anzahl der Seiten: 244
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar