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.
Arvustused
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
Muu info
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
Preface xi
Chapter
1. Introduction 1 PART 1.FUNDAMENTALS 9
Chapter
2.
Transitions in Deterministic Systems and the Melnikov Function 11 2.1 Flows
and Fixed Points.Integrable Systems.Maps: Fixed and Periodic Points 12 2.2
Homoclinic and Heteroclinic Orbits.Stable and Unstable Manifolds 20 2.3
Stable and Unstable Manifolds in the Three-Dimensional Phase Space 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
Chapter
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 Z2. 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
72
Chapter
4. Stochastic Processes 76 4.1 Spectral Density, Autocovariance,
Cross-Covariance 76 4.2 Approximate Representations of Stochastic Processes
87 4.3 Spectral Density of the Output of a Linear Filter with Stochastic
Input 94
Chapter
5. Chaotic Transitions in Stochastic Dynamical Systems and
the Melnikov Process 98 5.1 Behavior of a Fluidelastic Oscillator with
Escapes: Experimental and Numerical Results 100 5.2 Systems with Additive
and Multiplicative Gaussian Noise: Melnikov Processes and Chaotic Behavior
102 5.3 Phase Space Flux 106 5.4 Condition Guaranteeing Nonoccurrence of
Escapes in Systems Excited by Finite-Tailed Stochastic Processes. Example:
Dichotomous Noise 109 5.5 Melnikov-Based Lower Bounds for Mean Escape Time
and for Probability of Nonoccurrence of Escapes during a Specified Time
Interval 112 5.6 Effective Melnikov Frequencies and Mean Escape Time 119
5.7 Slowly Varying Planar Systems 122 5.8 Spectrum of a Stochastically
Forced Oscillator: Comparison between Fokker-Planck and Melnikov-Based
Approaches 122 PART
2. APPLICATIONS 127
Chapter
6. Vessel Capsizing 129
6.1 Model for Vessel Roll Dynamics in Random Seas 129 6.2 Numerical Example
132
Chapter
7. Open-Loop Control of Escapes in Stochastically Excited
Systems 134 7.1 Open-Loop Control Based on the Shape of the Melnikov Scale
Factor 134 7.2 Phase Space Flux Approach to Control of Escapes Induced by
Stochastic Excitation 140
Chapter
8. Stochastic Resonance 144 8.1
Definition and Underlying Physical Mechanism of Stochastic Resonance.
Application of the Melnikov Approach 145 8.2 Dynamical Systems and Melnikov
Necessary Condition for Chaos 146 8.3 Signal-to-Noise Ratio Enhancement for
a Bistable Deterministic System 147 8.4 Noise Spectrum Effect on
Signal-to-Noise Ratio for Classical Stochastic Resonance 148 8.5 System with
Harmonic Signal and Noise: Signal-to-Noise Ratio Enhancement through the
Addition of a Harmonic Excitation 152 8.6 Nonlinear Transducing Device for
Enhancing Signal-to-Noise Ratio 153 8.7 Concluding Remarks 154
Chapter
9.
Cutoff Frequency of Experimentally Generated Noise for a First-Order
Dynamical System 156 9.1 Introduction 156 9.2 Transformed Equation Excited
by White Noise 157
Chapter
10. Snap-Through of Transversely Excited Buckled
Column 159 10.1 Equation of Motion 160 10.2 Harmonic Forcing 161 10.3
Stochastic Forcing. Nonresonance Conditions. Melnikov Processes for Gaussian
and Dichotomous Noise 163 10.4 Numerical Example 164
Chapter
11.
Wind-Induced Along-Shor Currents over a Corrugated Ocean Floor 167 11.1
Offshore Flow Model 168 11.2 Wind Velocity Fluctuations and Wind Stresses
170 11.3 Dynamics of Unperturbed System 172 11.4 Dynamics of Perturbed
System 173 11.5 Numerical Example 174
Chapter
12. The Auditory Nerve Fiber
as a Chaotic Dynamical System 178 12.1 Experimental Neurophysiological
Results 179 12.2 Results of Simulations Based on the Fitzhugh-Nagumo Model.
Comparison with Experimental Results 182 12.3 Asymmetric Bistable Model of
Auditory Nerve Fiber Response 183 12.4 Numerical Simulations 186 12.5
Concluding Remarks 190 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 Z2 201 Appendix A5 Elements of Probability Theory 203
Appendix A6 Mean Upcrossing Rate for Gaussian Processes 211 Appendix A7
Mean Escape Rate for Systems Excited by White Noise 213 References 215
Index 221
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.
