Buch, Englisch, 246 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 377 g
Applications of Melnikov Processes in Engineering, Physics, and Neuroscience
Buch, Englisch, 246 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 377 g
Reihe: Princeton Series in Applied Mathematics
ISBN: 978-0-691-14434-4
Verlag: Princeton University Press
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.
Autoren/Hrsg.
Fachgebiete
- Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
- Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
- Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
- Naturwissenschaften Physik Angewandte Physik Statistische Physik, Dynamische Systeme
Weitere Infos & Material
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 Poincar? 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-
