Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach

✍ Scribed by Nils Berglund, Barbara Gentz

• Publisher: Springer
• Year: 2005
• Tongue: English
• Leaves: 283
• Series: Probability and its Applications
• Edition: 1st Edition.
• Category: Library

✦ Synopsis

Stochastic Differential Equations have become increasingly important in modelling complex systems in physics, chemistry, biology, climatology and other fields.  This book examines and provides systems for practitioners to use, and provides a number of case studies to show how they can work in practice.

✦ Table of Contents

Contents......Page 10
1.1 Stochastic Models and Metastability......Page 13
1.2 Timescales and Slow–Fast Systems......Page 18
1.3 Examples......Page 20
1.4 Reader's Guide......Page 25
Bibliographic Comments......Page 27
2 Deterministic Slow–Fast Systems......Page 29
2.1.1 Definitions and Examples......Page 30
2.1.2 Convergence towards a Stable Slow Manifold......Page 34
2.1.3 Geometric Singular Perturbation Theory......Page 36
2.2.1 Centre-Manifold Reduction......Page 39
2.2.2 Saddle–Node Bifurcation......Page 40
2.2.3 Symmetric Pitchfork Bifurcation and Bifurcation Delay......Page 46
2.2.4 How to Obtain Scaling Laws......Page 49
2.2.5 Hopf Bifurcation and Bifurcation Delay......Page 55
2.3.1 Convergence towards a Stable Periodic Orbit......Page 57
2.3.2 Invariant Manifolds......Page 59
Bibliographic Comments......Page 60
3 One-Dimensional Slowly Time-Dependent Systems......Page 62
3.1 Stable Equilibrium Branches......Page 64
3.1.1 Linear Case......Page 67
3.1.2 Nonlinear Case......Page 73
3.1.3 Moment Estimates......Page 77
3.2 Unstable Equilibrium Branches......Page 79
3.2.1 Diffusion-Dominated Escape......Page 82
3.2.2 Drift-Dominated Escape......Page 89
3.3 Saddle–Node Bifurcation......Page 95
3.3.1 Before the Jump......Page 98
3.3.2 Strong-Noise Regime......Page 101
3.3.3 Weak-Noise Regime......Page 107
3.4 Symmetric Pitchfork Bifurcation......Page 108
3.4.1 Before the Bifurcation......Page 110
3.4.2 Leaving the Unstable Branch......Page 112
3.4.3 Reaching a Stable Branch......Page 114
3.5.1 Transcritical Bifurcation......Page 116
3.5.2 Asymmetric Pitchfork Bifurcation......Page 119
Bibliographic Comments......Page 121
4 Stochastic Resonance......Page 122
4.1.1 Origin and Qualitative Description......Page 123
4.1.2 Spectral-Theoretic Results......Page 127
4.1.3 Large-Deviation Results......Page 135
4.1.4 Residence-Time Distributions......Page 137
4.2.1 Avoided Transcritical Bifurcation......Page 143
4.2.2 Weak-Noise Regime......Page 146
4.2.3 Synchronisation Regime......Page 149
4.2.4 Symmetric Case......Page 150
Bibliographic Comments......Page 152
5 Multi-Dimensional Slow–Fast Systems......Page 154
5.1 Slow Manifolds......Page 155
5.1.1 Concentration of Sample Paths......Page 156
5.1.2 Proof of Theorem 5.1.6......Page 162
5.1.3 Reduction to Slow Variables......Page 175
5.1.4 Refined Concentration Results......Page 177
5.2.1 Dynamics near a Fixed Periodic Orbit......Page 183
5.2.2 Dynamics near a Slowly Varying Periodic Orbit......Page 186
5.3.1 Concentration Results and Reduction......Page 189
5.3.2 Hopf Bifurcation......Page 196
Bibliographic Comments......Page 201
6 Applications......Page 203
6.1.1 The Overdamped Langevin Equation......Page 204
6.1.2 The van der Pol Oscillator......Page 206
6.2 Simple Climate Models......Page 209
6.2.1 The North-Atlantic Thermohaline Circulation......Page 210
6.2.2 Ice Ages and Dansgaard–Oeschger Events......Page 214
6.3 Neural Dynamics......Page 217
6.3.1 Excitability......Page 219
6.3.2 Bursting......Page 222
6.4.1 Ferromagnets and Hysteresis......Page 224
6.4.2 Josephson Junctions......Page 229
A.1 Brownian Motion......Page 233
A.2 Stochastic Integrals......Page 235
A.3 Strong Solutions......Page 239
A.4 Semigroups and Generators......Page 240
A.5 Large Deviations......Page 242
A.6 The Exit Problem......Page 244
Bibliographic Comments......Page 246
B.1 Doob's Submartingale Inequality and a Bernstein Inequality......Page 248
B.2 Using Tail Estimates......Page 249
B.3 Comparison Lemma......Page 250
B.4 Reflection Principle......Page 251
C.1 First Passage through a Curved Boundary......Page 252
C.2 Small-Ball Probabilities for Brownian Motion......Page 256
Bibliographic Comments......Page 257
References......Page 258
List of Symbols and Acronyms......Page 271
B......Page 278
F......Page 279
L......Page 280
P......Page 281
S......Page 282
W......Page 283