Noise-Induced Phenomena in Slow-Fast Dynamical Systems.
A Sample-Paths Approach
N. Berglund, B. Gentz
Springer, Probability and its Applications (2005)
ISBN: 1-84628-038-9
Publisher's website![]()
Contents
1 Introduction 1 1.1 Stochastic Models and Metastability 1 1.2 Timescales and Slow-Fast Systems 6 1.3 Examples 8 1.4 Reader's Guide 13 Bibliographic Comments 15 2 Deterministic Slow-Fast Systems 17 2.1 Slow Manifolds 18 2.1.1 Definitions and Examples 18 2.1.2 Convergence towards a Stable Slow Manifold 22 2.1.3 Geometric Singular Perturbation Theory 24 2.2 Dynamic Bifurcations 27 2.2.1 Centre-Manifold Reduction 27 2.2.2 Saddle-Node Bifurcation 28 2.2.3 Symmetric Pitchfork Bifurcation and Bifurcation Delay 33 2.2.4 How to Obtain Scaling Laws 36 2.2.5 Hopf Bifurcation and Bifurcation Delay 42 2.3 Periodic Orbits and Averaging 44 2.3.1 Convergence towards a Stable Periodic Orbit 45 2.3.2 Invariant Manifolds 47 Bibliographic Comments 48 3 One-Dimensional Slowly Time-Dependent Systems 51 3.1 Stable Equilibrium Branches 53 3.1.1 Linear Case 56 3.1.2 Nonlinear Case 62 3.1.3 Moment Estimates 66 3.2 Unstable Equilibrium Branches 68 3.2.1 Diffusion-Dominated Escape 71 3.2.2 Drift-Dominated Escape 78 3.3 Saddle-Node Bifurcation 84 3.3.1 Before the Jump 87 3.3.2 Strong-Noise Regime 90 3.3.3 Weak-Noise Regime 96 3.4 Symmetric Pitchfork Bifurcation 97 3.4.1 Before the Bifurcation 99 3.4.2 Leaving the Unstable Branch 101 3.4.3 Reaching a Stable Branch 103 3.5 Other One-Dimensional Bifurcations 105 3.5.1 Transcritical Bifurcation 105 3.5.2 Asymmetric Pitchfork Bifurcation 108 Bibliographic Comments 110 4 Stochastic Resonance 111 4.1 The Phenomenon of Stochastic Resonance 112 4.1.1 Origin and Qualitative Description 112 4.1.2 Spectral-Theoretic Results 116 4.1.3 Large-Deviations Results 124 4.1.4 Residence-Time Distributions 126 4.2 Stochastic Synchronisation: Sample-Paths Approach 132 4.2.1 Avoided Transcritical Bifurcation 132 4.2.2 Weak-Noise Regime 135 4.2.3 Synchronisation Regime 138 4.2.4 Symmetric Case 139 Bibliographic Comments 141 5 Multi-Dimensional Slow-Fast Systems 143 5.1 Slow Manifolds 144 5.1.1 Concentration of Sample Paths 145 5.1.2 Proof of Theorem 5.1.6 151 5.1.3 Reduction to Slow Variables 164 5.1.4 Refined Concentration Results 166 5.2 Periodic Orbits 172 5.2.1 Dynamics near a Fixed Periodic Orbit 172 5.2.2 Dynamics near a Slowly Varying Periodic Orbit 175 5.3 Bifurcations 178 5.3.1 Concentration Results and Reduction 178 5.3.2 Hopf Bifurcation 185 Bibliographic Comments 190 6 Applications 193 6.1 Nonlinear Oscillators 194 6.1.1 The Overdamped Langevin Equation 194 6.1.2 The Van der Pol Oscillator 196 6.2 Simple Climate Models 199 6.2.1 The North-Atlantic Thermohaline Circulation 200 6.2.2 Ice Ages and Dansgaard-Oeschger Events 204 6.3 Neural Dynamics 207 6.3.1 Excitability 209 6.3.2 Bursting 212 6.4 Models from Solid-State Physics 214 6.4.1 Ferromagnets and Hysteresis 214 6.4.2 Josephson Junctions 219 A A Brief Introduction to Stochastic Differential Equations 223 A.1 Brownian Motion 223 A.2 Stochastic Integrals 225 A.3 Strong Solutions 229 A.4 Semi-groups and Generators 230 A.5 Large Deviations 232 A.6 The Exit Problem 234 Bibliographic Comments 236 B Some Useful Inequalities 239 B.1 Doob's Submartingale Inequality and a Bernstein Inequality 239 B.2 Using Tail Estimates 240 B.3 Comparison Lemma 241 B.4 Reflection Principle 242 C First-Passage Times for Gaussian Processes 243 C.1 First Passage through a Curved Boundary 243 C.2 Small-Ball Probabilities for Brownian Motion 247 Bibliographic Comments 248 References 249 Index 263 List of Symbols and Acronyms 271