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