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