Noise-induced Phenomena in Slow-fast Dynamical Systems

by Berglund, Nils; Gentz, Barbara

• ISBN13: 9781846280382
• ISBN10: 1846280389
• Format: Hardcover
• Copyright: 2005-12-01
• Publisher: Springer Nature

Summary

This book covers stochastic differential equations involving two well-separated timescales (in other words, singularly perturbed, or slow-fast systems with noise). It's major goal is to provide constructive methods to characterise the typical paths of these equations, in order to obtain physically relevant information. The first part of the book is devoted to a systematic study of these singularly perturbed stochastic differential equations, starting with a one-dimensional study, then proceeding to multi-dimensional systems. Special attention is paid to the effect of noise on dynamic bifurcations near which transition phenomena are most likely to be observed. The rest of the book is devoted to case studies and applications. A whole chapter is devoted to stochastic resonance, and other applications including models of solid-state physcis and climatology.

Author Biography

Nils Berglund joined the research group "Classical and Quantum Dynamics" at the Centre de Physique Theorique (CNRS) in Marseille-Luminy in 2001. He teaches in the Mathematics Department of the Universite du Sud Toulon-Var. Barbara Gentz joined the research group "Interacting Random Systems" at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) in Berlin in 3998. She teaches in the Institute of Mathematics at the Technical University in Berlin.

Table of Contents

1 Introduction	1	(16)
1.1 Stochastic Models and Metastability	1	(5)
1.2 Timescales and Slow—Fast Systems	6	(2)
1.3 Examples	8	(5)
1.4 Reader's Guide	13	(2)
Bibliographic Comments	15	(2)
2 Deterministic Slow—Fast Systems	17	(34)
2.1 Slow Manifolds	18	(9)
2.1.1 Definitions and Examples	18	(4)
2.1.2 Convergence towards a Stable Slow Manifold	22	(2)
2.1.3 Geometric Singular Perturbation Theory	24	(3)
2.2 Dynamic Bifurcations	27	(18)
2.2.1 Centre-Manifold Reduction	27	(1)
2.2.2 Saddle—Node Bifurcation	28	(6)
2.2.3 Symmetric Pitchfork Bifurcation and Bifurcation Delay	34	(3)
2.2.4 How to Obtain Scaling Laws	37	(6)
2.2.5 Hopf Bifurcation and Bifurcation Delay	43	(2)
2.3 Periodic Orbits and Averaging	45	(3)
2.3.1 Convergence towards a Stable Periodic Orbit	45	(2)
2.3.2 Invariant Manifolds	47	(1)
Bibliographic Comments	48	(3)
3 One-Dimensional Slowly Time-Dependent Systems	51	(60)
3.1 Stable Equilibrium Branches	53	(15)
3.1.1 Linear Case	56	(6)
3.1.2 Nonlinear Case	62	(4)
3.1.3 Moment Estimates	66	(2)
3.2 Unstable Equilibrium Branches	68	(16)
3.2.1 Diffusion-Dominated Escape	71	(7)
3.2.2 Drift-Dominated Escape	78	(6)
3.3 Saddle—Node Bifurcation	84	(13)
3.3.1 Before the Jump	87	(3)
3.3.2 Strong-Noise Regime	90	(6)
3.3.3 Weak-Noise Regime	96	(1)
3.4 Symmetric Pitchfork Bifurcation	97	(8)
3.4.1 Before the Bifurcation	99	(2)
3.4.2 Leaving the Unstable Branch	101	(2)
3.4.3 Reaching a Stable Branch	103	(2)
3.5 Other One-Dimensional Bifurcations	105	(5)
3.5.1 Transcritical Bifurcation	105	(3)
3.5.2 Asymmetric Pitchfork Bifurcation	108	(2)
Bibliographic Comments	110	(1)
4 Stochastic Resonance	111	(32)
4.1 The Phenomenon of Stochastic Resonance	112	(20)
4.1.1 Origin and Qualitative Description	112	(4)
4.1.2 Spectral-Theoretic Results	116	(8)
4.1.3 Large-Deviation Results	124	(2)
4.1.4 Residence-Time Distributions	126	(6)
4.2 Stochastic Synchronisation: Sample-Paths Approach	132	(9)
4.2.1 Avoided Transcritical Bifurcation	132	(3)
4.2.2 Weak-Noise Regime	135	(3)
4.2.3 Synchronisation Regime	138	(1)
4.2.4 Symmetric Case	139	(2)
Bibliographic Comments	141	(2)
5 Multi-Dimensional Slow—Fast Systems	143	(50)
5.1 Slow Manifolds	144	(28)
5.1.1 Concentration of Sample Paths	145	(6)
5.1.2 Proof of Theorem 5.1.6	151	(13)
5.1.3 Reduction to Slow Variables	164	(2)
5.1.4 Refined Concentration Results	166	(6)
5.2 Periodic Orbits	172	(6)
5.2.1 Dynamics near a Fixed Periodic Orbit	172	(3)
5.2.2 Dynamics near a Slowly Varying Periodic Orbit	175	(3)
5.3 Bifurcations	178	(12)
5.3.1 Concentration Results and Reduction	178	(7)
5.3.2 Hopf Bifurcation	185	(5)
Bibliographic Comments	190	(3)
6 Applications	193	(30)
6.1 Nonlinear Oscillators	194	(5)
6.1.1 The Overdamped Langevin Equation	194	(2)
6.1.2 The van dcr Pol Oscillator	196	(3)
6.2 Simple Climate Models	199	(8)
6.2.1 The North-Atlantic Thermohaline Circulation	200	(4)
6.2.2 Ice Ages and Dansgaard–Oeschger Events	204	(3)
6.3 Neural Dynamics	207	(7)
6.3.1 Excitability	209	(3)
6.3.2 Bursting	212	(2)
6.4 Models from Solid-State Physics	214	(9)
6.4.1 Ferromagnets and Hysteresis	214	(5)
6.4.2 Josephson Junctions	219	(4)
A A Brief Introduction to Stochastic Differential Equations	223	(16)
A.1 Brownian Motion	223	(2)
A.2 Stochastic Integrals	225	(4)
A.3 Strong Solutions	229	(1)
A.4 Semigroups and Generators	230	(2)
A.5 Large Deviations	232	(2)
A.6 The Exit Problem	234	(2)
Bibliographic Comments	236	(3)
B Some Useful Inequalities	239	(4)
B.1 Doob's Submartingale Inequality and a Bernstein Inequality	239	(1)
B.2 Using Tail Estimates	240	(1)
B.3 Comparison Lemma	241	(1)
B.4 Reflection Principle	242	(1)
C First-Passage Times for Gaussian Processes	243	(6)
C.1 First Passage through a Curved Boundary	243	(4)
C.2 Small-Ball Probabilities for Brownian Motion	247	(1)
Bibliographic Comments	248	(1)
References	249	(14)
List of Symbols and Acronyms	263	(8)
Index	271