Geometric singular perturbation theory for stochastic differential equations

By N. Berglund, Barbara Gentz. 2002

117 citations recorded by the source corpus.

We consider slow–fast systems of differential equations, in which both the slow and fast variables are perturbed by noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold, which we construct explicitly. Depending on the dynamics of the reduced system, the results cover time spans which can be exponentially long in the noise intensity squared (that is, up to Kramers’ time). We obtain exponentially small upper and lower bounds on the probability of exceptional paths. If the slow manifold contains bifurcation points, we show similar concentration properties for the fast variables corresponding to non-bifurcating modes. We also give conditions under which the system can be approximated by a lower-dimensional one, in which the fast variables contain only bifurcating modes.

Subfields: Applied Mathematics, Global and Planetary Change, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3d4f07812bc223bca4610d2dbb30968500d7094b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. B. Schmalfuβ
2. K. Schneider
3. Anastasios Xepapadeas
4. Christian Kuehn

In Global and Planetary Change

1. Carl Folke
2. Anastasios Xepapadeas
3. Terry P. Hughes
4. B. Walker
5. Stephen Polasky
6. M. Troell
7. Gretchen Daily
8. Scott Barrett

In Statistical and Nonlinear Physics

1. B. Schmalfuβ
2. P. Ehrlich
3. K. Schneider
4. Anastasios Xepapadeas
5. Christian Kuehn
6. Simon A. Levin

Citations in the corpus

• A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics (2011, methodology): “As a first step we use the sample path approach [18, 16] and point out what it provides in the context of critical transitions.”
• Modeling Complex Systems (2010, methodology): “A closed-loop (or feedback) Nash equilibrium is an n-tuple of closed-loop strategies that satis…es conditions (2) for every possible initial state x0.”
• Social-ecological systems as complex adaptive systems: modeling and policy implications (2012, background): “Singular perturbation analysis (e.g., Wasow, 1965; Fenichel, 1979; Berglund and Gentz, 2003) can help analyze dynamical systems evolving in a fast–slow time framework.”
• Invariant Manifolds for Random Dynamical Systems with Slow and Fast Variables (2008, result): “A first answer in this direction has been given by Berglund and Gentz [3].”