Escape Rates in a Stochastic Environment with Multiple Scales

Abstract

We consider a stochastic environment with two time scales and outline a general theory that compares two methods to reduce the dimension of the original system. The first method involves the computation of the underlying deterministic center manifold followed by a naive replacement of the stochastic term. The second method allows one to more accurately describe the stochastic effects and involves the derivation of a normal form coordinate transform that is used to find the stochastic center manifold. The results of both methods are used along with the path integral formalism of large fluctuation theory to predict the escape rate from one basin of attraction to another. The general theory is applied to the example of a surface flow described by a generic, singularly perturbed, damped, nonlinear oscillator with additive, Gaussian noise. We show how both nonlinear reduction methods compare in escape rate scaling. Additionally, the center manifolds are shown to predict high pre-history probability regions of escape. The theoretical results are confirmed using numerical computation of the mean escape time and escape prehistory, and we briefly discuss the extension of the theory to stochastic control.

32 pages, 8 figures, Final revision to appear in SIAM Journal on Applied Dynamical Systems

References in corpus (6)

• Disease extinction in the presence of non-Gaussian noise
• Extinction of an infectious disease: a large fluctuation in a non-equilibrium system
• Normal form transforms separate slow and fast modes in stochastic dynamical systems
• Noise-induced cooperative dynamics and its control in coupled neuron models
• Dynamical epidemic suppression using stochastic prediction and control
• Scaling in activated escape of underdamped systems

Cited by in corpus (10)

• A primer on noise-induced transitions in applied dynamical systems
• An iterative action minimizing method for computing optimal paths in stochastic dynamical systems
• Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory
• Set-based corral control in stochastic dynamical systems: Making almost invariant sets more invariant
• Accurate Noise Projection for Reduced Stochastic Epidemic Models
• Large rare fluctuations in systems with delayed dissipation
• A Dynamical Systems Approach for Most Probable Escape Paths over Periodic Boundaries
• On stochastic parameterizing manifolds: Pullback characterization and Non-Markovian reduced equations
• Non-Markovian Reduced Systems for Stochastic Partial Differential Equations: The Additive Noise Case
• Stochastic switching in slow-fast systems: a large fluctuation approach