Most probable escape paths in perturbed gradient systems

TL;DR

This paper develops a method using large deviation theory and Melnikov analysis to compute the most probable escape paths in stochastic gradient systems with non-gradient perturbations, extending classical results.

Contribution

It introduces a novel approach combining large deviation theory and Melnikov analysis to identify escape paths in non-gradient stochastic systems.

Findings

Derived a condition for when the escape path is a heteroclinic orbit.

Applied the method to a numerical example demonstrating its effectiveness.

Extended classical gradient system results to more general perturbed systems.

Abstract

Stochastic systems are used to model a variety of phenomena in which noise plays an essential role. In these models, one potential goal is to determine if noise can induce transitions between states, and if so, to calculate the most probable escape path from an attractor. In the small noise limit, the Freidlin-Wentzell theory of large deviations provides a variational framework to calculate these paths. This work focuses on using large deviation theory to calculate such paths for stochastic gradient systems with non-gradient perturbations. While for gradient systems the most probable escape paths consist of time-reversed heteroclinic orbits, for general systems it can be a challenging calculation. By applying Melnikov theory to the resulting Euler-Lagrange equations recast in Hamiltonian form, we determine a condition for when the optimal escape path is the heteroclinic orbit for the perturbed system. We provide a numerical example to illustrate how the computed most probable escape path compares with the theoretical result.