Escape from attracting sets in randomly perturbed systems

TL;DR

This paper investigates how bounded random noise causes escape from attractors in dynamical systems, deriving analytical escape rate formulas and confirming them with numerical simulations.

Contribution

It introduces a novel approach linking escape dynamics under bounded noise to systems with a 'hole', providing analytical expressions for escape rates near the transition.

Findings

Escape occurs only above a minimum noise amplitude.

Analytical formulas for escape rate scaling are derived.

Numerical simulations confirm theoretical predictions.

Abstract

The dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science. Previous studies of escape in chaotic systems have rather focused on the case of unbounded noise, usually assumed to have Gaussian distribution. In this paper, we address the problem of escape induced by bounded noise. We show that the dynamics of escape from an attractor's basin is equivalent to that of a closed system with an appropriately chosen "hole". Using this equivalence, we show that there is a minimum noise amplitude above which escape takes place, and we derive analytical expressions for the scaling of the escape rate with noise amplitude near the escape transition. We verify our analytical predictions through numerical simulations of a two-dimensional map with noise.