Effect of noise on residence times of a heteroclinic cycle

Abstract

A heteroclinic cycle is an invariant set in a dynamical system consisting of saddle-type equilibria and heteroclinic connections between them. It is known that deterministic perturbations (inputs) to a heteroclinic cycle generally lead to periodic solutions. Addition of noise to such a system leads to a non-intuitive result: there is a range of noise levels for which the mean residence time near the equilibria of the heteroclinic cycle increases as the noise level increases to a given threshold. We explain how the interaction between noise and inputs gives rise to this by combining analytical results from constructing a Poincaré map with a simple stochastic system. We support our results with numerical simulations.