Escape over a saddle by coloured noise: theory and numerics

Abstract

Stochastic dynamical systems allow modelling of transitions induced by random disturbances, in particular from an attracting equilibrium and crossing the stable manifold of a saddle. While the small-noise limit is well described by the large deviation principle, existing computational methods often struggle with stochastic forcings any more complicated than non-degenerate Gaussian white noise, and with unbounded time-intervals. The primary innovations of this work are extending the framework to cater for coloured and degenerate forcing and unbounded time horizons: scenarios that are physically realistic but numerically challenging. We cater for degenerate noise by using the Hamiltonian optimal control method. We cater for a class of coloured noises by using linear filters on white noise. We cater for infinite time horizon by introducing the Method of Division (MOD), a novel computational approach for approximating rare transition events, including their most likely paths and the exponential scaling laws of their transition rates. The effectiveness of MOD and the above approaches to coloured noise, is demonstrated by illustration on two examples: an inverted double well potential and a simplified roll heave model for ship capsize.