Noise Induced Switching and Extinction in Systems with Delay

TL;DR

This paper analyzes how noise causes systems with fixed time delay to switch between states or go extinct, deriving rates through variational methods and confirming results with numerical simulations.

Contribution

It introduces a variational approach to calculate noise-induced switching and extinction rates in delayed systems, including acausal path equations and explicit solutions for small delays.

Findings

Rates are exponentially small for weak noise.

Analytical results match numerical simulations.

Explicit formulas derived for small delays.

Abstract

We consider the rates of noise-induced switching between the stable states of dissipative dynamical systems with delay and also the rates of noise-induced extinction, where such systems model population dynamics. We study a class of systems where the evolution depends on the dynamical variables at a preceding time with a fixed time delay, which we call hard delay. For weak noise, the rates of inter-attractor switching and extinction are exponentially small. Finding these rates to logarithmic accuracy is reduced to variational problems. The solutions of the variational problems give the most probable paths followed in switching or extinction. We show that the equations for the most probable paths are acausal and formulate the appropriate boundary conditions. Explicit general results are obtained for small delay compared to the relaxation rate. We also develop a direct variational method to find the rates. We find that the analytical results agree well with the numerical simulations for both switching and extinction rates.