Noise Driven Transitions between Stable Equilibria in Stochastic Dynamical Systems

In this paper we examine two specific models of dynamical systems in which noise plays a central role. The first is a stochastic differential equation (SDE) modeling a particle in a potential well; the second is a simplified version of the Morris-Lecar model of a neuron. In each case, we consider both the underlying deterministic dynamical system, which is a governed by an ordinary differential equation, as well as the randomly-perturbed dynamical system, whose solution is a stochastic process satisfying a stochastic integral equation. We investigate the how perturbations of the drift functions influence transitions between stable equilibria and what effects such perturbations may have on exit times. We compare the results from computer simulations to analytical derivations of expected exit times. These results contribute to the understanding of the forces driving transitions between stable equilibria in perturbed dynamical systems.