Noise-induced transitions between coexisting limit cycles beyond saddle-node bifurcations

Abstract

Nonlinear multistable systems with limit cycles may exhibit noise-induced transitions between oscillatory states, even in parameter regimes beyond the saddle-node bifurcations that determine the existence of these limit cycles. This behavior is associated with the ghost of a saddle-node bifurcation, a well-known feature of nonlinear systems near such bifurcation points. To investigate this phenomenon, we first consider a stochastic minimal model that supports two coexisting limit cycles within a parameter interval bounded by two saddle-node bifurcations. We numerically and theoretically estimate the probability of remaining near each stable attractor as a function of both the noise intensity and the distance to the ghost structure. The model is then extended by introducing spatially localized, non-homogeneous noise with enhanced intensity in a prescribed region of phase space. This modification is intended to emulate the asymmetric structure of the Hindmarsh-Rose model, in which bursting limit cycles approach each other closely in specific regions of phase space, thereby facilitating noise-induced transitions between them.