Noise-induced stabilization of saddle-node ghosts

Abstract

It is known that saddle-node (s-n) bifurcations leave a saddle remnant (or ghost) in the region of the phase space where the annihilation of the fixed points occurred. The corresponding time delay, td, found right after the bifurcation is known to follow the scaling law , where ϵ and ϵc are the control parameter and its critical value, respectively. While the properties of such delays are well understood for deterministic systems, much less is known about how intrinsic noise influences this phenomenon. As a first step towards analysing this issue, in this article we explore a model with autocatalysis and a two-species hypercycle to analyse the impact of noise on delayed transitions in one- and two-dimensional dynamical systems suffering a s-n bifurcation. The first model is investigated with Gillespie simulations and the diffusion approximation, focusing on the behaviour and properties close to the bifurcation. A Fokker–Planck equation is derived, together with the stochastic potential. We show that the slowing down of the dynamics remains robust to noise. In fact, we prove both analytically and numerically that increasing noise lengthens the delays after bifurcation threshold. Furthermore, the inverse square-root scaling law is not robust to fluctuations. By contrast, scaling properties are identified in the mean extinction times as criticality is approached from above the bifurcation. This noise-induced stabilisation of the delays is also found in the two-dimensional system.