Mean-square Stability and Bifurcations for Dissipative SDEs

TL;DR

This paper studies the stability and bifurcations of dissipative stochastic differential equations, providing a deterministic method to analyze how stochastic forcing affects equilibrium stability.

Contribution

It introduces a straightforward deterministic approach to assess mean-square stability and bifurcations in dissipative SDEs, linking nonlinear and linearized dynamics.

Findings

Established conditions for mean-square dissipativity of stochastic systems.

Provided criteria for boundedness of perturbations in linearized systems.

Demonstrated the method with numerical examples including bifurcation diagrams.

Abstract

We investigate the dynamics of dissipative systems with stochastic forcing and focus in particular on mean-square stability. First we show, under a natural condition on the drift and diffusion, that the stochastic system is mean-square dissipative. Next we examine the linearised system and state conditions ensuring that perturbations of a linear system with affine noise are bounded. We then relate the mean-square dynamics of the nonlinear and linearised systems. The approach gives a straightforward deterministic method to examine the effects of stochastic forcing on the stability of equilibria of deterministic systems and to obtain bifurcation diagrams that can be included into standard numerical continuation packages. The technique is illustrated numerically on some standard and non-standard examples.