Stability of equilibria of randomly perturbed maps

TL;DR

This paper establishes a sufficient condition for the stability in probability of equilibria in randomly perturbed maps, enabling stabilization of weakly unstable points through random forcing, supported by analytical and numerical examples.

Contribution

It introduces a new sufficient condition for stability in probability of equilibria under random perturbations, applicable to both linear and nonlinear maps.

Findings

Derived a stability condition for randomly perturbed maps.

Demonstrated stabilization of weakly unstable equilibria.

Validated results with numerical examples in 1D and 2D.

Abstract

We derive a sufficient condition for stability in probability of an equilibrium of a randomly perturbed map in ${\mathbb R}^d$. This condition can be used to stabilize weakly unstable equilibria by random forcing. Analytical results on stabilization are illustrated with numerical examples of randomly perturbed linear and nonlinear maps in one- and two-dimensional spaces.