The Hopf bifurcation with bounded noise

TL;DR

This paper investigates how bounded noise affects Hopf bifurcations in random differential equations, revealing that such noise causes a discontinuous change in the minimal forward invariant set during the bifurcation.

Contribution

It introduces the analysis of Hopf bifurcations under bounded noise, highlighting the discontinuous changes in invariant sets, which is a novel insight in stochastic bifurcation theory.

Findings

Bounded noise causes discontinuous changes in invariant sets during Hopf bifurcations.

The study extends understanding of stochastic effects on classical bifurcations.

Discontinuous bifurcation behavior contrasts with deterministic cases.

Abstract

We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.