Bifurcation Analysis of a Stochastically Driven Limit Cycle

TL;DR

This paper proves the occurrence of a bifurcation from a stable random equilibrium to chaos in a stochastic limit cycle, using Lyapunov exponents, extending deterministic results to stochastic systems.

Contribution

It establishes the existence of a bifurcation in stochastic limit cycles, addressing an open problem and extending prior deterministic bifurcation results to stochastic dynamics.

Findings

Bifurcation from random equilibrium to chaos identified

Sign change of the first Lyapunov exponent signals bifurcation

Extends deterministic bifurcation theory to stochastic systems

Abstract

We establish the existence of a bifurcation from an attractive random equilibrium to shear-induced chaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent. This addresses an open problem posed by Kevin Lin and Lai-Sang Young, extending results by Qiudong Wang and Lai-Sang Young on periodically kicked limit cycles to the stochastic context.