Stability analysis of a noise-induced Hopf bifurcation

TL;DR

This paper analyzes how noise can induce a bifurcation in a Duffing oscillator, leading to a transition from an absorbing state to oscillations, using both analytical and numerical methods.

Contribution

It provides a non-perturbative analysis of stochastic bifurcation via Lyapunov exponents and derives the phase diagram for the noise-induced transition.

Findings

Stochastic bifurcation occurs when the Lyapunov exponent becomes positive.

The phase diagram of the noise-induced transition is derived.

Physical observables like mean energy change across the bifurcation.

Abstract

We study analytically and numerically the noise-induced transition between an absorbing and an oscillatory state in a Duffing oscillator subject to multiplicative, Gaussian white noise. We show in a non-perturbative manner that a stochastic bifurcation occurs when the Lyapunov exponent of the linearised system becomes positive. We deduce from a simple formula for the Lyapunov exponent the phase diagram of the stochastic Duffing oscillator. The behaviour of physical observables, such as the oscillator's mean energy, is studied both close to and far from the bifurcation.