Long time fluctuations at critical parameter of Hopf's bifurcation

TL;DR

This paper analyzes the critical fluctuations of a dynamical system undergoing a supercritical Hopf bifurcation under stochastic perturbations, deriving a universal stochastic differential equation describing the fluctuations near equilibrium.

Contribution

It introduces a new approximation of the perturbed Hopf bifurcation system using a 2D process on the centre manifold and derives a universal SDE for the critical fluctuations.

Findings

Critical fluctuations are described by a universal SDE.

The system can be approximated by a 2D process on the centre manifold.

Fast angular dynamics are averaged out in the limit.

Abstract

A dynamical system that undergoes a supercritical Hopf's bifurcation is perturbed by a multiplicative Brownian motion that scales with a small parameter $\epsilon$. The random fluctuations of the system at the critical point are studied when the dynamics starts near equilibrium, in the limit as $\epsilon$ goes to zero. Under a space-time scaling the system can be approximated by a 2-dimensional process lying on the centre manifold of the Hopf's bifurcation and a slow radial component together with a fast angular component are identified. Then the critical fluctuations are described by a "universal" stochastic differential equation whose coefficients are obtained taking the average with respect to the fast variable.