Long time fluctuations at critical parameter of Hopf’s bifurcation

A dynamical system that undergoes a supercritical Hopf’s bifurcation is perturbed by a multiplicative Brownian motion that scales with a small parameter 𝜀. The random fluctuations of the system at the critical point are studied when the dynamics starts near equilibrium, in the limit as 𝜀 goes to zero. Under a space–time scaling the system can be approximated by a 2-dimensional process lying on the center 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.