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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
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