Long time fluctuations at critical parameter of Hopf’s bifurcation

M. Aleandri and P. Dai Pra

Stochastic Processes and their Applications, 2026, vol. 192, issue C

Abstract: A dynamical system that undergoes a supercritical Hopf’s bifurcation is perturbed by a multiplicative Brownian motion that scales with a small parameter ɛ. The random fluctuations of the system at the critical point are studied when the dynamics starts near equilibrium, in the limit as ɛ 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.

Keywords: Hopf’s bifurcation; Small noise perturbation; Averaging principle; Critical fluctuations; Normal forms theory (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1016/j.spa.2025.104785

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