Stochastic bifurcations: a perturbative study
Abstract
We study a noise-induced bifurcation in the vicinity of the threshold by using a perturbative expansion of the order parameter, called the Poincar\'e-Lindstedt expansion. Each term of this series becomes divergent in the long time limit if the power spectrum of the noise does not vanish at zero frequency. These divergencies have a physical consequence: they modify the scaling of all the moments of the order parameter near the threshold and lead to a multifractal behaviour. We derive this anomalous scaling behaviour analytically by a resummation of the Poincar\'e-Lindstedt series and show that the usual, deterministic, scalings are recovered when the noise has a low frequency cut-off. Our analysis reconciles apparently contradictory results found in the literature.
Cite
@article{arxiv.0807.4425,
title = {Stochastic bifurcations: a perturbative study},
author = {Sebastien Aumaitre and Kirone Mallick and Francois Petrelis},
journal= {arXiv preprint arXiv:0807.4425},
year = {2008}
}