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arXiv · 0807.4425

Stochastic bifurcations: a perturbative study

Sebastien Aumaitre·Kirone Mallick·Francois Petrelis
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Abstract

We study a noise-induced bifurcation in the vicinity of the threshold by using a perturbative expansion of the order parameter, called the Poincaré-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 Poincaré-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.

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Sebastien Aumaitre, Kirone Mallick, Francois Petrelis. 2008-07-28. Stochastic bifurcations: a perturbative study. https://arxiv.org/abs/0807.4425

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