Stochastic bifurcations: a perturbative study

TL;DR

This paper investigates how noise influences bifurcations near the threshold, revealing that non-vanishing low-frequency noise causes divergent series and multifractal scaling, which can be analytically resummed to recover deterministic behavior.

Contribution

It introduces a perturbative approach to analyze noise-induced bifurcations, showing how divergences affect scaling and providing a resummation method to recover deterministic results.

Findings

Divergences occur when noise spectrum doesn't vanish at zero frequency.

Non-vanishing low-frequency noise leads to multifractal scaling.

Resummation of series recovers classical scaling when noise has a low-frequency cutoff.

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.