The impact of white noise on a supercritical bifurcation in the Swift-Hohenberg equation

TL;DR

This paper investigates how additive Gaussian white noise affects the behavior of a supercritical bifurcation in the stochastic Swift-Hohenberg equation, providing insights into the noise's influence near stability changes.

Contribution

It introduces an approximation method using modulation equations to analyze the impact of small noise on bifurcation dynamics in the Swift-Hohenberg equation.

Findings

Identifies the noise order affecting the bifurcation

Develops a modulation equation-based analysis tool

Provides insights into noise influence near stability transition

Abstract

We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift-Hohenberg equation with polynomial nonlinearity. Here we identify the order where small noise first impacts the bifurcation. Using an approximation via modulation equations, we provide a tool to analyse how the noise influences the dynamics close to a change of stability.