Bifurcation theory for SPDEs: finite-time Lyapunov exponents and amplitude equations

TL;DR

This paper analyzes bifurcations in stochastic PDEs by studying finite-time Lyapunov exponents, using a reduction to simpler stochastic differential equations to detect stability changes near bifurcation points.

Contribution

It introduces a method to characterize bifurcation regions in stochastic PDEs through finite-time Lyapunov exponents and reduces complex dynamics to simpler SDEs near bifurcation points.

Findings

Finite-time Lyapunov exponents indicate bifurcation regions.

Reduction to simple SDEs simplifies analysis near bifurcation.

Identifies stability changes depending on noise and bifurcation proximity.

Abstract

We consider a stochastic partial differential equation close to bifurcation of pitchfork type, where a one-dimensional space changes its stability. For finite-time Lyapunov exponents we characterize regions depending on the distance from bifurcation and the noise strength where finite-time Lyapunov exponents are positive and thus detect bifurcations. One technical tool is the reduction of the essential dynamics of the infinite dimensional stochastic system to a simple ordinary stochastic differential equation, which is valid close to the bifurcation.