Finite-time Lyapunov exponents for SPDEs with fractional noise

TL;DR

This paper investigates how fractional Brownian motion influences the stability of solutions to stochastic partial differential equations near bifurcation points by estimating finite-time Lyapunov exponents.

Contribution

It provides the first analysis of finite-time Lyapunov exponents for SPDEs driven by fractional noise, characterizing stability regions based on bifurcation proximity, Hurst parameter, and noise strength.

Findings

Finite-time Lyapunov exponents are positive near bifurcations, indicating instability.

Regions of stability and instability depend on Hurst index and noise intensity.

Results extend understanding of stability in systems with fractional noise.

Abstract

We estimate the finite-time Lyapunov exponents for a stochastic partial differential equation driven by a fractional Brownian motion (fbm) with Hurst index $H\in(0,1)$ close to a bifurcation of pitchfork type. We characterize regions depending on the distance from bifurcation, the Hurst parameter of the fbm and the noise strength where finite-time Lyapunov exponents are positive and thus indicate a change of stability. The results on finite-time Lyapunov exponents are novel also for SDEs perturbed by fractional noise.