On the approximation of finite-time Lyapunov exponents for the stochastic Burgers equation
Alexandra Blessing, Dirk Blömker
TL;DR
This work analyzes finite-time Lyapunov exponents for SPDEs with quadratic nonlinearities near a stability change and develops a multiscale amplitude-equation reduction that captures the dominant dynamics on a finite-dimensional kernel. It proves rigorous upper and lower bounds linking the SPDE FTLE λ^U_T to the AE FTLE λ^a_T, with error terms governed by deviation measures K_X(T) and K_𝒩(T) and residuals from the AE approximation, enabling precise bifurcation analysis. The approach relies on a combination of multiscale decomposition, Itô-trick manipulations, and stopping-time arguments to control cross-terms arising from the quadratic nonlinearity, and is illustrated in detail for the stochastic Burgers equation. The results provide a practical framework for reducing complex SPDE FTLE computations to tractable finite-dimensional SDEs, facilitating both theoretical insight and potential numerical applications in stochastic bifurcation problems.
Abstract
We analyze stochastic partial differential equations (SPDEs) with quadratic nonlinearities close to a change of stability. To this aim we compute finite-time Lyapunov exponents (FTLEs), observing a change of sign based on the interplay between the distance towards the bifurcation and the noise intensity. A technical challenge is to provide a suitable control of the nonlinear terms coupling the dominant and stable modes of the SPDE and of the corresponding linearization. In order to illustrate our results we apply them to the stochastic Burgers equation.