Early-warning signs for pattern-formation in stochastic partial differential equations

Abstract

There have been significant recent advances in our understanding of the potential use and limitations of early-warning signs for predicting drastic changes, so called critical transitions or tipping points, in dynamical systems. A focus of mathematical modeling and analysis has been on stochastic ordinary differential equations, where generic statistical early-warning signs can be identified near bifurcation-induced tipping points. In this paper, we outline some basic steps to extend this theory to stochastic partial differential equations with a focus on analytically characterizing basic scaling laws for linear SPDEs and comparing the results to numerical simulations of fully nonlinear problems. In particular, we study stochastic versions of the Swift-Hohenberg and Ginzburg-Landau equations. We derive a scaling law of the covariance operator in a regime where linearization is expected to be a good approximation for the local fluctuations around deterministic steady states. We compare these results to direct numerical simulation, and study the influence of noise level, noise color, distance to bifurcation and domain size on early-warning signs.

Patterned vegetation, tipping points, and the rate of climate change

“…By contrast, very little work has been done on slow passage problems in PDE's with noise. However, see recent work [15] where some related questions are studied. In particular, we show that the spatial component of the PDE system, when combined with noise and parameter drift, plays a key role in preventing activation of the tipping point.…”

Forecasting critical transitions using data-driven nonstationary dynamical modeling

“…An interesting extension of the present work is critical transition modeling based on spatially extended data [38]. Some dimension reduction will then necessarily be involved in the analysis.…”

Detecting, anticipating, and predicting critical transitions in spatially extended systems

“…with the temporal correlation function C tem and the spatial correlation function C sp . We here assume, as in 20 , the noise to be white in time, that is,…”

“…For spatially extended systems, early-warning signs of critical transitions discussed in the literature include increasing autocorrelation 15 , spatial correlation 16 , spatial variance and skewness 17,18 as well as the patchiness of states 19 . A mathematically rigorous analysis of stochastic partial differential equations approaching bifurcations has been performed recently focusing on analytically deriving basic scaling laws of the covariance operator for linear stochastic partial differential equations and comparing the results to numerical simulations of fully nonlinear problems 20 .…”