Scaling Laws and Warning Signs for Bifurcations of SPDEs

TL;DR

This paper extends the theoretical understanding of warning signs for bifurcations in stochastic partial differential equations, including complex and degenerate spectra, aiding in predicting critical transitions in complex systems.

Contribution

It significantly advances the theory of bifurcation warning signs in SPDEs by addressing complex eigenvalues, degeneracies, and continuous spectra, broadening practical applicability.

Findings

Developed scaling laws for covariance growth in SPDEs with complex spectra

Extended bifurcation warning sign theory to degenerate eigenvalues

Provided comprehensive framework applicable to most practical SPDE bifurcation scenarios

Abstract

Critical transitions (or tipping points) are drastic sudden changes observed in many dynamical systems. Large classes of critical transitions are associated to systems, which drift slowly towards a bifurcation point. In the context of stochastic ordinary differential equations (SODEs), there are results on growth of variance and autocorrelation before a transition, which can be used as possible warning signs in applications. A similar theory has recently been developed in the simplest setting also for stochastic partial differential equations (SPDEs) for self-adjoint operators in the drift term. This setting leads to real discrete spectrum and growth of the covariance operator via a certain scaling law. In this paper, we develop this theory substantially further. We cover the cases of complex eigenvalues, degenerate eigenvalues as well as continuous spectrum. This provides a fairly comprehensive theory for most practical applications of warning signs for SPDE bifurcations.