Critical Slowing Down in Bifurcating Stochastic Partial Differential Equations with Red Noise

TL;DR

This work extends critical slowing down (CSD) and early-warning signal (EWS) concepts from finite-dimensional systems to bifurcating stochastic partial differential equations (SPDEs) driven by red noise. It presents a rigorous framework for SPDEs with three driving configurations—domain noise, continuous spectrum, and boundary noise—and derives time-asymptotic variance scaling laws near bifurcation ($p\to 0^-$) and for vanishing noise correlation time ($\kappa\to 0^+$). The results show that variance-based EWS persist for many probing functions, with divergence tied to the interaction between the observable and the critical modes, but non-stationary noise can produce false or muted warnings, especially in continuous-spectrum cases. Numerical experiments across cable-like, generalized-eigenvector, continuous-spectrum, and boundary-noise models validate the theory and illustrate when EWS are robust or deceptive, thereby broadening the applicability of CSD to real-world spatially extended systems while cautioning about noise structure and spectrum effects.

Abstract

The phenomenon of critical slowing down (CSD) has played a key role in the search for reliable precursors of catastrophic regime shifts. This is caused by its presence in a generic class of bifurcating dynamical systems. Simple time-series statistics such as variance or autocorrelation can be taken as proxies for the phenomenon, making their increase a useful early warning signal (EWS) for catastrophic regime shifts. However, the modelling basis justifying the use of these EWSs is usually a finite-dimensional stochastic ordinary differential equation, where a mathematical proof for the aptness is possible. Only recently has the phenomenon of CSD been proven to exist in infinite-dimensional stochastic partial differential equations (SPDEs), which are more appropriate to model real-world spatial systems. In this context, we provide an essential extension of the results for SPDEs under a specific noise forcing, often referred to as red noise. This type of time-correlated noise is omnipresent in many physical systems, such as climate and ecology. We approach the question with a mathematical proof and a numerical analysis for the linearised problem. We find that also under red noise forcing, the aptness of EWSs persists, supporting their employment in a wide range of applications. However, we also find that false or muted warnings are possible if the noise correlations are non-stationary. We thereby extend a previously known complication with respect to red noise and EWSs from finite-dimensional dynamics to the more complex and realistic setting of SPDEs.

Figures (2)

• Figure 1: Log-log plots of the variance in time obtained when projecting the SPDE solution along different modes. The limit $p\to 0^-$ is shown from right to left. Each panel corresponds to a different system: $(a)$ the cable equation on an interval with periodic boundary conditions, \ref{['eq:example_disc']}; $(b)$ the SDE \ref{['eq:example_gen']} with linear drift displaying generalized eigenvectors; $(c)$ the SPDE \ref{['eq:example_cont']} with a linear drift term with purely continuous spectrum; $(d)$ the boundary-driven system \ref{['eq:example_bound']} with red Dirichlet noise at the extremes of an interval. The values refer to the average of $10$ run samples. The dashed black lines indicate reference hyperbolic scaling laws, whereas the shaded grey regions represent twice the numerical standard deviation. The increase of variance in the projected modes is the manifestation of CSD in these SPDEs, where the drift term approaches a deterministic bifurcation point. Nonetheless, while the scaling law is hyperbolic in $(a)$ and $(d)$, we find examples of enhanced or silenced EWSs in $(b)$ and in $(c)$, respectively.
• Figure 2: Log-log plots of the variance in time obtained when projecting the SPDE solution along different modes. The limit $\kappa\to 0^-$ is shown from right to left. The subfigures $(a)-(d)$ correspond to the same systems as in Figure \ref{['Fig1']} and their values refer to the average of $10$ run samples. In contrast to the limit $p\to 0^-$, the variance exhibits hyperbolic divergence across all modes due to the noise structure. This is indicated by the alignment of all lines to the dashed black lines, which serve as a reference slope. The grey-shaded regions depict numerical uncertainties. The observed increase in variance, which depends on the increase in the noise correlation time $1/\kappa$, represents a false EWS in the context of CSD.

Theorems & Definitions (12)

• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• proof
• Theorem 3.5
• proof