The role of edge states for early-warning of tipping points

TL;DR

The paper tackles the challenge that early-warning signals (EWS) based on critical slowing down (CSD) are not universally visible in high-dimensional systems. It introduces and tests an edge-state/instanton framework: by identifying an unstable edge state separating a base attractor from an alternative attractor, one can project dynamics onto observables that exhibit maximal CSD as a tipping point is approached. Through conceptual gradient, Stommel, and five-box AMOC models, and a stochastic Veros global ocean model, the study shows that observables aligned with the edge-state direction reveal pronounced EWS, even when traditional indicators fail. The findings offer a system-specific, large-deviation–grounded strategy for selecting informative observables and underscore practical considerations about data availability and noise structure in detecting approaching tipping points in climate systems.

Abstract

Tipping points (TP) are often described as low-dimensional bifurcations, and are associated with early-warning signals (EWS) due to critical slowing down (CSD). CSD is an increase in amplitude and correlation of noise-induced fluctuations away from a reference attractor as the TP is approached. But for high-dimensional systems it is not obvious which variables or observables would display the critical dynamics and carry CSD. Many variables may display no CSD, or show changes in variability not related to a TP. It is thus helpful to identify beforehand which observables are relevant for a given TP. Here we propose this may be achieved by knowledge of an unstable edge state that separates the reference from an alternative attractor that remains after the TP. This is because stochastic fluctuations away from the reference attractor are preferentially directed towards the edge state along a most likely path (the instanton). As the TP is approached the edge state and reference attractor typically become closer, and the fluctuations can evolve further along the instanton. This can be exploited to find observables with substantial CSD, which we demonstrate using conceptual dynamical systems models and climate model simulations of a collapse of the Atlantic Meridional Overturning Circulation (AMOC).

Figures (14)

• Figure 1: a Potential defined by Eq. \ref{['eq:potential']} with $e=0.2$, as well as one realization of a noise-induced transition between the two stable fixed points using $\sigma_x = \sigma_y = 0.1$. Also shown is the basin boundary (black) and the instanton (purple), as well as a hypothetical more "direct" transition path of shorter length in phase space (dashed blue line). b Bifurcation diagram of the system described by Eq. \ref{['eq:gradient']} projected onto the variable $x$. The dashed line is the saddle.
• Figure 2: a-d Potential isolines of Eq. \ref{['eq:potential']}, along with the fixed points (colored symbols) of Eq. \ref{['eq:gradient']} for four values of the bifurcation parameter $e$ in increasing order until shortly before the bifurcation point. The green triangle is the edge state. Also shown in each panel is an ensemble of 100 simulations of Eq. \ref{['eq:gradient']} with additive Gaussian white noise (red trajectory), and the instanton (purple) computed by the method in KIK20. The duration of the simulations is $t=1500$. e-h Probability densities of residuals around the fixed point ($x^*$, $y^*$) aggregated from an ensemble of simulations with $N=1000$ realizations (fixed simulation time $t=1500$) of the model in Eq. \ref{['eq:gradient']} with added Gaussian white noise and $\sigma_x = \sigma_y = 0.1$. Each realization is initialized from the fixed point ($x^*$, $y^*$) with $x>0$, and simulations have been performed for four different values of the control parameter $e$. For the largest $e$ ( d), which is close to the bifurcation, noise-induced transitions occur, the effect of which on the residuals is removed by cutting the time series at the time when the $x$-value of the saddle is crossed for the last time before tipping to the other fixed point. The red dashed line is the identity line, associated with the observable $x+y$, and the green line is the vector pointing from the attractor to the edge state.
• Figure 3: From the same simulations as in Fig. \ref{['fig:gradient2D_results']}e-h we extract the distribution of fluctuations (around the mean value) of the observables $x+y$ ( a) and $x-y$ ( b) for four values of the control parameter $e$. Panel ( c) shows the evolution of the autocorrelation at lag 1 for the two observables towards the bifurcation, which has been averaged over all realizations of ensembles for a range of values of $e$. By "lag 1" we mean to indicate that the correlation of subsequent time series samples is calculated, where a sample spacing of 0.05 time units, i.e., every tenth time step of the numerical integration, is used. The vertical dashed line is the bifurcation point.
• Figure 4: a,b Bifurcation diagram of the Stommel model, projected on the variable $T$ ( a) as well as the observable $q=T-S$ ( b). The 'ON' ('OFF') state is shown in red (blue), and the saddle by the green dashed line. c,e Time series of $q$ and the observable $T+S$ for a simulation where $\eta_1$ is ramped up linearly across the bifurcation. d,f Early-warning signal of increased variance in a sliding window for the time series in ( c,e). The variance is normalized by the variance in the first 500 years, where $\eta_1$ was held fixed. The time points are the endpoints of the sliding windows. Panel d shows the variance for $q$ as well as the observable $2T-S$ (red), and f shows the variance for the observable $T+S$.
• Figure 5: a-d Phase portrait of the Stommel model for four values of $\eta_1$ progressing towards the bifurcation where the ON state loses stability. The two stable fixed points are shown by the red (ON) and blue (OFF) dots, and the saddle point is the green triangle. The basin boundary is in black. The red dashed line is the identity line $T=S$. The instanton from ON to OFF is in purple, and the one from OFF to ON is the blue dashed line. Both are computed by the method in KIK20. e-h Residuals with respect to the ON fixed point obtained by an ensemble of long simulations for four values of $e$ (same as in panels a-d). The different straight lines correspond to vectors defining observables discussed in the main text. The green solid line indicates the direction of the edge state.