Detecting and forecasting tipping points from sample variance alone

Abstract

Anticipating tipping points in complex systems is a fundamental challenge across domains. Traditional early warning signals (EWSs) based on critical slowing down, such as increasing sample variance, are widely used, but their ability to reliably indicate imminent bifurcations and forecast their timing remains limited. Here, we introduce TIPMOC (TIpping via Power-law fits and MOdel Comparison), a parametric framework designed to statistically detect the approach of a bifurcation and estimate its future location using only the sample variance. TIPMOC exploits the mathematical property that variance diverges with a characteristic power-law form near codimension-one bifurcations. By sequentially monitoring system variance as a control parameter changes, TIPMOC statistically adjudicates between linear and power-law divergence at each step. When evidence favors power-law divergence, TIPMOC forecasts the impending tipping point and estimates its position; otherwise, it avoids false positives. Through numerical simulations, we demonstrate TIPMOC's robustness and accuracy in both detection and timing prediction across different types of dynamics and bifurcation. TIPMOC shows low false positive rates and performs well even with uneven sampling and colored noise. This method thus enhances the interpretability and practical utility of classical EWSs, serving as both a transparent add-on and a stand-alone statistical tool for forecasting regime shifts in diverse complex systems.

Figures (4)

• Figure 1: Schematic of a power-law fit used by TIPMOC. The value of the control parameter at which the fitted power law diverges, $\hat{u}_{\text{c}}$, is the estimated bifurcation point.
• Figure 2: Detection of the saddle-node bifurcation point for the stochastic double-well system. (a) $\hat{V}$ as a function of $u$, shown as circles, for one simulation. An impending bifurcation is detected at $u_{\text{det}} = 2.702$, and the bifurcation point is estimated as $\hat{u}_{\text{c}} = 3.185$. The solid line represents the power-law fit to all the observed ($u$, $\hat{V}$) pairs. The blue dashed line represents the power-law fit at detection (i.e., fit using the data up to $u_{\text{det}}$). The red dashed line represents the least-square linear fit at detection. (b) $\Delta \text{AIC}_{\text{c}}$ as a function of $u$. The horizontal line represents the detection threshold, $\Delta \text{AIC}_{\text{c}} = -10$. We do not show $\Delta \text{AIC}_{\text{c}}$ for early $u$ values because we start the model fit and comparison at $\ell_0 = 8$ to avoid comparing between fits based on too few data points. (c) $\hat{V}$ as a function of $u$ for three simulation runs. Each color represents a simulation. The triangles and dotted lines represent $u_{\text{det}}$ and $\hat{u}_{\text{c}}$, respectively, with the same color convention. The dashed line represents the deterministic bifurcation point, $u = u_{\text{c}}$.
• Figure 3: Results of early detection of the bifurcation point for $100$ runs of the stochastic double-well dynamics. (a) Relationship between the control parameter at detection, i.e., $u_{\text{det}}$, and the bifurcation point predicted at $u = u_{\text{det}}$, i.e., $\hat{u}_{\text{c}}$. (b) Relationship between Kendall's $\tau$ and $\hat{u}_{\text{c}}$. Each circle corresponds to one run. The horizontal dashed lines show the deterministic saddle-node bifurcation point, $u_{\text{c}} \approx 3.079$, in the absence of dynamical noise. In panel (a), the solid line represents the identity line $\hat{u}_{\text{c}} = u_{\text{det}}$. In (a) and (b) each, we also show the Pearson correlation coefficient between the two quantities across the $100$ runs.
• Figure 4: $\hat{V}$ for three runs of different dynamical systems and detection of impending bifurcations. (a) Over-harvesting model with $K=10$, showing a saddle-node bifurcation. (b) Over-harvesting model with linear grazing, showing a transcritical bifurcation. (c) Rosenzweig-MacArthur model, showing a Hopf bifurcation. (d) Mutualistic-interaction model on a synthetic network with $100$ nodes having community structure. Each color denotes one run. Each panel has three runs. The triangles and dotted lines represent $u_{\text{det}}$ and $\hat{u}_{\text{c}}$, respectively. The dashed lines represent the bifurcation point in the deterministic case, $u_{\text{c}}$. In all runs shown in this figure, TIPMOC detects an impending bifurcation before reaching the deterministic bifurcation point. In (d), we have gradually decreased the control parameter, $u$, representing the coupling strength among different species, to emulate deterioration toward mass extinction. See Methods for the network used in (d).