nlin.CDAug 12, 2026
SaveKoopman early warning signals for bifurcation and rate-induced tipping
Organizations: Columbia University, USA · University of Edinburgh, UK · Los Alamos National Laboratory, USA · Pacific Northwest National Laboratory, USA ·
Abstract
Abrupt transitions in complex systems are often preceded by early warning signals. However, most indicators rely on the notion of critical slowing down and do not generally extend to rate-induced tipping where transitions can occur without local loss of stability. This is problematic in stochastic, nonautonomous systems where internal variability and time-varying variables interact to shape tipping onset. We use Koopman operator theory to develop a unified early warning framework for both bifurcation and rate-induced tipping in stochastic systems. Our approach builds on residual Koopman mode decomposition that measures discrepancies between dynamics and their finite-dimensional approximation, and extends it to the control setting by augmenting the observable space with time-varying control variables. In idealized examples, the resulting indicators recover expected signatures near bifurcation points and improve detection in rate-induced regimes where classical indicators fail. We further show that learned embeddings through deep learning outperform prescribed dictionaries, especially in a high-dimensional setting. Applied to simulations of the Atlantic Meridional Overturning Circulation, our Koopman-based indicators distinguish tipping from non-tipping trajectories and reveal interpretable spectral signatures prior to critical transition.
Figures & tables
Figure 1: Schematic illustration of Koopman-based early warning mechanisms. ( a ) In B-tipping, slow Koopman modes move toward the stability boundary as the system approaches critical slowing down, reducing the spectral gap and increasing dynamical memory. ( b ) Slower recovery causes stochastic perturbations to persist longer, leading to variance growth. ( c ) In R-tipping, the frozen dynamics can remain locally stable, but the system can fail to track a moving attracting branch. The current and future attracting states, and , indicate how the branch shifts as the control changes. The branch drift measures this shift, while the tracking error measures the departure of the current state from the instantaneous attracting state. Together, these spectral properties and tracking error mechanisms motivate our unified Koopman-based EWS.
Figure 2: Koopman EWS benchmarked against traditional metrics across tipping scenarios: In ( a ) the Rosenzweig-MacArthur model, panels show the ensemble behavior of five indicators computed on pre-transition sliding windows of length : Spectral gap, AR1, variance, ResKMD (RFF), and ResKMD (MLP observable with a hidden size of (16, 8, 4)). In ( b ) the Bautin oscillator, panels show the ensemble behavior of five indicators computed on pre-transition sliding windows: Spectral gap, AR1, and control-aware ResKMD (TDF, MLP observable with a hidden size of (8, 4, 2)). The shaded region denotes the interquantile range around the median.
Figure 3: Koopman EWS across empirical systems: ( a ) vocal phonation exhibiting fold transition, ( b ) paleoclimate time series of calcium carbonate associated with the end of greenhouse Earth, and ( c ) electricity blackout at Western Interconnect in 1996. The grey vertical dashed line indicates identified critical transitions. We find that our Koopman EWS indicator is able to detect critical transitions earlier (e.g., voice onset and electricity blackout) and more accurately (e.g., paleoclimate) than classical CSD metrics, such as variance. For all indicators, lighter dots represent earlier time.
Figure 4: Idealized AMOC box-model trajectories. ( a ) B-tipping and ( b ) R-tipping. Each panel shows left : phase-space evolution in , center : AMOC strength , and right : the corresponding hosing or control schedule.
Figure 5: EWS for the B- and R-tipping setups in AMOC box-model. For each setup, the panels show the ensemble behavior of five indicators computed on pre-transition sliding windows of length . ( a ) Spectral gap, AR1, variance, ResKMD (RFF), and ResKMD (MLP observable with a hidden size of (8, 4, 2)). ( b ) Spectral gap, variance, ResKMD (TDF), and control-aware ResKMD (TDF, MLP observable with a hidden size of (8, 4, 2)). The shaded region denotes the interquantile range around the median.
Figure 6: AMOC tipping (red) and non-tipping (blue) runs at N. ( a ) Depth profiles of Atlantic overturning streamfunction for representative coupled AMOC trajectories: the first two panels show non-tipping runs that recover to the stronger branch, while the latter two show tipping runs that evolve toward a weaker circulation state. ( b ) AMOC strength trajectories and ( c ) Denmark Strait sea-ice flux used as the control variable in the control-aware Koopman analysis. Light lines denote individual ensemble members and thick lines denote ensemble means. AMOC strength is computed as the maximum overturning streamfunction below 500 m, and Denmark Strait sea-ice flux is smoothed with a 10-year running mean.
Figure 7: Koopman EWS and spectral diagnostics for the coupled AMOC ensemble at N. (a) EWS for the coupled AMOC ensemble at N. The panels show the ensemble behaviour of AR1, variance, and control-aware ResKMD using TDF with delay size of 20 and MLP observables with a hidden size of of (16, 8, 4). (b) discrete-time Koopman eigenvalues in the unit disk with colored annulus representing -percentile distribution. (c) e-folding times of the slowest decaying non-constant generator mode for each ensemble member. (d) rolling group means of the three generator eigenvalues with smallest , obtained from EDMD fits on rolling windows of length of the full record. Red denotes tipping members and blue denotes non-tipping members.
Figure 8: Quantitative benchmarking of early warning indicators using AUROC. Panels correspond to ( a ) May’s harvesting model, ( b ) Rosenzweig–MacArthur model, ( c ) forced saddle-node system, ( d ) Bautin oscillator, ( e ) B-tipping AMOC box-model, and ( f ) R-tipping AMOC box-model. Higher AUROC values indicate improved discrimination between tipping and non-tipping trajectories. Across both tipping regimes, Koopman-based indicators generally outperform classical CSD indicators, with control-aware ResKMD outperforms in the R-tipping regimes.
Appendix figures & tables6 assets
Supplementary material from the paper’s appendix.
Appendix
Figure S1: Representative B-tipping ( a , b ) and R-tipping ( c , d ) examples. In each panel, the upper plot shows tipping (red) and non-tipping (blue) trajectories, and the lower plot shows the corresponding parameter schedule.
Figure S2: EWS for B-tipping using May’s harvesting model. Panels show the ensemble behavior of five indicators computed on pre-transition sliding windows of length : Spectral gap, AR1, variance, ResKMD with RFF, and ResKMD with MLP observable with a hidden size of (16, 8, 4). The shaded region denotes the interquantile range around the median.
Figure S3: EWS for the R-tipping in saddle-node. Panels show the ensemble behavior of five indicators computed on pre-transition sliding windows of length : Spectral gap, AR1, variance, and control-aware ResKMD (TDF, MLP observable with a hidden size of (8, 4, 2)). The shaded region denotes the interquantile range around the median.
Figure S4: Koopman EWS across empirical systems: ( a ) fold bifurcation in a microcosm experiment where cyanobacteria population undergoes dilution perturbation and increasing light stress, and ( b ) cytosolic ATP dynamics in living plant cells.
Figure S5: Supplementary coupled AMOC results at N. ( a ) Depth profile of overturning streamfunction for representative coupled AMOC trajectories at N: the first two columns show non-tipping runs that recover to the stronger branch, whereas the latter two show tipping runs that evolve toward a much weaker circulation state. ( b ) EWS on pre-transition sliding windows of length . ( c ) Discrete-time Koopman eigenvalues. ( d ) E-folding times of the slowest decaying non-constant mode. ( e ) Rolling generator eigenvalues with smallest .
Figure S6: Ablation of forcing variables in control-aware ResKMD for the coupled AMOC runs. Columns correspond to alternative forcing inputs used in the control channel: Labrador Sea MLD maximum, Irminger Sea MLD maximum, Scotland–Iceland Basin MLD maximum, and Greenland–Iceland–Norway Seas MLD maximum. The first row shows the corresponding 10-year rolling-average forcing trajectories, grouped by tipping (red) and non-tipping (blue) ensemble members. The second and third rows show the resulting control-aware ResKMD with TDF observables using sliding windows of length for AMOC strength at N and N, respectively.