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Koopman early warning signals for bifurcation and rate-induced tipping

Juan Nathaniel, Carla Roesch, Derek DeSantis, Parvathi Kooloth, Hang Fan, Valerio Lucarini, Anastasia Romanou, Pierre Gentine

arXiv:2608.14716v1nlin.CDcs.LG

TL;DR

Existing early warning signals often address stability loss or tracking failure separately, limiting their coverage of stochastic systems with interacting tipping mechanisms. This paper develops a unified Koopman-based framework and finds useful warning behavior across bifurcation and rate-induced tipping, including AMOC simulations.

  • Problem

    Existing early warning signals typically target either stability loss or tracking failure, despite their coexistence in stochastic, externally forced systems.

  • Method

    The paper extends residual Koopman mode decomposition with time-varying control variables to provide a unified spectral framework for bifurcation and rate-induced tipping.

  • Results

    Across model hierarchies and AMOC simulations, the indicators recover bifurcation warnings, discriminate rate-induced tipping where classical signals can be weak, and provide interpretable spectral diagnostics.

  • Takeaways & Limitations

    Koopman residuals offer a common, physically interpretable language for studying stability loss and tracking loss in stochastic, high-dimensional, strongly driven systems.

  • Takeaways & Limitations

    For rate-induced tipping, the residual need not diverge and can remain zero when augmented state-control dynamics are exactly represented in the observable space.

Abstract

from arXiv · show

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.

Introduction · Generalized Koopman framework for early warning of tipping points

The paper develops a unified Koopman-based early-warning framework for bifurcation- and rate-induced tipping in stochastic, nonautonomous systems. It uses residual Koopman representations, augmented with controls for tracking failure, to connect spectral deterioration and unresolved branch drift with impending transitions.

  • Introduction: Critical transitions can arise from loss of local stability or failure to track changing external conditions, while stochastic fluctuations and interacting timescales complicate early warning.These mechanisms are especially relevant in climate systems with internal variability, nonstationary forcing, and multiscale feedbacks.
  • Introduction: Classical early-warning signals primarily target bifurcation-induced tipping through critical slowing down, typically observed as increasing lag-1 autocorrelation and variance.Such indicators are not generally designed to detect rate-induced tipping without local stability loss.
  • Generalized Koopman framework for early warning of tipping points: The framework uses Koopman operator theory and residual Koopman mode decomposition to measure deterioration of finite-dimensional spectral representability near tipping.Far from tipping, dynamics may be captured by a compact set of modes; near tipping, this representation deteriorates through slow-mode accumulation or tracking failure.
  • Generalized Koopman framework for early warning of tipping points: For bifurcation-induced tipping, the residual separates truncation and stochastic contributions, both of which increase as slow Koopman modes approach the stability boundary and fluctuations become amplified.ResKMD therefore generalizes classical critical-slowing-down indicators such as AR1 and variance within a spectral operator-theoretic framework.
  • Generalized Koopman framework for early warning of tipping points: For rate-induced tipping, the Koopman representation is augmented with time-dependent control variables, producing a residual on the joint state-control space.This control-aware residual retains truncation and stochastic components while representing the changing forcing explicitly.
  • Generalized Koopman framework for early warning of tipping points: Rate-induced tipping is characterized by growth of tracking error when branch drift is not sufficiently compensated by local restoration, even while the dynamics remain locally stable.The tracking error measures deviation from the instantaneous attracting branch associated with the control variable.
  • Generalized Koopman framework for early warning of tipping points: The control-aware residual increases when branch drift contains components unresolved by the retained Koopman representation, distinguishing loss of tracking from spectral slowing near instability.Other unresolved dynamics can also increase the residual, so unresolved branch drift is not its only possible source.
  • Introduction: The resulting residual quantities serve as Koopman-based early-warning signals for bifurcation- and rate-induced tipping, respectively.The framework is evaluated across prototypical, reduced-order, and fully coupled AMOC simulations, with learned deep-learning embeddings outperforming prescribed observables such as RFF and TDF.

Koopman EWS skills on prototypical systems · Koopman EWS skills on empirical observations · Tipping of the Atlantic Meridional Overturning Circulation

Koopman residual-based indicators provide early warnings for both bifurcation and rate-induced tipping, including cases where classical critical-slowing-down metrics fail. Across empirical observations and AMOC simulations, control-aware residuals distinguish tipping trajectories and expose interpretable spectral changes before transition.

  • Koopman EWS skills on prototypical systems: On prototypical stochastic systems, Koopman residuals increase more clearly for bifurcation-tipping trajectories than for non-tipping trajectories as slow modes approach the stability boundary.The Koopman spectral gap is defined as γ = −Re(λL,1) and is expected to decrease toward zero near critical slowing down.
  • Koopman EWS skills on prototypical systems: For rate-induced tipping, AR1, variance, and spectral gap fail to consistently separate trajectories, whereas control-aware Koopman residuals increase substantially more for tipping cases.Classical indicators can increase in both tipping and non-tipping trajectories, consistent with unresolved branch drift under changing controls.
  • Koopman EWS skills on empirical observations: In more complex or noisy empirical settings, residual-based indicators detect transitions earlier or more accurately, including voice onset, electricity blackout, and paleoclimate transitions.Figure 3 specifically reports earlier detection for voice onset and electricity blackout and more accurate detection for paleoclimate.
  • Koopman EWS skills on empirical observations: Across heterogeneous empirical observations, Koopman indicators remain informative, with residual trends reflecting dynamical memory and spectral slowing in classical critical-slowing-down systems.The evaluated observations span cyanobacteria, vocal phonation, plant-cell hypoxia, paleoclimate, and electricity-blackout dynamics.
  • Tipping of the Atlantic Meridional Overturning Circulation: In the AMOC box model, classical indicators increase before B-tipping but fail for R-tipping, while Koopman residuals better distinguish tipping from non-tipping trajectories.The AMOC framework uses a reduced two-dimensional salinity model and supplies diagnosed AMOC strength Q as the scalar observable.
  • Tipping of the Atlantic Meridional Overturning Circulation: In the coupled NASA GISS ModelE ensemble, control-aware Koopman indicators provide the clearest separation, whereas AR1 is highly variable and variance gives weaker separation.At 48°N, eight members remain on a stronger overturning branch and two evolve toward a markedly weaker state.
  • Tipping of the Atlantic Meridional Overturning Circulation: Spectral diagnostics show tipping trajectories with eigenvalues closer to the unit circle, longer dominant e-folding times, and slow modes moving faster toward Re(λL,j) = 0.These patterns indicate spectral densification and a reduced spectral gap in the tipping ensemble.

Discussion

The paper presents a unified Koopman-based early-warning framework for bifurcation and rate-induced tipping in stochastic systems, using spectral approximation error to study both loss of stability and loss of tracking. Across examples including AMOC simulations, the indicators recover expected bifurcation warnings and help discriminate rate-induced tipping, while performance depends on modeling choices and finite-sample approximations.

  • Contributions: The framework extends residual Koopman mode decomposition from bifurcation tipping to rate-induced tipping through control-aware observables that capture loss of tracking.The residual combines truncation error and stochastic fluctuation within a single spectral quantity in the bifurcation setting.
  • Results: Across prototypical systems, reduced box models, and fully coupled AMOC simulations, Koopman indicators recover expected bifurcation-warning behavior and discriminate rate-induced regimes where classical signals can be weak or inconsistent.Classical examples include AR1 and variance.
  • Results: AMOC applications provide interpretable spectral diagnostics, including Te, spectral gaps, and slow Koopman-mode distributions, alongside tipping detection.These quantities support interpretation of resilience loss and timescale separation.
  • Limitations: Performance depends on observables, truncation level, and supplied control or driver variables, with some AMOC control choices substantially more informative than others.Selecting the dynamically relevant forcing channel remains a nontrivial modeling problem.
  • Limitations: Rate-induced residual increases are problem dependent and arise through unresolved branch drift, while finite-sample EDMD and ResDMD errors, dictionary dependence, and high-dimensional robustness require further study.The bifurcation case has a clearer theoretical divergence result than the rate-induced case.
  • Novelty: The study claims the first unified operator-theoretic early-warning framework spanning bifurcation and rate-induced tipping in stochastic systems.ResKMD treats loss of stability and loss of tracking through spectral approximation error rather than separate heuristic indicators.

Methods

The methods formulate stochastic bifurcation- and rate-induced tipping through local stability, moving-branch tracking, and Koopman operators. Control-aware residuals capture unresolved branch drift and approximation or stochastic effects, while acknowledging that R-tipping need not produce universal residual divergence.

  • Stochastic formulation: The stochastic dynamics assume zero conditional-mean, finite-second-moment forcing, with independent Gaussian noise used in synthetic experiments.Expectations and covariances are defined under the probability law induced by the noise sequence.
  • Bifurcation-induced tipping: B-tipping is characterized by a stable equilibrium branch whose Jacobian spectral radius approaches the unit circle, weakening restoring stability and producing critical slowing down.When multiple modes decay on comparable timescales, AR1 and variance are effective scalar summaries rather than exact spectral diagnostics.
  • Rate-induced tipping: R-tipping is formulated as loss of tracking of a moving frozen-equilibrium branch, caused by competition between local restoration and control-induced branch drift.The frozen dynamics can remain locally stable at every fixed control value while the nonautonomous trajectory departs from the moving branch.
  • Koopman framework: The stochastic Koopman framework represents nonlinear evolution with operators on observables, including first- and second-moment operators to connect spectra with variability and variance.Data-driven observable dictionaries include random Fourier features, time-delay features, and deep embeddings based on MLP autoencoders.
  • Residual indicators: Near B-tipping, residual Koopman mode decomposition can increase through poorer finite-dimensional spectral approximation and larger stochastic fluctuations as omitted or nonconstant eigenvalues approach the unit circle.The truncation error is bounded by omitted Koopman modes, while persistent perturbations amplify stochastic fluctuations.
  • Control-aware residuals: For R-tipping, no universal residual divergence is expected, but unresolved branch drift provides a lower-bound contribution to the control-aware residual alongside other sources of residual growth.If augmented state-control dynamics are represented exactly, the residual can remain zero even when trajectories lose track of the moving equilibrium branch.

Declarations · Supplementary Information for Koopman early warning signals for bifurcation and rate-induced tipping

The supplementary information lists the paper’s nine authors and their affiliations across six institutions in the USA and UK.

  • Supplementary Information for Koopman early warning signals for bifurcation and rate-induced tipping: The author list includes Juan Nathaniel, Carla Roesch, Derek DeSantis, Parvathi Kooloth, and Hang Fan.These authors are listed first in the supplementary information.
  • Supplementary Information for Koopman early warning signals for bifurcation and rate-induced tipping: The remaining authors are Valerio Lucarini, Anastasia Romanou, and Pierre Gentine.These authors continue the supplementary information’s author list.
  • Supplementary Information for Koopman early warning signals for bifurcation and rate-induced tipping: The affiliations span Columbia University, University of Edinburgh, Los Alamos National Laboratory, Pacific Northwest National Laboratory, University of Leicester, and NASA Goddard Institute for Space Studies.The listed institutions are located in the USA and UK.

S1 Spectral view of classical indicators

Classical early warning indicators are derived from equilibrium-centered linearization, where slowing recovery near bifurcation produces increasing autocorrelation and variance. As the dominant eigenvalue approaches +1 or −1, lag-one correlation approaches +1 or −1, while variance increases and diverges in the linear approximation.

  • Equilibrium-centered interpretation: Classical CSD analysis linearizes dynamics around a stable fixed point and assumes the local stability structure remains nearly fixed over the analysis window.This equilibrium-centered assumption underpins the interpretation of AR1 and variance as early warning indicators.
  • Spectral signatures: As the dominant Jacobian eigenvalue approaches unit magnitude, perturbations decay more slowly and persistence increases.For a real dominant eigenvalue, the absolute lag-one correlation approaches 1.
  • Spectral signatures: Lag-one correlation approaches +1 near a saddle-node bifurcation as the leading eigenvalue approaches +1, but approaches −1 as the leading eigenvalue approaches −1.The sign of the limiting correlation follows the sign of the real dominant eigenvalue.
  • Variance response: As the spectral radius approaches 1, the variance term increases and diverges within the linear approximation.This variance growth follows from stationarity and uncorrelated innovations in the derivation.

S2 Derivation and Proof

The derivation decomposes one-step prediction error into squared truncation error and conditional stochastic fluctuation, then integrates this identity to prove Proposition 1. Subsequent arguments show that residuals detect unresolved dynamics and rate-induced tracking failure even when frozen dynamics remain stable.

  • Proposition 1 proof: The squared prediction error separates into truncation error and conditional fluctuation terms.The cross term vanishes, and the first term is exactly the squared truncation error at (ω, u).
  • Proposition 1 proof: The stochastic fluctuation contribution equals the trace of the conditional covariance.Expectations and covariances are taken over the one-step stochastic transition and integrated over state-control space against measure µ.
  • Proposition 1 proof: Integrating the error identity over Ω× R^U with respect to µ proves Proposition 1.The control-aware expectations and covariances are defined for the current state-control pair (ω, u).
  • Forced-system example: For the forced system ω_t+1 = aω_t + u_t, u_t+1 = u_t + r with 0 < a < 1, sufficiently large r prevents tracking despite stable frozen dynamics.The example defines a tracking error and uses an observable dictionary whose augmented dynamics close exactly; deterministic truncation and stochastic fluctuation therefore vanish.
  • Residual lower bound: The residual is bounded below by truncation error because the stochastic fluctuation term is nonnegative.If each component of ϵ lies in the chosen observable space, then Q_mϵ = 0 and the corresponding special case follows immediately.

S3 Prototypical Experiments

The prototypical experiments compare tipping and non-tipping trajectories in bifurcation-induced and rate-induced systems. They evaluate classical and Koopman-based early warning indicators using pre-transition sliding-window ensembles.

  • Representative trajectories: The experiments present representative tipping and non-tipping trajectories for both B-tipping and R-tipping systems, alongside their parameter schedules.These examples cover bifurcation-induced and rate-induced tipping cases.
  • B-tipping: For B-tipping, five indicators are evaluated on pre-transition sliding windows of length 50%: Spectral gap, AR1, variance, ResKMD with RFF, and ResKMD with MLP.The MLP observable uses hidden size (16, 8, 4), and shaded regions show the interquantile range around the median.
  • R-tipping: For R-tipping in a saddle-node system, five indicators are evaluated on pre-transition sliding windows of length 50%, including control-aware ResKMD.The indicator set comprises Spectral gap, AR1, variance, and control-aware ResKMD with a TDF and MLP observable of hidden size (8, 4, 2).

S4 Empirical Experiments

Empirical experiments show that Koopman-based early warning signals can identify critical transitions across distinct biological systems, including a cyanobacterial fold bifurcation and cytosolic ATP dynamics in living plant cells.

  • S4 Empirical Experiments: Koopman-based early warning signals correctly identify critical transitions in empirical systems.The experiments include a cyanobacteria microcosm undergoing dilution perturbation and increasing light stress, and cytosolic ATP dynamics in living plant cells.

S5 Additional Experiments · S5.1 Coupled AMOC results at 26◦N

Supplementary coupled AMOC experiments at 26◦N reproduce the main conclusions, with control-aware ResKMD separating tipping trajectories earlier than AR1 and variance. Forcing-variable ablations show that indicator performance depends on whether the control captures the dominant AMOC-weakening mechanism.

  • S5.1 Coupled AMOC results at 26◦N: At 26◦N, coupled AMOC results yield similar results and conclusions to the main-text analysis at 48◦N.The supplementary analysis examines representative coupled AMOC trajectories and early warning signals before transition.
  • S5.1 Coupled AMOC results at 26◦N: Control-aware ResKMD provides better discrimination and earlier early-warning-signal detection than AR1 and variance at 26◦N.The comparison is made on pre-transition sliding windows of length 20%.
  • S5.1 Coupled AMOC results at 26◦N: At 26◦N, non-tipping trajectories recover to the stronger circulation branch, whereas tipping trajectories evolve toward a much weaker circulation state.The figure also reports discrete-time Koopman eigenvalues and e-folding times of the slowest decaying non-constant mode.
  • S5.2 AMOC control variable ablation: The control-variable ablation replaces Denmark Strait sea-ice flux with maximum mixed-layer depth in four AMOC-relevant regions.The regions are the Labrador Sea, Irminger Sea, Scotland-Iceland Basin, and Greenland-Iceland-Norway Seas.
  • S5.2 AMOC control variable ablation: Control-aware ResKMD performance is sensitive to forcing choice, with some forcings producing clearer tipping-versus-non-tipping separation than others.The ablation compares forcing trajectories and corresponding control-aware residuals across the two ensembles.
  • S5.2 AMOC control variable ablation: The findings support using control variables that capture the dominant mechanism associated with AMOC weakening.This interpretation explains why alternative forcing channels differ in the clarity of their residual-based separation.
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