Koopman early warning signals for bifurcation and rate-induced tipping

2026-08-11

AI Summary

This study addresses the lack of effective early warning signals for rate-induced tipping in stochastic non-autonomous systems by proposing a unified framework grounded in Koopman operator theory. By integrating residual mode decomposition, observable space augmentation, and deep learning embeddings, this approach overcomes the failure of traditional indicators under rate-induced mechanisms, demonstrating that data-driven embeddings outperform preset dictionaries. Validated through Atlantic Meridional Overturning Circulation (AMOC) simulations, the method successfully distinguishes tipping trajectories and reveals interpretable spectral features preceding critical transitions. Consequently, this work enables both effective monitoring and mechanistic interpretation of abrupt changes in complex dynamical systems, offering a robust solution for predicting rate-dependent bifurcations where conventional metrics prove inadequate.

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.

Problem

• Early warning signals
• Rate-induced tipping
• Bifurcation
• Stochastic nonautonomous systems
• Critical transitions

Innovation

• Koopman operator theory
• rate-induced tipping
• residual Koopman mode decomposition
• deep learning embeddings
• early warning signals