Koopman early warning signals for bifurcation and rate-induced tipping

• URL: http://arxiv.org/abs/2608.14716v1
• Date: Wed, 12 Aug 2026 01:33:22 GMT
• Title: Koopman early warning signals for bifurcation and rate-induced tipping
• Abstract summary: We use Koopman operator theory to develop a unified early warning framework for both bifurcation and rate-induced tipping in systems.<n>Our approach builds on residual Koopman mode decomposition that measures discrepancies between dynamics and their finite-dimensional approximation.<n>In idealized examples, the resulting indicators recover expected signatures near bifurcation points and improve detection in rate-induced regimes.
• Score: 3.0353973023015346
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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