Choosing observables that capture critical slowing down before tipping points: A Fokker-Planck operator approach

TL;DR

This paper introduces a data-driven method using diffusion maps to identify optimal observables that effectively capture critical slowing down, aiding early warning of tipping points in complex systems.

Contribution

It proposes a novel approach to select observables based on the eigenfunctions of the Fokker-Planck operator, improving early warning signals for tipping points.

Findings

Eigenfunction-based observables enhance early warning detection.

Method reduces false positives/negatives in critical slowing down signals.

Approach applicable to high-dimensional complex systems.

Abstract

Tipping points (TP) are abrupt transitions between metastable states in complex systems, most often described by a bifurcation or crisis of a multistable system induced by a slowly changing control parameter. An avenue for predicting TPs in real-world systems is critical slowing down (CSD), which is a decrease in the relaxation rate after perturbations prior to a TP that can be measured by statistical early warning signals (EWS) in the autocovariance of observational time series. In high-dimensional systems, we cannot expect a priori chosen scalar observables to show significant EWS, and some may even show an opposite signal. Thus, to avoid false negative or positive early warnings, it is desirable to monitor fluctuations only in observables that are designed to capture CSD. Here we propose that a natural observable for this purpose can be obtained by a data-driven approximation of the first non-trivial eigenfunction of the backward Fokker-Planck (or Kolmogorov) operator, using the diffusion map algorithm.