Early-warning indicators for rate-induced tipping

TL;DR

This paper investigates early-warning signals for rate-induced tipping in dynamical systems, revealing delays in indicators like variance and autocorrelation relative to the actual tipping event, through analysis of a saddle-node model with drift and noise.

Contribution

It demonstrates that common early-warning indicators exhibit a delay relative to the tipping point and provides a systematic analysis of the most likely tipping time using variational methods.

Findings

Early-warning indicators show a delay before tipping occurs.

Most likely tipping path crosses the threshold with a delay.

Systematic analysis of tipping time based on parameters and noise.

Abstract

A dynamical system is said to undergo rate-induced tipping when it fails to track its quasi-equilibrium state due to an above-critical-rate change of system parameters. We study a prototypical model for rate-induced tipping, the saddle-node normal form subject to time-varying equilibrium drift and noise. We find that both most commonly used early-warning indicators, increase in variance and increase in autocorrelation, occur not when the equilibrium drift is fastest but with a delay. We explain this delay by demonstrating that the most likely trajectory for tipping also crosses the tipping threshold with a delay and therefore the tipping itself is delayed. We find solutions of the variational problem determining the most likely tipping path using numerical continuation techniques. The result is a systematic study of the most likely tipping time in the plane of two parameters, distance from tipping threshold and noise intensity.