Probability of noise- and rate-induced tipping

TL;DR

This paper develops an approximation to estimate the probability of tipping caused by the interaction of parameter change speed and noise, aiding in early-warning detection and understanding of different tipping mechanisms.

Contribution

It introduces a new approximation method for calculating tipping probabilities under combined rate and noise effects, applicable to various systems and scenarios.

Findings

Valid for small to moderate drift speeds

Helps estimate false positive/negative rates in early warnings

Provides an asymptotic formula for critical ramp speed without noise

Abstract

We propose an approximation for the probability of tipping when the speed of parameter change and additive white noise interact to cause tipping. Our approximation is valid for small to moderate drift speeds and helps to estimate the probability of false positives and false negatives in early-warning indicators in the case of rate- and noise-induced tipping. We illustrate our approximation on a prototypical model for rate-induced tipping with additive noise using Monte-Carlo simulations. The formula can be extended to close encounters of rate-induced tipping and is otherwise applicable to other forms of tipping. We also provide an asymptotic formula for the critical ramp speed of the parameter in the absence of noise for a general class of systems undergoing rate-induced tipping.