Inverse-square law between time and amplitude for crossing tipping thresholds

TL;DR

This paper derives an inverse-square law relating the maximum parameter deviation and duration above a threshold to prevent tipping in dynamical systems, with applications to stochastic models like the Indian Summer Monsoon.

Contribution

It introduces a simple criterion linking parameter overshoot and duration to avoid tipping, including stochastic effects, validated through numerical simulations.

Findings

Inverse-square law between parameter peak and time above threshold

Approximate probability of tipping with stochastic forcing

Validation with a high-dimensional climate model

Abstract

A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses} near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher dimensional system, a model for the Indian Summer Monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.