The dynamic saddle-node bifurcation with noise on the slow variable

TL;DR

This paper investigates how Gaussian white noise added to the slow variable affects a slow-fast system near a saddle-node bifurcation, revealing that noise generally increases the slow variable on average.

Contribution

It provides the first analysis of noise effects on the slow variable in saddle-node bifurcations, deriving explicit formulas involving Airy functions.

Findings

Noise increases the expected value of the slow variable after bifurcation.

Explicit formulas for expectation and variance are derived using Airy functions.

Numerical simulations confirm the analytical predictions for significant noise levels.

Abstract

In this work, we analyse the effect of adding Gaussian white noise to the slow variable of a slow--fast system passing through a saddle--node (or fold) bifurcation. This problem is mainly motivated by applications to non-equilibrium energy sinks. While the effect of adding noise to the fast variable, which is important for noise-induced tipping, has been previously analysed in detail, the case where the slow variable is perturbed by noise has not been considered before. Our main result is that the noise increases the slow variable on average. We compute the effect of the noise, to lowest order, on the expectation and variance of the slow variable after the bifurcation. The contribution of the noise can be explicitly expressed in terms of Airy functions. We also provide numerical simulations, which show that the expansion to lowest order matches the observations for fairly large values of the noise intensity.