Noise-induced phase slips, log-periodic oscillations, and the Gumbel distribution

TL;DR

This paper investigates how weak noise causes phase slips in synchronized oscillators, revealing that their locations and durations follow specific statistical distributions related to extreme-value theory and log-periodic oscillations.

Contribution

It demonstrates that phase slip locations and durations converge to Gumbel and geometric distributions in the low-noise limit, linking transition path and extreme-value theories.

Findings

Phase slip locations converge to a Gumbel distribution.

Phase slip durations also follow a Gumbel distribution.

Results connect log-periodic oscillations with extreme-value theory.

Abstract

When two synchronised phase oscillators are perturbed by weak noise, they display occasional losses of synchrony, called phase slips. The slips can be characterised by their location in phase space and their duration. We show that when properly normalised, their location converges, in the vanishing noise limit, to the sum of an asymptotically geometric random variable and a Gumbel random variable. The duration also converges to a Gumbel variable with different parameters. We relate these results to recent works on the phenomenon of log-periodic oscillations and on links between transition path theory and extreme-value theory.