Noise-Induced Phase Slips, Log-Periodic Oscillations, and the Gumbel Distribution

N. Berglund

2016, v.22, №3, 467-505

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.

Keywords: synchronization, phase slip, stochastic exit problem, large deviations, random Poincar\'e map, log-periodic oscillations, cycling, transition-path theory, extreme-value theory, Gumbel distribution%

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