Frequency and Phase Synchronization in Stochastic Systems

TL;DR

This paper explores how frequency and phase synchronization occur in noisy stochastic systems, comparing different phase definitions, analyzing noise effects, and applying methods to various models like oscillators and bistable systems.

Contribution

It introduces a robust framework for analyzing phase synchronization in stochastic systems, including exact expressions and a novel Rice frequency method for noisy potential systems.

Findings

Analytic expressions for mean frequency and phase diffusivity in a thermal two-state system.

A new method to quantify frequency synchronization using Rice crossing rates.

Insights into stochastic resonance and phase coherence in driven bistable oscillators.

Abstract

The phenomenon of frequency and phase synchronization in stochastic systems requires a revision of concepts originally phrased in the context of purely deterministic systems. Various definitions of an instantaneous phase are presented and compared with each other with special attention payed to their robustness with respect to noise. We review the results of an analytic approach describing noise-induced phase synchronization in a thermal two-state system. In this context exact expressions for the mean frequency and the phase diffusivity are obtained that together determine the average length of locking episodes. A recently proposed method to quantify frequency synchronization in noisy potential systems is presented and exemplified by applying it to the periodically driven noisy harmonic oscillator. Since this method is based on a threshold crossing rate pioneered by S.O. Rice the related phase velocity is termed Rice frequency. Finally, we discuss the relation between the phenomenon of stochastic resonance and noise-enhanced phase coherence by applying the developed concepts to the periodically driven bistable Kramers oscillator.