Transition to Collective Oscillations in Finite Kuramoto Ensembles

TL;DR

This paper investigates finite-size effects on synchronization in the Kuramoto model, identifying how sample kurtosis and skewness influence collective oscillations and their properties.

Contribution

It introduces an approach linking sample kurtosis and skewness to collective oscillation conditions, extending understanding to the thermodynamic limit.

Findings

Sample kurtosis affects the amplitude of collective oscillations.

Sample skewness determines the frequency of the collective mode.

Effects of kurtosis and skewness persist in infinite ensembles.

Abstract

We present an alternative approach to finite-size effects around the synchronization transition in the standard Kuramoto model. Our main focus lies on the conditions under which a collective oscillatory mode is well defined. For this purpose, the minimal value of the amplitude of the complex Kuramoto order parameter appears as a proper indicator. The dependence of this minimum on coupling strength varies due to sampling variations and correlates with the sample kurtosis of the natural frequency distribution. The skewness of the frequency sample determines the frequency of the resulting collective mode. The effects of kurtosis and skewness hold in the thermodynamic limit of infinite ensembles. We prove this by integrating a self-consistency equation for the complex Kuramoto order parameter for two families of distributions with controlled kurtosis and skewness, respectively.