Transition to Collective Oscillations in Finite Kuramoto 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.

arXiv comment11 pages, 8 figures, Editors' Suggestion PRE

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• Dynamics of oscillator populations with disorder in the coupling phase shifts
• Microscopic Cross-Correlations in the Finite-Size Kuramoto Model of Coupled Oscillators
• Exponentially long transient time to synchronization of coupled chaotic circle maps in dense random networks
• Synchronization transitions in Kuramoto oscillators with mixed interactions