Approximate solution for frequency synchronisation in a finite-size Kuramoto model

TL;DR

This paper derives an approximate analytical solution for frequency synchronization in finite-size Kuramoto models, applicable to any frequency distribution and coupling strength, and provides a stability criterion for the synchronized state.

Contribution

It introduces a master solution for the finite-size Kuramoto model that is valid for any frequency distribution and coupling strength, without restrictions.

Findings

Provides an approximate analytical solution for phase angles.

Derives a stability criterion for frequency synchronization.

Applicable to any natural frequency distribution.

Abstract

Scientists have been considering the Kuramoto model to understand the mechanism behind the appearance of collective behaviour, such as frequency synchronisation (FS) as a paradigm, in real-world networks with a finite number of oscillators. A major current challenge is to obtain an analytical solution for the phase-angles. Here, we provide an approximate analytical solution for this problem by deriving a master solution for the finite-size Kuramoto model, without imposing any restriction on the distribution of the natural frequencies of the oscillators. The master solution embodies all particular solutions of the finite-size Kuramoto model for any frequency distribution and coupling strength larger than the critical one. Furthermore, we present a criterion to determine the stability of the FS solution. This allows one to analytically infer the relationship between the physical parameters and the stable behaviour of networks.