On the complete phase synchronization for the Kuramoto model in the mean-field limit

TL;DR

This paper proves complete phase synchronization in the Kuramoto model for identical oscillators and extends the results to non-atomic initial data, exploring entropy's role in convergence for non-identical frequencies.

Contribution

It provides a new proof of complete synchronization for identical oscillators and extends the results to measure-valued initial data in the continuum limit.

Findings

Complete frequency synchronization for all initial data in identical case

Complete phase synchronization for non-atomic measure initial data

Relation between entropy boundedness and incoherent state convergence

Abstract

We study the Kuramoto model for coupled oscillators. For the case of identical natural frequencies, we give a new proof of the complete frequency synchronization for all initial data; extending this result to the continuous version of the model, we manage to prove the complete phase synchronization for any non-atomic measure-valued initial datum. We also discuss the relation between the boundedness of the entropy and the convergence to an incoherent state, for the case of non identical natural frequencies.