# The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit

## TL;DR

This paper rigorously justifies the mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limits, including practical graph types like small-world and nearest-neighbor graphs.

## Contribution

It extends Neunzert's method to prove the mean field approximation for the Kuramoto model on complex graph sequences with non-Lipschitz limits.

## Key findings

- Established rigorous justification of the mean field equation on non-Lipschitz graph limits.
- Applied the method to graphs like small-world and nearest-neighbor networks.
- Provided a mathematical foundation for analyzing synchronization on complex networks.

## Abstract

The Kuramoto model (KM) of coupled phase oscillators on graphs provides the most influential framework for studying collective dynamics and synchronization. It exhibits a rich repertoire of dynamical regimes. Since the work of Strogatz and Mirollo, the mean field equation derived in the limit as the number of oscillators in the KM goes to infinity, has been the key to understanding a number of interesting effects, including the onset of synchronization and chimera states. In this work, we study the mathematical basis of the mean field equation as an approximation of the discrete KM. Specifically, we extend the Neunzert's method of rigorous justification of the mean field equation to cover interacting dynamical systems on graphs. We then apply it to the KM on convergent graph sequences with non-Lipschitz limit. This family of graphs includes many graphs that are of interest in applications, e.g., nearest-neighbor and small-world graphs.