Continuum limit of the Kuramoto model with random natural frequencies on uniform graphs

TL;DR

This paper extends the continuum limit analysis of the Kuramoto model to cases with random natural frequencies on uniform graphs, showing that the limit still accurately predicts stable dynamics.

Contribution

It proves the validity of the continuum limit for the Kuramoto model with random natural frequencies, especially uniformly distributed, on various uniform graphs.

Findings

Continuum limit accurately predicts stable solutions with random frequencies.

Asymptotic stability in the continuum limit corresponds to stability in the finite model.

Numerical simulations confirm theoretical predictions.

Abstract

We study the Kuramoto model (KM) having random natural frequencies and defined on uniform graphs that may be complete, random dense or random sparse. The natural frequencies are assumed to be independent and identically distributed on a bounded interval. In the previous work, the corresponding continuum limit (CL) was proven to approximate stable motions in the KM well when the natural frequencies are deterministic, even if the graph is not uniform, although it may not do so for unstable motions and bifurcations. We show that the method of CLs is still valid even when the natural frequencies are random, especially uniformly distributed. In particular, an asymptotically stable family of solutions to the CL is proven to behave in the $L^2$ sense as if it is an asymptotically stable one in the KM, under an appropriate uniform random permutation. We demonstrate the theoretical results by numerical simulations for the KM with uniformly distributed random natural frequencies.