On a generalized Kuramoto model with relativistic effects and emergent dynamics

TL;DR

This paper introduces a generalized Kuramoto model incorporating relativistic effects, analyzes its emergent synchronization behaviors, and compares approximate models with analytical results through numerical simulations.

Contribution

It develops a relativistic Kuramoto model derived from the relativistic Cucker-Smale model and provides frameworks for synchronization and non-relativistic limits.

Findings

Conditions for complete synchronization based on initial data and parameters

Relativistic Kuramoto model reduces to classical Kuramoto model in non-relativistic limit

Numerical simulations validate analytical predictions

Abstract

We propose a generalized Kuramoto model with relativistic effects and investigate emergent asymptotic behaviors. The proposed generalized Kuramoto model incorporates relativistic Kuramoto(RK) type models which can be derived from the relativistic Cucker-Smale (RCS) on the unit sphere under suitable approximations. We present several sufficient frameworks leading to complete synchronization in terms of initial data and system parameters. For the relativistic Kuramoto model, we show that it can be reduced to the Kuramoto model in any finite time interval in a non-relativistic limit. We also provide several numerical examples for two approximations of the relativistic Kuramoto model, and compare them with analytical results.