Sufficient conditions for asymptotic phase-locking to the generalized Kuramoto model

Abstract

In this paper, we introduce a generalized Kuramoto model and provide several sufficient conditions leading to asymptotic phase-locking. The proposed generalized Kuramoto model incorporates relativistic Kuramoto type models which can be derived from the relativistic Cucker-Smale (RCS) on the unit sphere via suitable approximations. For asymptotic phase-locking, 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 reduces to the Kuramoto model in a finite time interval, as the speed of light tends to infinity. Moreover, for some admissible initial data, nonrelativistic limit can be made uniformly in time. We also provide several numerical examples for two approximations of the relativistic Kuramoto model, and compare them with analytical results.