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.
2020 Mathematics Subject Classification. 34D06, 70F10, 70G60, 92D25. Key words and phrases. Complete phase-locking, complete synchronization, the relativistic Kuramoto model, non-relativistic limit, order parameters, phase-locked state. The work of C. Min was supported by the new faculty research fund of Ajou University and the work of S.-Y. Ha is partially supported by National Research Foundation of Korea Grant NRF-2020R1A2C3A01003881). \u2217Corresponding author: Myeongju Kang.
