Complete and Partial Synchronization of Two-Group and Three-Group Kuramoto Oscillators

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1720-1765, September 2024.
Abstract.This paper is to investigate synchronization theories of a two-group Kuramoto model and a three-group Kuramoto model. In the settings of these models, every oscillator directly interacts with each other in the same group. In each group, only one oscillator directly interacts with one oscillator in another group. We prove that if the coupling strength is large and the initial configuration of each group is confined to a sector with the arc length less than [math], then all oscillators achieve a complete frequency synchronization asymptotically. We emphasize that there is no need to impose any initial condition on the connection between different groups. If, in addition, the natural frequencies in one group are the same, then partial phase synchronization occurs. Moreover, if all natural frequencies are identical, we prove that all oscillators either achieve a complete phase synchronization asymptotically or tend to a bipolar phase-locking state. We also provide several numerical simulations to support the main results.