Synchronization of discrete time Kuramoto oscillators with delayed states

This paper investigates the synchronization problem of a kind of discrete-time Kuramoto oscillators based on delayed position states. For this kind of Kuramoto model, the oscillators can only obtain delayed phase and frequency states relative to each other at a series of discrete time instants. Some simple sufficient conditions on the time step, the time delay, the natural frequencies, the coupling strength, and the historical and initial values of the phases and frequencies are derived analytically to ensure the phase synchronization of homogeneous oscillators and the frequency synchronization of heterogeneous oscillators, respectively. Finally, some numerical simulations are performed to verify the correctness of the theoretical results.