Frequency synchronization of Kuramoto-like model subject to time delays and phase shifts

This paper investigates the frequency synchronization of a generalized Kuramoto-like model incorporating both time delays and phase shifts for. A simple condition related to the maximum time delays, phase shifts, coupling strength, natural frequencies, and initial states are performed to ensure the boundedness of the maximum phase difference among all oscillators, which further indicates the exponential frequency synchronization of the heterogeneous Kuramoto-like oscillators. Moreover, the results also show that the coupling strength cannot be arbitrarily large enough due to the characteristics of the model. Finally, a straightforward simulation illustrates the exponential decay of the maximum phase and frequency differences, corroborating the validity of the theoretical findings.