Formation of phase-locked states in a population of locally interacting Kuramoto oscillators

In this paper, we present an asymptotic formation of phase-locked states from the ensemble of Kuramoto oscillators with a symmetric and connected interaction topology. For a limited interaction topology that does not have an all-to-all interaction, Lyapunov type approaches based on phase and frequency diameters do not work due to the lack of completeness. Thus, we employ an energy method together with the connectedness of underlying interaction topologies to determine the complete synchronization estimates. Our synchronization estimation method consists of two parts. First we establish that the uniform boundedness of fluctuations yields the asymptotic formation of phase-locked states using Łojasiewicz gradient inequality. Second, we show that for the initial configurations lying in the half circle, the uniform boundedness of fluctuations can be derived by a comparison with solutions to the linear Gronwall's differential inequality for the total phase variance.