Phase-locking between Kuramoto oscillators: Robustness to time-varying natural frequencies

In this paper we analyze the robustness of phase-locking in the Kuramoto system with arbitrary bidirectional interconnection topology. We show that the effects of time-varying natural frequencies encompass the heterogeneity in the ensemble of oscillators, the presence of exogenous disturbances, and the influence of unmodeled dynamics. The analysis, based on a Lyapunov function for the incremental dynamics of the system, provides a general methodology to build explicit bounds on the region of attraction, on the size of admissible inputs, and on the input-to-state gains. As an illustrative application of this method, we show that, in the particular case of the all-to-all coupling, the synchronized state is exponentially input-to-state stable provided that all initial phase differences lie in the same half circle. The approach provides an explicit bound on the convergence rate, thus extending recent results on the exponential synchronization of the finite Kuramoto model. Furthermore, the proposed Lyapunov function for the incremental dynamics allows for a new characterization of the robust asymptotically stable phase-locked states of the unperturbed dynamics in terms of its isolated local minima.