Synchronization and semistability analysis of the Kuramoto model of coupled nonlinear oscillators

An interesting problem in synchronization is the study of coupled oscillators, wherein oscillators with different natural frequencies synchronize to a common frequency and equilibrium phase difference. In this paper, we investigate the stability and convergence in a network of coupled oscillators described by the Kuramoto model. We consider networks with finite number of oscillators, arbitrary interconnection topology, non-uniform coupling gains and non-identical natural frequencies. We show that such a network synchronizes provided the underlying graph is connected and certain conditions on the coupling gains are satisfied. In the analysis, we consider as states the phase and angular frequency differences between the oscillators, and the resulting dynamics possesses a continuum of equilibria. The synchronization problem involves establishing the Lyapunov stability of the fixed points and showing convergence of trajectories to these points. The synchronization result is established in the framework of semistability theory.