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An interesting problem in synchronization is the study of coupledoscillators, wherein oscillators with different natural frequencies synchronizeto a common frequency and equilibrium phase difference. In this paper, weinvestigate the stability and convergence in a network of coupled oscillatorsdescribed by the Kuramoto model. We consider networks with finite number ofoscillators, arbitrary interconnection topology, non-uniform coupling gains andnon-identical natural frequencies. We show that such a network synchronizesprovided the underlying graph is connected and certain conditions on thecoupling gains are satisfied. In the analysis, we consider as states the phaseand angular frequency differences between the oscillators, and the resultingdynamics possesses a continuum of equilibria. The synchronization probleminvolves establishing the Lyapunov stability of the fixed points and showingconvergence of trajectories to these points. The synchronization result isestablished in the framework of semistability theory.

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