Synchronization analysis of kuramoto oscillators

In this paper, we study the original Kuramoto oscillators and the generalized Kuramoto oscillators with directed coupling topology. For the original Kuramoto model with identical oscillators, we obtain that frequency synchronization can occur for all initial phase configurations distributed over the whole circle, which is proved by means of a new method based on the Lojasiewicz inequality for gradient systems of analytic functions. This improves the corresponding result in [S.-Y. Ha, T. Ha, and J.-H. Kim, Physica D 239, 1692-1700, 2010], where the authors only considered initial phase configurations distributed over the open half circle. For the generalized Kuramoto model with directed coupling topology, we show that when the phases of oscillators are distributed over the half circle and the coupling strength is sufficiently large, frequency synchronization is guaranteed. This improves and extends the previous results in [N. Chopra and M. W. Spong, IEEE Trans. Automat. Control 54, 353-357, 2009], [S.Y. Ha, T. Ha, and J.H. Kim, Physica D 239, 1692-1700, 2010], and [Y.P. Choi, S.Y. Ha, S. Jung, and Y. Kim, Physica D 241, 735-754, 2012], where the corresponding results hold in the original Kuramoto model for initial phase configurations whose diameters are smaller than π/2 or π. Finally, we extend the result to the case of switching topology.