On Synchronization of Kuramoto Oscillators

Synchronization is a key concept to the under-standing of self-organization phenomena occurring in coupled oscillators of the dissipative type. In this paper we study one of the most representative models of coupled phase oscillators, the Kuramoto model. The traditional Kuramoto model (all-to-all connectivity) is said to synchronize if the angular frequencies of all oscillators converge to the mean frequency of the group and the oscillators get phase locked. Recently, Jadbabaie et. al. calculated a lower bound on the coupling gain which is necessary for the onset of synchronization in the traditional Kuramoto model. It was also shown that there exists a large enough coupling gain so that the phase differences are locally asymptotically stable. Furthermore, the authors demonstrated that the convergence is exponential when all oscillators have the same natural frequency. In this paper we assume that the natural frequencies of all oscillators are arbitrarily chosen from the set of reals. We develop a tighter lower bound on the coupling gain, as compared to the one proposed by Jadbabaie et. al., which is necessary for the onset of synchronization in the traditional Kuramoto model. Our main result says that it is possible to find a coupling gain such that the angular frequencies of all oscillators locally exponentially synchronize to the mean frequency of the group. To the best of our knowledge, this is the first result which demonstrates that in the traditional Kuramoto model, with all-to-all coupling and different natural frequencies, the oscillators locally exponentially synchronize.