Phase Locked Synchronization for Kuramoto Model, with Attractive and Repulsive Interconnections

We consider Kuramoto model of phase oscillators with attractive-repulsive interconnections. We generalize results on existence of phase locked solutions for a family of input-output weighted sign-symmetric interconnection digraphs. Necessary and sufficient conditions for existence of phase locked solutions are formulated and their local asymptotically stability is analysed. In particular, we show that all systems from their family have the same synchronization frequency and each of them has only one asymptotically stable phase locked solution. Finally, our results are illustrated through a demonstrative example.