Spanning-Tree-Based Synchronization Conditions for Second-Order Kuramoto Networks

This brief studies the synchronization of second-order phase-coupled Kuramoto oscillators. Under the effects of non-uniformity among the system parameters, we propose novel spanning-tree-based conditions for the oscillators to remain phase cohesive over time, which further guarantees the synchronization. First, by choosing a spanning tree in the underlying network graph, we consider the phase cohesiveness as a 2-norm bound on the tree-induced phase differences and an infinity norm bound on the relative phases that are excluded from the spanning tree. Energy functions are constructed to derive two sets of conditions on achieving these two types of boundedness, respectively. Then, we prove that presented phase cohesive conditions guarantee the asymptotic synchronization for a second-order Kuramoto network. Our method is less conservative than those in literature on estimating a region of permissible initial states for oscillators to achieve synchronization, which is verified by numerical examples.