On frequency synchronization of Kuramoto model with non-symmetric interconnection structure

In this paper we consider a problem of frequency synchronization for a system of Kuramoto oscillators in the case where interconnection graph is not necessarily symmetric. We show that similar to “all-to-all” case, problem of frequency synchronization is equivalent to the problem of existence of solutions for a non-linear system of algebraic equations. It is well known that for an undirected graph of interconnections, synchronization frequency is equal to the average of natural frequencies. However, in the case of directed graphs this is not true, moreover for fixed natural frequencies and given coupling structure, frequency of synchronization can also depend on the coupling strength. We give an analytical expression for the limit of synchronization frequency as the coupling strength goes to infinity. Finally, two examples are presented in order to illustrate our results.