Asymptotic phase synchronization of Kuramoto model with weighted non-symmetric interconnections: A case study

We consider a synchronization problem for the Kuramoto oscillators using a linear modeling framework. For the general case of Kuramoto model with directed and weighted graph of interconnections we show that the problem of existence of phase locked solutions is equivalent to an algebraic problem of existence of a corresponding matrix with a single purely imaginary eigenvalue. In the case of complete input- output weighted graphs this approach allows to give analytical expressions both for the synchronization frequencies and the phase locked solutions and, on the next step, to analyze their stability properties.