Certification of almost global phase synchronization of all-to-all coupled phase oscillators

Abstract

Coupled oscillators may exhibit almost global phase synchronization, namely their phases tend to asymptotically overlap for almost all initial conditions. We consider certification of this property using Rantzer’s dual Lyapunov approach with sum of squares (SOS) programming. To this aim, we use a stereographic transformation from a hypertorus to an Euclidean space. For the case of all-to-all coupling, this transformation converts the problem of certifying stability into the problem of certifying divergence of almost all solutions to infinity. We show that the latter can be solved using a polynomial Lyapunov density, which can be constructed via SOS programming. This leads to the certification of almost global phase synchronization of all-to-all coupled phase oscillators. We apply our method to an example of coupled phase oscillators and to an example of coupled van der Pol oscillators, and show that it can support the existing tools of local stability analysis by ensuring almost global phase synchronization.