Linear reformulation of the Kuramoto model: Asymptotic mapping and stability properties

Problems of synchronization of coupled nonlinear oscillators and, in particular, problem of synchronization of Kuramoto model, have attracted attention of researchers from diverse fields including physics, biology, neuroscience, and mathematics during several last decades. In this paper for the classical “all-to-all” Kuramoto model we construct an auxiliary linear model that preserves information on the natural frequencies and interconnection gains of the original Kuramoto model and depends on the phase locked soltions of the Kuramoto model. Stability properties of this model are analysed and we show that solutions of the new linear system exponentially converge to a stable periodic limit cycle for almost all initial conditions. Finally we show that asymptotically, in the time limit, this model maps on the original Kuramoto model.