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

For the classical “all-to-all” Kuramoto model, we construct a family of auxiliary linear models that preserves information on the natural frequencies and interconnection gains of the original Kuramoto model and depends on its phase-locked solutions. Stability properties of the family of linear systems are analyzed, we show that there is only one system in this family which is stable and for almost all initial conditions its solutions exponentially converge to a stable periodic limit cycle. Finally, we show that asymptotically, in the time limit, this linear system maps on the original Kuramoto model.