Kuramoto oscillators with inertia: A fast-slow dynamical systems approach

We present a fast-slow dynamical systems theory for a Kuramoto type model with inertia. The fast part of the system consists of N-decoupled pendulum equations with constant friction and torque as the phase of individual oscillators, whereas the slow part governs the evolution of order parameters that represent the amplitude and phase of the centroid of the oscillators. In our new formulation, order parameters serve as orthogonal observables in the framework of Artstein-Kevrekidis-Slemrod-Titi's unified theory of singular perturbation. We show that Kuramoto's order parameters become stationary regardless of the coupling strength. This generalizes an earlier result (Ha and Slemrod (2011)) for Kuramoto oscillators without inertia.