A fast–slow dynamical systems theory for the Kuramoto type phase model

We present a fast–slow dynamical systems theory for the Kuramoto type phase model. When the order parameters are frozen, the fast system consists of independent oscillator equations, whereas the slow system describes the evolution of order parameters. We average out the slow system over the fast manifold to derive a weak form of an amplitude–angle coupled system for the evolution of Kuramotoʼs order parameters. This yields the slow evolution of order parameters to be constant values which gives a rigorous proof to Kuramotoʼs original assumption in his self-consistent mean-field theory.