We provide a complete analysis of the Kuramoto model of coupled nonlinear oscillators with uncertain natural frequencies and arbitrary interconnection topology. Our work extends and supersedes existing, partial results for the case of an all-to-all connected network. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value all the oscillators synchronize, resulting in convergence of all phase di#erences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity.

Introduction Coupled oscillator systems are good models to describe various nonlinear phenomena in the field of natural science and there are many studies on mutual synchronization of oscillators[1]--[6]. Concerning the synchronization of N van der Pol oscillators coupled by capacitors or inductors, we can generally observe the coexistence of in-phase synchronization and N - phase synchronization. However, N-phase synchronization occurs only when N is a prime number. In case of N =4,two pairs of anti-phase synchronization occur independently. Hence, stable four-phase synchronization cannot be observed in the system of van der Pol oscillators coupled by capacitors or inductors [1]--[4]. Recently, It is found that stable four-phase synchronization occurs in three-dimensional oscillators coupled by capacitors[5], [6]. In the system, in-phase, independent anti-phase, four-phase, and in- and anti-phase syn