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.

Abstract. We consider a system of N phase oscillators having randomly distributed natural frequencies and diagonalizable interactions among the oscillators. We show that in the limit of N → ∞, all solutions of such a system are incoherent with probability one for any strength of coupling, which implies that there is no sharp transition from incoherence to coherence as the coupling strength is increased, in striking contrast to Kuramoto’s (special) oscillator system. Key words. Network of phase oscillators, Kuramoto model AMS subject classifications. 34C15, 37N25, 37N20 1. Introduction. Synchronization

Abstract. We consider a system of N phase oscillators having randomly distributed natural frequencies and diagonalizable interactions among the oscillators. We show that, in the limit of N → ∞, all solutions of such a system are incoherent with probability one for any strength of coupling, which implies that there is no sharp transition from incoherence to coherence as the coupling strength is increased, in striking contrast to Kuramoto’s (special) oscillator system. Key words. Network of phase oscillators, Kuramoto model AMS subject classifications. 34C15, 37N25, 37N20 1. Introduction. Synchronization