System of phase oscillators with diagonalizable interaction

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 going to infinity, 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.