Synchronization and clustering of phase oscillators with heterogeneous coupling

We generalize Kuramoto's theory for the synchronization transition of globally coupled phase oscillators to populations where each oscillator has a different coupling strength. We show that, beyond the transition, even those oscillators with very small couplings may participate in the synchronized ensemble, provided that their natural frequencies are close enough to the synchronization frequency. In finite systems, numerical realizations reveal that the transition is preceded by a regime of clustering where the population splits into internally synchronized groups of various sizes. The appearance of these clusters may be qualitatively understood in terms of fluctuations in the distribution of natural frequencies.