Bifurcation Study of the Kuramoto Transition in Random Oscillator Networks

Abstract

We study the frequency synchronization of phase oscillators in random networks. We investigate the kinetic equations of the oscillator distribution function using the continuum approximation. From a linear analysis of this model, we find that the unstable eigenfunction is proportional to k. We also derive an amplitude equation of the unstable modes employing center manifold reduction. We find that the coefficients of the Taylor expansion of the amplitude equation always diverge for random scale-free networks. The results of numerical studies are consistent with these analyses.