Spontaneous synchronization of coupled oscillator systems with frequency adaptation.

We study the synchronization of Kuramoto oscillators with all-to-all coupling in the presence of slow, noisy frequency adaptation. In this paper, we develop a model for oscillators, which adapt both their phases and frequencies. It is found that this model naturally reproduces some observed phenomena that are not qualitatively produced by the standard Kuramoto model, such as long waiting times before the synchronization of clapping audiences. By assuming a self-consistent steady state solution, we find three stability regimes for the coupling constant k , separated by critical points k{1} and k{2}: (i) for kk{2}, the incoherent state becomes unstable and only the synchronized state exists; and (iii) for k{1}coherent transition can be predicted by using Kramer's escape time formula and grows exponentially with the number of oscillators; (b) when the incoherent state is unstable (k>k{2}), the average waiting time grows logarithmically with the number of oscillators.