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We study the synchronization of Kuramoto oscillators with all-to-all couplingin the presence of slow, noisy frequency adaptation. In this paper we develop anew model for oscillators which adapt both their phases and frequencies. It isfound that this model naturally reproduces some observed phenomena that are notqualitatively produced by the standard Kuramoto model, such as long waitingtimes before the synchronization of clapping audiences. By assuming aself-consistent steady state solution, we find three stability regimes for thecoupling constant k, separated by critical points k1 and k2: (i) for k<k1, onlythe stable incoherent state exists; (ii) for k>k2, the incoherent state becomesunstable and only the synchronized state exists; (iii) for k1<k<k2, both theincoherent and synchronized states are stable. In the bistable regimespontaneous transitions between the incoherent and synchronized states areobserved for finite ensembles. These transitions are well described as astochastic process on the order parameter r undergoing fluctuations due to thesystem's finite size, leading to the following conclusions: (a) in the bistableregime, the average waiting time of an incoherent-to-coherent transition can bepredicted by using Kramer's escape time formula and grows exponentially withthe number of oscillators; (b) when the incoherent state is unstable (k>k2),the average waiting time grows logarithmically with the number of oscillators.

Spontaneous synchronization of coupled oscillator systems with frequency adaptation