  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
new 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 k1 and k2: (i) for k<k1, only
the stable incoherent state exists; (ii) for k>k2, the incoherent state becomes
unstable and only the synchronized state exists; (iii) for k1<k<k2, both the
incoherent and synchronized states are stable. In the bistable regime
spontaneous transitions between the incoherent and synchronized states are
observed for finite ensembles. These transitions are well described as a
stochastic process on the order parameter r undergoing fluctuations due to the
system's finite size, leading to the following conclusions: (a) in the bistable
regime, the average waiting time of an incoherent-to-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>k2),
the average waiting time grows logarithmically with the number of oscillators.
