我们研究了仓本振子的同步与所有的所有耦合在存在缓慢的，嘈杂的频率适应。在本文中，我们开发了一个新的模型，适应他们的相位和频率的振荡器。结果发现，该模型自然地再现了一些观察到的现象，而不是定性产生的标准仓本模型，如长时间的等待时间之前，鼓掌观众的同步。通过自洽定态解的假设，我们发现了耦合常数k的三个稳定区域，它们由临界点k1和k2分开：（i）当k<k1时，只存在稳定的非相干态;（ii）当k>k2时，非相干态变得不稳定，只存在同步态;（iii）当k1<k<k2时，非相干态和同步态都是稳定的。在非相干态和同步态之间的自发跃迁观察到有限的合奏。这些跃迁被描述为一个关于序参量r的随机过程，由于系统的有限尺寸，这些随机过程经历了涨落，从而得到以下结论：（a）在非相干区，非相干到相干跃迁的平均等待时间可以用Kramer的逃逸时间公式来预测，并且随着振子的数目呈指数增长;（b）当非相干态不稳定时（k> k ~ 2），平均等待时间随振子个数的增加而呈指数增长。
英文摘要：
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.
