The Kuramoto model is a canonical model for understanding phase-locking phenomenon. It is well-understood that, in the usual mean-field scaling, full phase-locking is unlikely and that it is partially phase-locked states that are important in applications. Despite this, while there has been much attention given to existence and stability of fully phase-locked states in the finite N Kuramoto model, the partially phase-locked states have received much less attention. In this paper we present two related results. Firstly, we derive an analytical criteria that, for sufficiently strong coupling, guarantees the existence of a partially phase-locked state by proving the existence of an attracting ball around a fixed point of a subset of the oscillators. We also derive a larger invariant ball such that any point in it will asymptotically converge to the attracting ball. Secondly, we consider the large N behavior of the finite N Kuramoto system with randomly distributed frequencies. In the case where the frequencies are independent and identically distributed we use a result of De Smet and Aeyels on partial entrainment to derive a condition giving (with high probability) the existence of a partially entrained cluster. We also derive upper and lower bounds on the size of the largest entrained cluster, together with a lower bound on the order parameter. Interestingly in a series of numerical experiments we find that the observed size of the largest entrained cluster is predicted extremely well by the upper bound.

Kuramoto 模型是用于理解锁相现象的典型模型。众所周知，在通常的平均场缩放中，完全锁相是不可能的，部分锁相状态在应用中很重要。尽管如此，尽管人们对有限N中完全锁相状态的存在性和稳定性给予了很多关注Kuramoto 模型中，部分锁相状态受到的关注要少得多。在本文中，我们展示了两个相关的结果。首先，我们推导出一个分析标准，对于足够强的耦合，通过证明在振荡器子集的固定点周围存在吸引球来保证部分锁相状态的存在。我们还推导出了一个更大的不变球，使得其中的任何点都将渐近收敛到吸引球。其次，我们考虑有限N的大N行为随机分布频率的 Kuramoto 系统。在频率独立且同分布的情况下，我们使用 De Smet 和 Aeyels 在部分夹带上的结果来推导出一个条件，给出（以高概率）部分夹带集群的存在。我们还推导出最大夹带簇大小的上限和下限，以及顺序参数的下限。有趣的是，在一系列数值实验中，我们发现观察到的最大夹带簇的大小非常好地由上限预测。