We analyze two classes of Kuramoto models on spheres that have been introduced in previous studies. Our analysis is restricted to ensembles of identical oscillators with the global coupling. In such a setup, with an additional assumption that the initial distribution of oscillators is uniform on the sphere, one can derive equations for order parameters in closed form. The rate of synchronization in a real Kuramoto model depends on the dimension of the sphere. Specifically, synchronization is faster on higher-dimensional spheres. On the other side, real order parameter in complex Kuramoto models always satisfies the same ODE, regardless of the dimension. The derivation of equations for real order parameters in Kuramoto models on spheres is based on recently unveiled connections of these models with geometries of unit balls. Simulations of the system with several hundreds of oscillators yield perfect fits with the theoretical predictions, that are obtained by solving equations for the order parameter.

我们分析了之前研究中引入的两类 Kuramoto 球体模型。我们的分析仅限于具有全局耦合的相同振荡器的集合。在这样的设置中，额外假设振荡器的初始分布在球体上是均匀的，可以推导出封闭形式的阶参数方程。真实 Kuramoto 模型中的同步率取决于球体的尺寸。具体来说，在高维球体上同步速度更快。另一方面，复杂 Kuramoto 模型中的实阶参数始终满足相同的 ODE，无论维度如何。Kuramoto 球体模型中实阶参数方程的推导基于最近公布的这些模型与单位球几何的联系。