From the flashing of fireflies to autonomous robot swarms, synchronization phenomena are ubiquitous in nature and technology. They are commonly described by the Kuramoto model that, in this paper, we generalise to networks over n-dimensional spheres. We show that, for almost all initial conditions, the sphere model converges to a set with small diameter if the model parameters satisfy a given bound. Moreover, for even n, a special case of the generalized model can achieve phase synchronization with nonidentical frequency parameters. These results contrast with the standard n = 1 Kuramoto model, which is multistable (i.e., has multiple equilibria), and converges to phase synchronization only if the frequency parameters are identical. Hence, this paper shows that the generalized network Kuramoto models for n ≥ 2 displays more coherent and predictable behavior than the standard n = 1 model, a desirable property both in flocks of animals and for robot control.

从萤火虫的闪烁到自主机器人群，同步现象在自然界和技术中无处不在。它们通常由 Kuramoto 模型描述，在本文​​中，我们将其推广到n维球体上的网络。我们表明，对于几乎所有的初始条件，如果模型参数满足给定的界限，球体模型就会收敛到一个小直径的集合。此外，对于偶数n，广义模型的一个特殊情况可以实现具有不同频率参数的相位同步。这些结果与标准n = 1 Kuramoto 模型，它是多稳态的（即具有多个平衡点），并且仅当频率参数相同时才收敛到相位同步。因此，本文表明，n  ≥ 2的广义网络 Kuramoto 模型显示出比标准n  = 1 模型更连贯和可预测的行为，这是动物群和机器人控制的理想特性。