The Kuramoto model has many higher-dimensional generalizations with similar synchronization behaviour, such as phase oscillators on the $n$-sphere which under suitable conditions can synchronize in any dimension $n$, with applications to swarming and flocking phenomena as well as to consensus and opinion formation. We consider further generalizations in which trajectories lie on the double sphere ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^{m-1}$ for any dimension $m,n$ with synchronization properties that depend on the underlying parameters, and reduce to previously studied models for specific $m$ or $n$. In such systems the synchronization properties on one sphere can be controlled from the other sphere by parameters such as the frequencies of oscillation and the connectivity coefficients. We show in particular that for these models the conformal group $\mathrm{SO}(n,m)$ acts as a time-evolution matrix, for example in the special case of the Kuramoto model there is an $\mathrm{SO}(2,1)$ matrix at each node which maps the initial value to the final value at any time $t$. For uniform coupling and identical frequencies, the time evolution of the Kuramoto model is controlled over all nodes by a single $\mathrm{SO}(2,1)$ matrix, a property which follows from the Watanabe–Strogatz reduction. This generalizes to the double sphere model with uniform coupling and identical frequency matrices, which we show is partially integrable for any $m,n$, and the system again evolves by means of a single time-evolution matrix in $\mathrm{SO}(n,m)$ which acts over all nodes. We explicitly perform the partial integration by means of a unit map which generalizes linear fractional transformations, equivalently Möbius maps ${\mathbb{S}}^{n-1}\to {\mathbb{S}}^{n-1}$ for the case $m=1$, which have been previously used to reduce the $n$-sphere equations. As with the Kuramoto model, there exist conserved cross ratios which restrict all solutions to lie in a low-dimensional manifold. We parametrize $\mathrm{SO}(n,m)$ and so derive reduced equations, independent of $N$ in number, which exactly solve the double sphere equations for any $N$. These models, and their further extensions to unit sphere systems with fifth-order nonlinearities, furnish a wide range of exactly reducible synchronization models which can be used to investigate systems for very large $N$.

仓本模型具有许多具有类似同步行为的高维概括，例如 $ñ$球形，在合适的条件下可以在任何维度上同步 $ñ$，适用于蜂拥而至的群体现象以及共识和意见的形成。我们考虑轨迹在双球面上的进一步概括${\mathbb{小号}}^{ñ-\mathrm{1个}}\times {\mathbb{小号}}^{米-\mathrm{1个}}$ 对于任何尺寸 $米，ñ$ 具有取决于基础参数的同步属性，并简化为先前研究的特定模型 $米$ 要么 $ñ$。在这样的系统中，可以通过诸如振荡频率和连通性系数之类的参数来控制一个球面上的同步特性。我们特别表明，对于这些模型，保形群$\mathrm{所以}（ñ，米）$ 充当时间演化矩阵，例如，在仓本模型的特殊情况下， $\mathrm{所以}（2，\mathrm{1个}）$ 每个节点上的矩阵，可随时将初始值映射到最终值 $Ť$。为了获得均匀的耦合和相同的频率，仓本模型的时间演化由一个节点控制所有节点$\mathrm{所以}（2，\mathrm{1个}）$矩阵，其属性来自于Watanabe–Strogatz减少。这可以推广到具有均匀耦合和相同频率矩阵的双球模型，我们证明该模型对任何$米，ñ$，系统再次借助单个时间演化矩阵进行演化 $\mathrm{所以}（ñ，米）$它作用于所有节点。我们通过单元图明确执行部分积分，该单元图推广了线性分数变换，等效为莫比乌斯图${\mathbb{小号}}^{ñ-\mathrm{1个}}\to {\mathbb{小号}}^{ñ-\mathrm{1个}}$ 对于这种情况 $米=\mathrm{1个}$，以前已用于减少 $ñ$球方程。与仓本模型一样，存在守恒的交叉比率，该比率限制所有解决方案都位于低维流形中。我们参数化$\mathrm{所以}（ñ，米）$ 从而得出简化的方程式，与 $ñ$ 在数量上，它可以精确地解决任何一个问题的双球方程 $ñ$。这些模型以及它们对具有五阶非线性的单位球体系统的进一步扩展，提供了范围广泛的可精确还原的同步模型，这些模型可用于研究超大型系统$ñ$。