论文标题
关于库拉马托方程的典型和非典型解决方案
On the typical and atypical solutions to the Kuramoto equations
论文作者
论文摘要
Kuramoto模型是一个动力学系统，该系统模拟了耦合振荡器的相互作用。有很多工作可以有效地将平衡的数量绑定到给定网络的库拉莫托模型。通过将Kuramoto方程制定为代数方程系统，我们首先将Kuramoto方程的复杂根计数与基础网络的组合物相关联，通过表明复杂的根计数通常等于网络相应的邻接式邻接式多层镜头的归一化体积。然后，我们给出明确的代数条件，在该条件下，该界限是严格的，并表明在某些网络中，库拉马托方程具有无限的许多平衡。
The Kuramoto model is a dynamical system that models the interaction of coupled oscillators. There has been much work to effectively bound the number of equilibria to the Kuramoto model for a given network. By formulating the Kuramoto equations as a system of algebraic equations, we first relate the complex root count of the Kuramoto equations to the combinatorics of the underlying network by showing that the complex root count is generically equal to the normalized volume of the corresponding adjacency polytope of the network. We then give explicit algebraic conditions under which this bound is strict and show that there are networks where the Kuramoto equations have infinitely many equilibria.
