Department of Mathematics, Western University, London, Ontario N6A 3K7, Canada.
The Salk Institute for Biological Studies, La Jolla, California 92037, USA.
Chaos. 2022 Mar;32(3):031104. doi: 10.1063/5.0078791.
One of the simplest mathematical models in the study of nonlinear systems is the Kuramoto model, which describes synchronization in systems from swarms of insects to superconductors. We have recently found a connection between the original, real-valued nonlinear Kuramoto model and a corresponding complex-valued system that permits describing the system in terms of a linear operator and iterative update rule. We now use this description to investigate three major synchronization phenomena in Kuramoto networks (phase synchronization, chimera states, and traveling waves), not only in terms of steady state solutions but also in terms of transient dynamics and individual simulations. These results provide new mathematical insight into how sophisticated behaviors arise from connection patterns in nonlinear networked systems.
在非线性系统研究中，最简单的数学模型之一是 Kuramoto 模型，它描述了从昆虫群到超导体等系统中的同步。我们最近发现了原始的实值非线性 Kuramoto 模型与相应的复值系统之间的联系，该系统允许用线性算子和迭代更新规则来描述系统。现在，我们使用这种描述方法来研究 Kuramoto 网络中的三种主要同步现象（相位同步、嵌合体状态和行波），不仅涉及稳态解，还涉及瞬态动力学和个体模拟。这些结果为复杂行为如何从非线性网络系统的连接模式中产生提供了新的数学见解。
