Engelbrecht Jan R, Mirollo Renato
Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA.
Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467, USA.
Chaos. 2014 Mar;24(1):013114. doi: 10.1063/1.4858458.
We present a complete classification of attractors for networks of coupled identical Kuramoto oscillators. In such networks, each oscillator is driven by the same first-order trigonometric function, with coefficients given by symmetric functions of the entire oscillator ensemble. For [Formula: see text] oscillators, there are four possible types of attractors: completely synchronized fixed points or limit cycles, and fixed points or limit cycles where all but one of the oscillators are synchronized. The case N = 3 is exceptional; systems of three identical Kuramoto oscillators can also posses attracting fixed points or limit cycles with all three oscillators out of sync, as well as chaotic attractors. Our results rely heavily on the invariance of the flow for such systems under the action of the three-dimensional group of Möbius transformations, which preserve the unit disc, and the analysis of the possible limiting configurations for this group action.
我们给出了耦合相同Kuramoto振子网络吸引子的完整分类。在这样的网络中，每个振子由相同的一阶三角函数驱动，其系数由整个振子集合的对称函数给出。对于(N)个振子，存在四种可能类型的吸引子：完全同步的不动点或极限环，以及除一个振子外所有振子都同步的不动点或极限环。(N = 3)的情况是特殊的；三个相同Kuramoto振子的系统还可以具有所有三个振子都不同步的吸引不动点或极限环，以及混沌吸引子。我们的结果在很大程度上依赖于此类系统的流在保持单位圆盘的三维莫比乌斯变换群作用下的不变性，以及对该群作用可能的极限构型的分析。
