Exact results for the Kuramoto model with a bimodal frequency distribution
Exact results for the Kuramoto model with a bimodal frequency distribution
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DOI：
10.1103/physreve.79.026204
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发表时间：
2009-02-01
影响因子：
2.4
通讯作者：
Antonsen, T. M.
中科院分区：
文献类型：
--
作者：
Martens, E. A.;Barreto, E.;Antonsen, T. M.
我们分析了一个大型的全球耦合相振荡器，该系统的固有频率是双峰分布的。该系统的动态一直是长期关注的主题。 1984年，库拉莫托提出了几种关于其行为的猜想。十年后，克劳福德（Crawford）通过当地中心歧管计算获得了第一个分析结果。然而，许多问题仍然开放，尤其是关于全球分叉的可能性。在这里，我们在双峰分布由两个同等加权的洛伦兹人组成的特殊情况下得出了系统的稳定图。使用Ott和Antonsen最近发现的ANSATZ，我们表明，在这种情况下，无限维度问题完全减少到四个维度的流动。根据参数和初始条件，长期动力学会演变为三个状态之一：不一致，在所有振荡器都均无异步；部分同步，其中一组宏观的相锁振荡器与一系列干燥的振荡器共存。还有一个常规状态，其中两个反向旋转的振荡器的反向旋转组出现。对这些状态之间的分叉边界提出了分析结果。对于通过两个高斯人的总和给出了双峰分布的情况，也获得了类似的结果。
We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.