Martens E A, Barreto E, Strogatz S H, Ott E, So P, Antonsen T M
Department of Theoretical & Applied Mechanics, Cornell University, Ithaca, New York 14853, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Feb;79(2 Pt 2):026204. doi: 10.1103/PhysRevE.79.026204. Epub 2009 Feb 6.
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.
我们分析了一个全局耦合相位振子的大型系统，其固有频率呈双峰分布。该系统的动力学一直是长期关注的主题。1984年，Kuramoto对其行为提出了几个猜想；十年后，Crawford通过局部中心流形计算获得了首个解析结果。然而，许多问题仍然悬而未决，尤其是关于全局分岔的可能性。在此，我们针对双峰分布由两个等权重洛伦兹分布组成的特殊情况，推导了该系统的稳定性图。利用Ott和Antonsen最近发现的一个假设，我们表明在这种情况下，无穷维问题精确地简化为四维流。根据参数和初始条件，长期动力学演化至三种状态之一：非相干态，即所有振子都不同步；部分同步态，即宏观的锁相振子群与大量不同步振子共存；以及驻波态，即出现两组反向旋转的锁相振子。给出了这些状态之间分岔边界的解析结果。对于双峰分布由两个高斯分布之和给出的情况，也获得了类似结果。
