Sarkar Mrinal, Gupte Neelima
Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India.
Phys Rev E. 2021 Mar;103(3-1):032204. doi: 10.1103/PhysRevE.103.032204.
We study a system of Kuramoto oscillators arranged on a two-dimensional periodic lattice where the oscillators interact with their nearest neighbors, and all oscillators have the same natural frequency. The initial phases of the oscillators are chosen to be distributed uniformly between (-π,π]. During the relaxation process to the final stationary phase, we observe different features in the phase field of the oscillators: initially, the state is randomly oriented, then clusters form. As time evolves, the size of the clusters increases and vortices that constitute topological defects in the phase field form in the system. These defects, being topological, annihilate in pairs; i.e., a given defect annihilates if it encounters another defect with opposite polarity. Finally, the system ends up either in a completely phase synchronized state in case of complete annihilation or a metastable phase locked state characterized by presence of vortices and antivortices. The basin volumes of the two scenarios are estimated. Finally, we carry out a duality transformation similar to that carried out for the XY model of planar spins on the Hamiltonian version of the Kuramoto model to expose the underlying vortex structure.
我们研究了一个排列在二维周期晶格上的Kuramoto振子系统，其中振子与最近邻相互作用，且所有振子具有相同的固有频率。振子的初始相位被选择为在((-\pi,\pi])之间均匀分布。在向最终稳定相位的弛豫过程中，我们在振子的相位场中观察到不同特征：最初，状态是随机取向的，然后形成簇。随着时间演化，簇的大小增加，并且在系统中形成构成相位场拓扑缺陷的涡旋。这些缺陷由于是拓扑性的，会成对湮灭；即，给定的缺陷如果遇到具有相反极性的另一个缺陷就会湮灭。最后，如果完全湮灭，系统最终处于完全相位同步状态，或者处于以存在涡旋和反涡旋为特征的亚稳相位锁定状态。估计了两种情况的相体积。最后，我们在Kuramoto模型的哈密顿量版本上进行了类似于对平面自旋的XY模型所进行的对偶变换，以揭示潜在的涡旋结构。
