Lee Tony E, Tam Heywood, Refael G, Rogers Jeffrey L, Cross M C
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Sep;82(3 Pt 2):036202. doi: 10.1103/PhysRevE.82.036202. Epub 2010 Sep 2.
We study synchronization in the two-dimensional lattice of coupled phase oscillators with random intrinsic frequencies. When the coupling K is larger than a threshold K{E} , there is a macroscopic cluster of frequency-synchronized oscillators. We explain why the macroscopic cluster disappears at K{E} . We view the system in terms of vortices, since cluster boundaries are delineated by the motion of these topological defects. In the entrained phase (K>K{E}) , vortices move in fixed paths around clusters, while in the unentrained phase (K<K{E}) , vortices sometimes wander off. These deviant vortices are responsible for the disappearance of the macroscopic cluster. The regularity of vortex motion is determined by whether clusters behave as single effective oscillators. The unentrained phase is also characterized by time-dependent cluster structure and the presence of chaos. Thus, the entrainment transition is actually an order-chaos transition. We present an analytical argument for the scaling K{E}∼K{L} for small lattices, where K{L} is the threshold for phase locking. By also deriving the scaling K{L}∼log N , we thus show that K{E}∼log N for small N , in agreement with numerics. In addition, we show how to use the linearized model to predict where vortices are generated.
我们研究了具有随机固有频率的耦合相位振子二维晶格中的同步现象。当耦合强度(K)大于阈值(K_{E})时，会出现频率同步振子的宏观簇。我们解释了为什么宏观簇在(K_{E})处消失。由于簇边界由这些拓扑缺陷的运动所界定，我们从涡旋的角度来观察该系统。在同步阶段（(K>K_{E})），涡旋围绕簇沿着固定路径移动，而在非同步阶段（(K<K_{E})），涡旋有时会偏离。这些异常涡旋导致了宏观簇的消失。涡旋运动的规律性由簇是否表现为单个有效振子决定。非同步阶段还具有随时间变化的簇结构和混沌现象。因此，同步转变实际上是一个从有序到混沌的转变。对于小晶格，我们给出了(K_{E}\sim K_{L})的标度关系的解析论证，其中(K_{L})是锁相阈值。通过推导(K_{L}\sim\log N)也成立，我们证明了对于小的(N)，(K_{E}\sim\log N)，这与数值结果一致。此外，我们展示了如何使用线性化模型来预测涡旋产生的位置。
