Vortices and the entrainment transition in the two-dimensional Kuramoto model

Abstract

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.