Vortices and the entrainment transition in the 2D Kuramoto model

TL;DR

This paper investigates the transition to synchronization in a 2D lattice of oscillators, revealing a vortex-driven order-chaos transition at a critical coupling that scales logarithmically with system size.

Contribution

It introduces a vortex-based framework to explain the entrainment transition and derives scaling laws linking the critical coupling to system size and phase-locking thresholds.

Findings

Vortices move in fixed paths in the synchronized phase.

Disappearance of the macroscopic cluster is due to deviant vortices.

The transition is characterized as an order-chaos transition.

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 $K_E\sim K_L$ for small lattices, where $K_L$ is the threshold for phase-locking. By also deriving the scaling $K_L\sim\log N$, we thus show that $K_E\sim\log N$ for small $N$, in agreement with numerics. In addition, we show how to use the linearized model to predict where vortices are generated.