Phase and frequency entrainment in locally coupled phase oscillators with repulsive interactions

TL;DR

This paper investigates how locally coupled oscillators with repulsive interactions organize into spatial patterns, revealing geometry-dependent phase arrangements, domain formation, and synchronization phenomena in one- and two-dimensional lattices.

Contribution

It provides a detailed analysis of phase and frequency entrainment in repulsively coupled Kuramoto oscillators on various lattice geometries, highlighting the influence of lattice structure on pattern formation.

Findings

Stable phase patterns depend on lattice geometry.

No transition to order in the thermodynamic limit in 1D.

Domains with different helicities can form and freeze at high coupling.

Abstract

Recent experiments in one and two-dimensional microfluidic arrays of droplets containing Belousov -Zhabotinsky reactants show a rich variety of spatial patterns [J. Phys. Chem. Lett. 1, 1241-1246 (2010)]. The dominant coupling between these droplets is inhibitory. Motivated by this experimental system, we study repulsively coupled Kuramoto oscillators with nearest neighbor interactions, on a linear chain as well as a ring in one dimension, and on a triangular lattice in two dimensions. In one dimension, we show using linear stability analysis as well as numerical study, that the stable phase patterns depend on the geometry of the lattice. We show that a transition to the ordered state does not exist in the thermodynamic limit. In two dimensions, we show that the geometry of the lattice constrains the phase difference between two neighbouring oscillators to 120 degrees. We report the existence of domains with either clockwise or anti-clockwise helicity, leading to defects in the lattice. We study the time dependence of these domains and show that at large coupling strengths, the domains freeze due to frequency synchronization. Signatures of the above phenomena can be seen in the spatial correlation functions.