Twisted states in low-dimensional hypercubic lattices

TL;DR

This paper extends the concept of twisted states from rings to finite-sized hypercubic lattices, revealing new phase-synchronized states with non-zero winding numbers in higher dimensions, and analyzes their emergence with heterogeneous oscillators.

Contribution

It introduces and characterizes twisted states in 2D and 3D hypercubic lattices, generalizing known ring states and exploring their behavior with nonidentical oscillators.

Findings

New phase-synchronized states with non-zero winding numbers in 2D and 3D lattices.

States reduce to ring twisted states when considering subsets of oscillators.

States appear in random configurations above critical coupling strengths.

Abstract

Twisted states with non-zero winding numbers composed of sinusoidally coupled identical oscillators have been observed in a ring. The phase of each oscillator in these states constantly shifts, following its preceding neighbor in a clockwise direction, and the summation of such phase shifts around the ring over $2\pi$ characterizes the winding number of each state. In this work, we consider finite-sized $d$-dimensional hypercubic lattices, namely square ($d=2$) and cubic ($d=3$) lattices with periodic boundary conditions. For identical oscillators, we observe new states in which the oscillators belonging to each line (plane) for $d=2$ ($d=3$) are phase synchronized with non-zero winding numbers along the perpendicular direction. These states can be reduced into twisted states in a ring with the same winding number if we regard each subset of phase-synchronized oscillators as one single oscillator. For nonidentical oscillators with heterogeneous natural frequencies, we observe similar patterns with slightly heterogeneous phases in each line $(d=2)$ and plane $(d=3)$. We show that these states generally appear for random configurations when the global coupling strength is larger than the critical values for the states.