# 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.

## Key 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.