Twisted states in low-dimensional hypercubic lattices

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.