Twisted states in low-dimensional hypercubic lattices

Twisted states with nonzero 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 2π 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 nonzero 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.