Winding-number excitation in one-dimensional oscillators with variable interaction range

At how long of an interaction range does an oscillator system behave as a fully-connected one? To answer this question, we consider a system of nonlocally-coupled phase oscillators in one dimension, and explore the effects of a variable interaction range L on collective dynamics. In particular, we investigate the winding-number distribution, paying particular attention to the existence of a twisted wave in the system, and observe that the twisted state vanishes when the interaction range exceeds a critical value. Finite-size scaling of the width of the winding-number distribution reveals that the transition occurs at 2L/N ≈ 0.6, regardless of the system size N. We also show that at the same transition point for the topological twisted state, the phase synchrony in the system becomes partial.