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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.

We revisit the globally-coupled kinetic Ising model in the presence of an oscillating external magnetic field with an amplitude h0 and a driving angular frequency Ω. In the low-frequency region Ω → 0, the dynamic phase transition is shown to become discontinuous, and the phase boundary is found to be described by h0 ∼ (1 − T ) with the temperature T and a critical temperature of unity in the limit h0 → 0. Our result is complementary to h0 ∼ (1 − T ), previously known for the other limit (1 − T ) Ω. We extend the existing analytic calculation for the continuous transition to the second-order perturbation expansion and find that the dynamic phase transition occurs at lower temperature than the static one, which was not captured by the existing first-order calculation. The stochastic resonance behavior is also discussed in relation to the nature of the dynamic phase transition.