Long-time persistence of the modulation in the order parameter’s temporal decay

Ascertain whether the weak, slowly varying modulation observed in the time-dependence of the Kuramoto order parameter in the two-dimensional lattice of identical oscillators with frustrated nearest-neighbor couplings persists as steady oscillations or decays away in the long-time limit.

Background

Beyond the leading reciprocal-logarithmic relaxation of the order parameter, the authors observe a slow modulation superimposed on the decay. This modulation appears in both time dynamics and, less visibly, in finite-size behavior.

Determining the asymptotic fate of this modulation is necessary to fully characterize the late-time dynamical behavior and whether additional oscillatory components survive in the thermodynamic and long-time limits.