Uniform-in-time transition from discrete to continuous dynamics in the Kuramoto synchronization

Numerical simulations for continuous dynamical systems are based on their discretized models in time and space, and we take it for granted that the simulation results for the discrete dynamical systems as what is happening in the corresponding continuous dynamical systems. In general, this is valid in any finite time window under suitable conditions on system parameters. In this paper, we present a smooth transition from discrete dynamics to continuous dynamics for the ensemble of Kuramoto oscillators which is valid in the whole time interval by utilizing the exponential synchronization estimates and confining the initial phase in a compact set of the state space. We perform this uniform-in-time transition for the discrete Kuramoto model and its augmented second-order extension.