Uniform-in-time stability and continuous transition of the time-discrete infinite Kuramoto model

We study a continuous transition from the discrete infinite Kuramoto model to the continuous counterpart in a whole time interval. The discrete infinite Kuramoto model corresponds to the discretization of the infinite Kuramoto model [18] via the first-order Euler discretization algorithm. For the proposed discrete infinite Kuramoto model, we study the emergent dynamics and uniform (-in-time) stability with respect to initial data under a suitable framework which is formulated in terms of system parameters and initial data. For a homogeneous ensemble with the same natural frequencies, we identify sufficient conditions for the existence of “quasi-stationary state” and complete synchronization. In contrast, for a heterogeneous ensemble, we also provide a weak emergent dynamics, namely “practical synchronization”. For the continuous transition in a zero time-step limit, we provide an improved truncation error estimate compared to the error estimate which can be obtained from the general theory for first-order discretized model using the uniform stability and emergent dynamics.