Emergent behaviors of the Kuramoto model with a time delay on a general digraph

Abstract

In this paper, we study the interplay of time-delayed interactions and network structure on the collective behaviors of Kuramoto oscillators. For the continuous and discrete Kuramoto models with time delay on a general digraph, we provide two frameworks leading to the complete synchronization in terms of system parameters (maximal time delay and depth of spanning tree) and initial data. In related literature on Cucker-Smale flocking, the linear structure of the velocity coupling is crucially used in the matrix theory-based approach. However, the phase coupling function in the Kuramoto model is sinusoidal so that we cannot use the matrix theory directly as it is. To circumvent this difficulty, we lift the Kuramoto model to the second-order system by introducing an auxiliary frequency variable. In this second-order formulation, the Kuramoto model enjoys several similar mathematical structures as for the Cucker-Smale flocking model. Then we use a network structure containing a spanning tree and inductive argument on the propagation of frequency information along a spanning tree in the network to derive a complete synchronization estimate. © 2020 Society for Industrial and Applied Mathematics