Synchronization in a Kuramoto model with delay-dependent couplings

This paper aims to provide bifurcation analysis for a Kuramoto model with time-delay and random coupling strength. A delay differential equation governing the system is obtained on the Ott-Antonsen's manifold, and the bifurcation analysis is proceeded by using the characteristic equation and the normal form method. The general case where the coupling strength is chosen as a function of delay is investigated. Afterwards, the synchronization of the model with three different distributions of time delay including degenerate distribution, two-point distribution and Gamma distribution, is discussed respectively. Particularly, the coupled system of which the coupling strength and the delays are divided into two groups is studied in detail and the bifurcation results are obtained both theoretically and numerically.