Synchronization of fractional-order fuzzy complex networks with time-varying couplings and proportional delay

The topic of synchronization of fractional-order fuzzy complex networks with time-varying couplings and proportional delay is addressed in this paper. By combining graph theory, the Lyapunov method, and the Razumikhin method, the synchronization criteria are obtained by designing a new time-varying graph-theoretical Lyapunov function. Different from the typical quadratic Lyapunov function, the time-varying graph-theoretical Lyapunov function contains the product of three-term functions and poses the difficulty of fractional derivative due to the unavailability of the general Leibniz formula and the chain rule. A new inequality about the fractional derivative rule of the time-varying graph-theoretical Lyapunov function is established based on the convex function creatively. In addition, compared with relevant literatures, the synchronization criteria are related to the proportionality coefficient and the order of fractional derivative. Then, the synchronization of fractional-order power system established on fuzzy complex networks with time-varying couplings and proportional delay is studied for the first time by applying the feedback control. Lastly, some simulations are exhibited to demonstrate the effectiveness of the theoretical results.