Quasi-Synchronization of Heterogeneous Fractional-Order Dynamical Networks With Time-Varying Couplings

This paper addresses the problem of quasi-synchronization for a class of heterogeneous fractional-order dynamical networks with time-varying couplings. Our proposed approach, called delay-dependent hybrid impulsive control, considers the network’s topology and utilizes Lyapunov functions for synchronization analysis. We employ a graph-theoretic method to construct these Lyapunov functions, including a novel time-varying graph-theoretic Lyapunov function (TGLF) that changes with the network’s structure and state variables. However, applying fractional derivatives to the TGLF poses challenges, as the general Leibniz formula and chain rule do not apply. To overcome this obstacle, we establish a new fractional derivative rule for time-varying-coefficient convex functions. We also introduce a new lemma for a fractional-order delayed impulsive differential inequality to analyze the quasi-stability and instability of impulse-free systems. By combining the TGLF and the fractional-order delayed impulsive differential inequality, we derive criteria for quasi-synchronization and estimate the allowable error bounds. Finally, we provide numerical examples to demonstrate the effectiveness of our proposed approach.
Note to Practitioners
—Most quasi-synchronization results of complex dynamical networks are mainly supported by the traditional quadratic Lyapunov function or the graph-theoretic Lyapunov function. However, these functions are only effective for networks with fixed couplings, which limits the application scope of the Lyapunov method. Aiming at the case of time-varying couplings, this paper proposes a novel time-varying graph-theoretic Lyapunov function that changes with the network’s structure and state variables. By applying the new fractional derivative rule of this function, we shall obtain quasi-synchronization criteria of heterogeneous fractional-order complex dynamical networks with time-varying couplings by using delay-dependent hybrid impulsive control. To verify the application of theoretical results, a generalization of the fractional-order power system is provided, and we realize its quasi-synchronization theoretically and numerically.