Global Mittag-Leffler stability and synchronization of discrete-time fractional-order complex-valued neural networks with time delay

Abstract

Without decomposing complex-valued systems into real-valued systems, this paper investigates existence, uniqueness, global Mittag-Leffler stability and global Mittag-Leffler synchronization of discrete-time fractional-order complex-valued neural networks (FCVNNs) with time delay. Inspired by Lyapunov's direct method on continuous-time systems, a class of discrete-time FCVNNs is further discussed by employing the fractional-order extension of Lyapunov's direct method. Firstly, by means of contraction mapping theory and Cauchy's inequality, a sufficient condition is presented to ascertain the existence and uniqueness of the equilibrium point for discrete-time FCVNNs. Then, based on the theory of discrete fractional calculus, discrete Laplace transform, the theory of complex functions and discrete Mittag-Leffler functions, a sufficient condition is established for global Mittag-Leffler stability of the proposed networks. Additionally, by applying the Lyapunov's direct method and designing a effective control scheme, the sufficient criterion is derived to ensure the global Mittag-Leffler synchronization of discrete-time FCVNNs. Finally, two numerical examples are also presented to manifest the feasibility and validity of the obtained results.

Keywords: Discrete-time; Fractional-order complex-valued neural networks; Lyapunov’s direct method; Mittag-Leffler stability; Synchronization; Time delay.