Global stability analysis of delayed complex-valued fractional-order coupled neural networks with nodes of different dim
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ORIGINAL ARTICLE
Global stability analysis of delayed complex‑valued fractional‑order coupled neural networks with nodes of different dimensions Manchun Tan1 · Qi Pan1 Received: 12 November 2016 / Accepted: 6 December 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2017
Abstract In this paper, the delayed fractional-order complex-valued coupled neural networks (FCCNNs) with nodes of different dimensions are investigated. Firstly, stability theorems for linear fractional-order systems with multiple delays are presented. Secondly, by using the homeomorphism theory, the existence and uniqueness of the equilibrium point for delayed FCCNNs are proved. Then, the global stability criteria for delayed FCCNNs are derived by comparison theorem. Finally, numerical examples are given to illustrate the effectiveness of the presented results. Keywords Fractional-order complex-valued coupled neural networks · Nodes of different dimensions · Stability · Existence and uniqueness of equilibrium · Comparison theorem
1 Introduction During the past decades, various neural networks have attracted considerable attention due to their wide applications in science and engineering (see [1–4] and references therein). It is well known that implementations of neural networks depend heavily on the dynamical behaviors of the networks, and stability analysis is one of the hottest topics [5–10]. Linearly coupled systems are a large class of dynamical systems with continuous time and state, as well as discrete space, for describing coupled oscillators [11, 12]. Recently, there is increasing interest in coupled neural networks that consist of sub-networks of neurons. As the mutual connections among neurons, coupled neural networks can exhibit a variety of interesting behaviors that are qualitatively different from their isolated behavior [13–15]. The study of dynamics of coupled neural networks can help people understand brain science better and design neural networks for practical use. As a generalization of integer-order differentiation and integration to arbitrary non-integer order, fractional-order system has proved to be more propriate in modeling some of the real-world problems. Fractional calculus can provide * Manchun Tan [email protected] 1
Department of Mathematics, Jinan University, Guangzhou 510632, People’s Republic of China
an excellent instrument for the description of memory and hereditary properties of various materials and processes [16]. In recent years, fractional calculus has been well integrated into artificial neural networks. The dynamics of fractional-order NNs have been investigated, since the fractional-order NNs provide more accurate results than their integer-order counterparts [17–26]. Complex-valued neural networks with complex-valued state, output, connection weight, and activation function processes information in the complex plane and can solve some problems which cannot be solved with their real-valued counterparts [27–33]. In [28] several sufficient conditions were given for the asymptotic stab
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