Synchronization of Coupled Neural Networks with Nodes of Different Dimensions
A class of coupled neural networks with nodes of different coupling time-delays and different state dimensions is investigated in this paper. Based on Lyapunov stability theory, some sufficient conditions for synchronization of coupled neural networks are
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Introduction
Over the last decades, the dynamical behaviors of neural networks (NNs) have been extensively investigated (see [1–5] and references therein). Particularly, the combination of a set of neural networks could achieve higher level of information processing, and the dynamical behaviors of coupled networks are more complex than those of any subsystem. The stabilization and synchronization in an array of coupled neural networks have attracted increasing attention of many researchers in the past few years [6–15]. Most of studies (for instance, [9–11]) on synchronization of coupled NNs are based on Lyapunov functional approach integrated with LMI technique. In [9], neural networks with hybrid coupling and interval time-varying delay are analyzed, by constructing the augmented functional with multiple Kronecker product operations. Instead of using Kroneker product method, which often treats the coupled complex networks as a whole dynamical system, in [10], the authors deal with the isolated neural networks directly. In [11], the author transforms both of the problems of global synchronization of dynamics and convergence of dynamics into solving a corresponding homogeneous system of linear algebraic equations. In [12], the authors analyze the finite-time synchronization in an array of coupled neural networks with discontinuous activation functions. In [13], the synchronization of linearly coupled dynamical systems with discontinuous identical nodes and time delay is studied. In [14], the authors investigate the stability and synchronization of memristor-based coupling neural networks with time-varying delays via intermittent control. c Springer International Publishing Switzerland 2016 L. Cheng et al. (Eds.): ISNN 2016, LNCS 9719, pp. 135–142, 2016. DOI: 10.1007/978-3-319-40663-3 16
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M. Tan and D. Xu
Most of the aforementioned works have assumed that the state vectors have the same dimensions [16–18]. Recently, there are more and more studies on complex networks with non-identical nodes, because most dynamical networks in engineering have different nodes [19–24]. If a network is constructed by nodes with different state dimension, the network will exhibit different dynamical behaviors and the previous methods of stabilization will be invalid [25–29]. In reality, complex dynamical networks are more likely to have different time-delay coupling for different nodes. It is necessary to investigate the dynamics of neural networks with different coupling time-delays. Motivated by the above discussion, in this paper, we consider the synchronization of coupled neural networks with non-identical nodes, which may possess different coupling time-delays and different state dimensions. Some sufficient conditions guaranteeing the synchronization of NNs are developed. Numerical simulations will be given to illustrate the validity of the theoretical results.
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Model Description and Preliminaries
In this section, the function projective synchronization is considered between two coupled neural networks. The drive neural netw
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