Dynamical behaviors of Cohen–Grossberg neural networks with delays and reaction–diffusion terms

doi:10.1016/j.neucom.2005.11.009

Neurocomputing

Volume 70, Issues 1–3, December 2006, Pages 536-543

https://doi.org/10.1016/j.neucom.2005.11.009 Get rights and content

Abstract

In this paper, we study Cohen–Grossberg neural networks with delays and reaction–diffusion terms. By employing homotopic mapping theory and constructing suitable Lyapunov functional method we present some sufficient conditions ensuring the existence, uniqueness and globally exponential stability (GES) of the equilibrium point. These conditions obtained have important leading significance in the designs and applications of GES for reaction–diffusion neural circuit systems with delays. Finally, we show a numerical example to verify the theoretical analysis.

Introduction

Cohen–Grossberg neural networks (CGNNs) were first introduced by Cohen–Grossberg [4]. CGNNs include Hopfield neural networks as special cases [1], [15], [17], [18], [3], [20], [21]. The class of networks has good application in parallel computation, associative memory and optimization problems, which has been one of the most active areas of research and has received much attention. One can refer to the articles [19], [2], [16], [14], [9] for detailed discussion on these aspects. Thus, from the viewpoint of application, the dynamical study for CGNNs is quite important and significant, and cannot be replaced with the dynamical study for the traditional neural networks such as Hopfield neural networks and cellular neural networks. Furthermore, in the analysis of dynamical CGNNs for parallel computation and optimization, to increase the rate of convergence to the equilibrium point of the networks, it is necessary to ensure a desired exponential convergence rate of the networks trajectories, starting from arbitrary initial states to the equilibrium point which corresponds to the optimal solution. From the viewpoint of mathematics and engineering, it is required that the equilibrium point of CGNNs is globally exponential stability (GES). As a consequence, many authors have paid much effort to the research on GES of equilibrium point for CGNNs in Refs. [19], [2], [16], [14], [9]. However, strictly speaking, diffusion effects cannot be avoided in neural network models when electrons are moving in asymmetric electromagnetic field, thus we must consider the space is varying with the time. Refs. [11], [12], [13], [7] have considered the stability of neural networks with diffusion terms, which are expressed by partial differential equations. It is also common to consider the diffusion in biological systems (such as immigration) [8], [6], [10]. To the best our knowledge, few authors study GES of Cohen–Grossberg neural networks with delays and reaction–diffusion terms.

Motivated by the above discussions, in this paper we analyze further problem of GES of Cohen–Grossberg neural networks with delays and reaction–diffusion terms, and give a set of sufficient conditions ensuring the existence, uniqueness and GES of the equilibrium point by using homotopic mapping theory and employing Lyapunov functional method.

Section snippets

Preliminary

Let $C ≜ C ([- τ, 0] \times R^{m}, R^{n})$ be the Banach space of continuous functions which map $[- τ, 0] \times R^{m}$ into $R^{n}$ with the topology of uniform convergence. $Ω$ be an open bounded domain in $R^{m}$ with smooth boundary $\partial Ω$ , and $mes Ω > 0$ denotes the measure of $Ω$ . $L^{2} (Ω)$ is the space of real functions on $Ω$ which are $L^{2}$ for the Lebesgue measure. It is a Banach space for the norm $∥ u (t) ∥ = \sum_{i = 1}^{n} ∥ u_{i} (t) ∥_{2},$ where $u (t) = (u_{1} (t), \dots, u_{n} (t))^{T}, ∥ u_{i} (t) ∥_{2} = {(\int_{Ω} | u_{i} (t, x) |^{2} d x)}^{1 / 2} .$

For any $ϕ (t, x) \in C [[- τ, 0] \times Ω, R^{n}]$ , we define $∥ ϕ ∥ = \sum_{i = 1}^{n} ∥ ϕ_{i} ∥_{2},$ where $ϕ (t, x) = (ϕ_{1} (t, x),$

Existence and GES of the equilibrium point

In the paper, we always assume that

(H1)
There exist $m_{i}$ and $M_{i}$ , such that $0 < m_{i} ⩽ a_{i} (u_{i}) ⩽ M_{i}, i = 1, \dots, n .$
(H2)
$b_{i} (\cdot)$ is differentiable, $α_{i} ≜ \inf_{u_{i} \in R} {\dot{b_{i}} (u_{i})} > 0$ , where $\dot{b_{i}} (\cdot)$ is the derivative of $b_{i} (\cdot)$ , $b_{i} (0) = 0, i = 1, \dots, n$ .
(H3)
There exist constants $β_{j} (j = 1, \dots, n)$ , such that $| s_{j} (x) - s_{j} (y) | ⩽ β_{j} | x - y |, \forall x, y \in R .$
(H4)
There exist constants $p_{ij}$ , $q_{ij}$ , $p_{ij}^{*}$ , $q_{ij}^{*}$ , such that $- 2 m_{i} α_{i} + \sum_{j = 1}^{n} M_{i} | a_{ij} |^{2 p_{ij}} β_{j}^{2 q_{ij}} + \sum_{j = 1}^{n} M_{j} | a_{ji} |^{2 - 2 p_{ji}} β_{i}^{2 - 2 q_{ji}} + \sum_{k = 0}^{K} \sum_{j = 1}^{n} M_{i} | t_{ij}^{(k)} |^{2 p_{ij}^{*}} β_{j}^{2 q_{ij}^{*}} + \sum_{k = 0}^{K} \sum_{j = 1}^{n} M_{j} | t_{ji}^{(k)} |^{2 - 2 p_{ji}^{*}} β_{i}^{2 - 2 q_{ji}^{*}} < 0 .$

In order to study the existence and uniqueness

Example

Example

Consider Cohen–Grossberg neural networks with delays and reaction–diffusion terms $\{\begin{matrix} \frac{\partial u_{1} (t, x)}{\partial t} = \frac{\partial}{\partial x} (D_{11} \frac{\partial u_{1} (t, x)}{\partial x}) - a_{1} (u_{1} (t, x)) \\ [b_{1} (u_{1} (t, x)) - \sum_{j = 1}^{2} a_{1 j} s_{j} (u_{j}) \\ - \sum_{j = 1}^{2} \sum_{k = 0}^{1} t_{1 j}^{(k)} s_{j} (u_{j} (t - τ_{k}, x))] \\ \frac{\partial u_{2} (t, x)}{\partial t} = \frac{\partial}{\partial x} (D_{21} \frac{\partial u_{2} (t, x)}{\partial x}) - a_{2} (u_{2} (t, x)) \\ [b_{2} (u_{2} (t, x)) - \sum_{j = 1}^{2} a_{2 j} s_{j} (u_{j}) \\ - \sum_{j = 1}^{2} \sum_{k = 0}^{1} t_{2 j}^{(k)} s_{j} (u_{j} (t - τ_{k}, x))] . \end{matrix}$

Let $D_{11} > 0, D_{21} > 0$ , $a_{i} = 4 + \sin u_{i}$ , $b_{i} (u_{i}) = 4 u_{i}$ , $s_{1} (u_{1}) = \arctan u_{1}$ , $s_{2} = u_{2}$ . Clearly, $a_{i}$ satisfies (H1) with $m_{i} = 3$ , $M_{i} = 5$ , $s_{j}$ satisfies (H2) with $β_{j} = 1$ , $b_{i}$ satisfies (H3) with $α_{i} = 4$ , $i, j = 1, 2$ . Moreover, we choose $a_{11} = \frac{1}{12}$ , $a_{12} = \frac{1}{12}$ , $a_{21} = \frac{1}{6}$ , $a_{22} =$

Conclusions

In this paper, the dynamics of Cohen–Grossberg neural networks model with delays and reaction–diffusion is studied. By employing homotopic mapping theory and constructing Lyapunov functional method, some sufficient conditions have been obtained which guarantee the model to be GES. The given algebra conditions are useful in design and applications of reaction–diffusion Cohen–Grossberg neural networks. Moreover, our methods in the paper may be extended for more complex networks.

Acknowledgment

The authors would like to thank the Editor and the reviewers for their helpful comments and constructive suggestions, which have been very useful for improving the presentation of this paper.

Hongyong Zhao received the Ph.D. degree from Sichuan University, Chengdu, China, and the Post-Doctoral Fellow in the Department of Mathematics at Nanjing University, Nanjing, China.

He was with Department of Mathematics at Nanjing University of Aeronautics and Astronautics, Nanjing, China. He is currently a Professor of Nanjing University of Aeronautics and Astronautics, Nanjing, China. He also is the author or coauthor of more than 40 journal papers. His research interests include nonlinear

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Hongyong Zhao received the Ph.D. degree from Sichuan University, Chengdu, China, and the Post-Doctoral Fellow in the Department of Mathematics at Nanjing University, Nanjing, China.

Kunlun Wang is with Department of Computer Science and Technology in Hefei University, Hefei, China. He is currently a professor of Department of Computer Science and Technology in Hefei University, Hefei, China. He also is the author or coauthor of more than 30 journal papers. His research interests include Artificial neural networks, Intelligent control and Speech Recognition.

^☆: This research was supported by the National Natural Science Foundation of Nanjing University of Aeronautics and Astronautics, and also supported by the National Natural Science Foundation of the Education Department of Anhui, China.

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Dynamical behaviors of Cohen–Grossberg neural networks with delays and reaction–diffusion terms☆

Abstract

Introduction

Section snippets

Preliminary

Existence and GES of the equilibrium point

Example

Conclusions

Acknowledgment

Phys. Lett. A

J. Math. Anal. Appl.

IEEE Trans. Syst. Man Cybernet.

Phys. Lett. A

Nonlinear Anal. TMA

Neural Networks

Chaos Solitons Fractals

Phys. Lett. A

Neurocomputing

Comput. Math. Appl.