Journal of Applied Mathematics

Volume 2012 (2012), Article ID 136048, 19 pages

http://dx.doi.org/10.1155/2012/136048

## Dynamics of Fuzzy BAM Neural Networks with Distributed Delays and Diffusion

^{1}Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang, Guizhou 550004, China^{2}Department of Mathematics, Hunan City University, Yiyang, Hunan 413000, China^{3}Department of Computer and Information Science, Hunan Institute of Technology, Hengyang, Hunan 421002, China

Received 23 May 2011; Accepted 17 November 2011

Academic Editor: Carlos J. S. Alves

Copyright © 2012 Qianhong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Constructing a new Lyapunov functional and employing inequality technique, the existence, uniqueness, and global exponential stability of the periodic oscillatory solution are investigated for a class of fuzzy bidirectional associative memory (BAM) neural networks with distributed delays and diffusion. We obtained some sufficient conditions ensuring the existence, uniqueness, and global exponential stability of the periodic solution. The results remove the usual assumption that the activation functions are differentiable. An example is provided to show the effectiveness of our results.

#### 1. Introduction

The bidirectional associative memory (BAM) neural network was first introduced by Kosko [1, 2]. These models generalize the single-layer autoassociative Hebbian correlator to a two layer pattern-matched heteroassociative circuits. BAM neural network is composed of neurons arranged in two-layers, the *X*-layer and the *Y*-layer. The BAM neural network has been used in many fields such as image processing, pattern recognition, and automatic control [3]. Recently, the stability and the periodic oscillatory solutions of BAM neural networks have been studied (see, e.g., [1–25]). In 2002, Cao and Wang [7] derived some sufficient conditions for the global exponential stability and existence of periodic oscillatory solution of BAM neural networks with delays. Some authors [16, 19, 25] studied the BAM neural networks with distributed delays, which are more appropriate when neural networks have a multitude of parallel pathways with a variety of axon sizes and lengths.

However, strictly speaking, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields. So we must consider that activations vary in space as well as in time. Song et al. [25] have considered the stability of BAM neural networks with diffusion effects, which are expressed by partial differential equations. In this paper, we would like to integrate fuzzy operations into BAM neural networks. Speaking of fuzzy operations, T. Yang and L. B. Yang [26] first introduced fuzzy cellular neural networks (FCNNs) combining those operations with cellular neural networks. So far, researchers have founded that FCNNs are useful in image processing, and some results have been reported on stability and periodicity of FCNNs [26–32]. However, to the best of our knowledge, few authors consider the global exponential stability and existence of periodic solutions for fuzzy BAM neural networks with distributed delays and diffusion terms. In [33], Li studied the global exponential stabilities of both the equilibrium point and the periodic solution for a class of BAM fuzzy neural networks with delays and reaction-diffusion terms. Motivated by the above discussion, in this paper, by constructing a suitable Lyapunov functional and employing inequality technique, we will derive some sufficient conditions of the global exponential stability and existence of periodic solutions for fuzzy BAM neural networks with distributed delays and diffusion terms.

#### 2. System Description and Preliminaries

In this paper, we consider the globally exponentially stable and periodic fuzzy BAM neural networks with distributed delays and diffusion terms described by partial differential equations with delays: where and correspond to the number of neurons in -layer and -layer, respectively. For , is a bounded compact set with smooth boundary and in space and are the state of the th neuron and the th neurons at time and in space , respectively. , and they denote the rate with which the th neuron and th neuron will reset its potential to the resting state in isolation when disconnected from the network and external inputs; and are constants, denoting the connection weights. , and are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template in -layer, respectively; , and are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template in -layer, respectively; and denote the fuzzy AND and fuzzy OR operations, respectively; denote external input of the th neurons in -layer and external input of the th neurons in -layer, respectively; and denote the external inputs on the th neurons in -layer and the th neurons in -layer at time , respectively; and are continuously periodic functions with periodic., that is, and are delay kernels functions. and are signal transmission functions of th neurons and th neurons at time and in space . Smooth functions and correspond to the transmission diffusion operators along the th neurons and the th neurons, respectively.

The boundary conditions and the initial conditions are given by where and are continuous bounded functions defined on , respectively.

Throughout the paper, we give the following assumptions.(A1)The signal transmission functions are Lipschtiz continuous on with Lipschtiz constants and , namely, for any ,(A2)The delay kernels are nonnegative continuous functions that satisfy the following conditions:(i), (ii), (iii) there exists a positive constant such that

To be convenient, we introduce some notations. Let . Let and be the equilibrium of system (2.1). We denote

*Definition 2.1. *The equilibrium of the delay fuzzy BAM neural networks (2.1) is said to be globally exponentially stable, if there exist positive constants such that
for every solution of the delay fuzzy BAM neural networks (2.1) with the initial conditions (2.2) and (2.3) for all .

*Definition 2.2. *If * is a *continuous function, then the upper right derivative of * is *defined as

Lemma 2.3 (see [26]). *Suppose and are two states of system (2.1), then one has*

The remainder of this paper is organized as follows. In Section 3, we will study global exponential stability of fuzzy BAM neural networks (2.1). In Section 4, we present the existence of periodic solution for fuzzy BAM neural networks (2.1). In Section 5, an example will be given to illustrate effectiveness of our results obtained. We will give a general conclusion in Section 6.

#### 3. Global Exponential Stability

In this section, we will discuss the global exponential stability of fuzzy BAM neural networks (2.1) by constructing suitable functional.

Theorem 3.1. *Under assumptions (A1) and (A2), if there exist such that
**
where then the equilibrium of system (2.1) is the globally exponentially stable.*

* Proof. *By using (3.1), we can choose a small number such that
It is well known that the bounded functions always guarantee the existence of an equilibrium point for system (2.1). The uniqueness of the equilibrium for system (2.1) will follow from the global exponential stability to be established below.

Suppose is any solution of system (2.1).

Rewrite (2.1) as follows
Multiply both sides of (3.3) by and integrate, we have
Applying the boundary condition (2.2) and the Gauss formula, we get
in which is the gradient operator and
From (3.5), (3.6), hypothesis (A1), Lemma 2.3, and the Holder inequality, we have
Multiply both sides of (3.4) by , similarly, we get
We construct a Lyapunov functional
Calculating the upper right derivative of along the solution of (3.3) and (3.4), from (3.8) and (3.9), we get
Estimating the right of (3.11) by elemental inequality , we obtain that
Therefore,
Noting that
On the other hand, we have
Let
then and
This implies that the equilibrium point of system (2.1) is globally exponentially stable. The proof is completed.

Corollary 3.2. *Suppose (A1) and (A2) hold. Then, the equilibrium of system (2.1) is the globally exponentially stable, if the following conditions are satisfied:
*

#### 4. Periodic Oscillatory Solutions of Fuzzy BAM Neural Networks

In this section, we consider the existence and uniqueness of periodic oscillatory solutions for system (2.1).

Theorem 4.1. *Suppose that assumption (A1) and (A2) hold. Then, there exists exactly an -periodic solution of system (2.1), with the initial values (2.2) and (2.3), and all other solutions converge exponentially to it as . If there exists such that
**
where .*

*Proof. *Let
For any , we define
Then, is the Banach space of continuous functions.

For any , we denote the solutions of system (2.1) through , and as
respectively. And define that
then for . Let . Therefore, it follows from system (2.1) that
Considering the following Lyapunov functional
By a minor modification of the proof of Theorem 3.1, we can easily obtain
for all . We can choose a positive integer such that
and define a Poincare mapping by
It follows that
Letting , we, from (4.8) to (4.11), obtain that
It implies that is a contraction mapping. Hence, there exist a unique equilibrium point such that
Note that
Let be the solution of system (2.1) through , then is also the solution of system (2.1). It is clear that
for all . It follows from (4.15) that
which shows that is exactly an periodic solution of system (2.1) with the initial conditions (2.2) and (2.3) and other solutions of system (2.1) with the initial conditions (2.2) and (2.3) converge exponentially to it as .

#### 5. An Illustrative Example

In this section, we will give an example to illustrate feasible our result.

*Example 5.1. * Consider the following system
where
It is obvious that satisfy assumption (A1) and satisfy assumption (A2), moreover . Let
By simply calculating, we can get
Since all the conditions of Corollary 3.2 are satisfied, therefore the system (5.1) has a unique equilibrium point which is globally exponentially stable.

#### 6. Conclusion

In this paper, we have studied the exponential stability of the equilibrium point and the existence of periodic solutions for fuzzy BAM neural networks with distributed delays and diffusion. Some sufficient conditions set up here are easily verified, and the conditions which are only correlated with parameters of the system (2.1) are independent of time delays. The results obtained in this paper remove the assumption about the activation functions with differentiable and only require the activation functions are bounded and Lipschitz continuous. Thus, it allows us even more flexibility in choosing activation functions.

#### Acknowledgments

The authors would like to thank the editor and anonymous reviewers for their helpful comments and valuable suggestions, which have greatly improved the quality of this paper. This work is partially supported by the Scientific Research Foundation of Guizhou Science and Technology Department([2011]J2096).

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