International Journal of Differential Equations

Volume 2013 (2013), Article ID 543947, 12 pages

http://dx.doi.org/10.1155/2013/543947

Research Article

Periodic Solutions for Impulsive Stochastic BAM Neural Networks with Time-Varying Delays in Leakage Terms

Li Yang and Yongkun Li

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received 27 April 2013; Accepted 18 July 2013

Academic Editor: Yong Zhou

Copyright © 2013 Li Yang and Yongkun Li. 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.

By using an integral inequality, we establish some sufficient conditions for the existence and p-exponential stability of periodic solutions for a class of impulsive stochastic BAM neural networks with time-varying delays in leakage terms. Moreover, we present an example to illustrate the feasibility of our results.

1. Introduction

Since it was proposed by Kosko (see [1]), the bidirectional associative memory (BAM) neural networks have attracted considerable attentions due to their extensive applications in classification of patterns, associative memories, image processing, and other areas. In the past few years, many scholars have obtained lots of good results on the dynamical behaviors analysis of BAM neural networks. The reader may see [2–8] and the references therein.

But in a real nervous system, it is usually unavoidably affected by external perturbations which are in many cases of great uncertainty and hence may be treated as random. As pointed out by Haykin [9], in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. And the stability of neural networks could be stabilized or destabilized by some stochastic inputs [10]. Therefore, it is significant and of prime importance to consider the dynamics of stochastic neural networks. With respect to stochastic neural networks, there are many works on the stability. For example, in  [11–17], the scholars studied the stability of different classes of stochastic neural networks. For other results on stochastic neural networks, the reader may see  [18–23] and the references therein.

However, the above results are mainly on the stability of considered stochastic neural networks. And it is well known that studies on neural dynamical systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior. On the other hand, the neural networks are often subject to impulsive effects that in turn affect dynamical behaviors of the systems. Moreover, a leakage delay, which is the time delay in the leakage term of the systems and a considerable factor affecting dynamics for the worse in the systems, is being put to use in the problem of stability for neural networks. However, so far, very little attention has been paid to neural networks with time delay in the leakage (or “forgetting”) term. Such time delays in the leakage term are difficult to handle but have great impact on the dynamical behavior of neural networks. Therefore, it is meaningful to consider neural networks with time delays in the leakage term [24–32].

But to the best of our knowledge, there are few papers published on studying the existence of periodic solutions of impulsive stochastic neural networks with time delay in the leakage term. Motivated by the previous discussions, in this paper, we consider the following impulsive stochastic BAM neural networks:

where , ( and are the number of neurons in layers) and and denote the activations of the th neuron and the th neuron at time ; and represent the rate with which the th neuron and th neuron will reset their potential to the resting state in isolation when they are disconnected from the network and the external inputs at time ; and denote the leakage delays; and are the input-output functions (the activation functions); and are elements of feedback templates at time ; and denote biases of the th neuron and the th neuron at time , ; and are -dimensional Brownian motions defined on complete probability space ; here, we denote by the associated -algebra generated by with the probability measure ,  ;   and are Borel measurable functions; and are diffusion coefficient matrices; and are impulses at moment , which describe that the evolution processes experience abrupt change of state at the moments of time , where and .

Our main purpose in this paper is using an integral inequality, which is from a lemma in  [33], to establish some sufficient conditions on the existence and -exponential stability of the periodic solutions of (1).

Let be a complete probability space with a filtration satisfying the usual conditions; that is, is right continuous and contains all -null sets. Denote by the family of -measurable, valued random variables , where is an piecewise-continuous stochastic process; that is, is continuous for all but at most countable points and at these points , and exist, . For , define the norm , where is an integer, ; stands for the correspondent expectation operator with respect to the given probability measure .

For convenience, for an -periodic function , denote ,  . The initial condition of (1) is

where , ,  ,  .

Throughout this paper, we assume that the following conditions hold:

,  ,  , , , , , and are all periodic continuous functions with period for ,  ,  ; , , , and are Lipschitz-continuous with Lipschitz constants , , , and , ,  ; and are real sequences and and are -periodic, , ,  .

Remark 1. From , there exist constants  , and such that

where ,  ,  and .

This paper is organized as follows. In Section 2, we introduce some definitions and state some preliminary results which are needed in later sections. In Section 3, we state and prove our results. In Section 4, we give an example to illustrate the feasibility of our results obtained in the previous section.

2. Preliminaries

In this section, we introduce some definitions and state some preliminary results.

Definition 2. A stochastic process is said to be periodic with period if its finite dimensional distributions are periodic with period ; that is, for any positive integer and any moments of time , the joint distribution of the random variables is independent of ,  .

Lemma 3 (see [34]). If is an -periodic stochastic process, then its mathematical expectation and variance are -periodic.

Definition 4. The solution of (1) is said to be

(i)-uniformly bounded if for each , , there exists a positive constant which is independent of such that implies ,  , (ii)-point dissipative if there exists a constant such that for any point , there exists such that for each , , there exists a positive constant which is independent of such that ,  .

Lemma 5 (see [35]). Under conditions , assume that the solution of (1) is -uniformly bounded and -point dissipative for ; then (1) has an -periodic solution.

Lemma 6 (see [36]). For any and ,

Definition 7. The periodic solution with initial value of (1) is said to be -exponential stable if there are constants and such that any solution with initial value of (1) satisfies

Lemma 8 (see [33]). Let be a solution of the delay integral inequality

where , and ,   is a constant vector, and . If , then there are constants and such that

where satisfies .

Lemma 9 (see [33]). Assume that all conditions of Lemma 9 hold. If , then all solutions of inequality of (6) exponentially converge to zero.

By Lemmas 8 and 9, we have the following corollary.

Corollary 10. Let be a solution of the delay integral inequality

where , and ,   is a constant, and . If , then there are constants and such that

where satisfies . Moreover, if , then all solutions of inequality of (8) exponentially converge to zero.

Under our assumptions, we consider the following system:

where ,  .

Similar to Lemma 2.1 in paper  [33], we have the following lemma.

Lemma 11. Let hold. Then,

(i) if is a solution of (10), then , is a solution of (1); (ii) if ,,,,, is a solution of (1), then , is a solution of (10).

3. Main Results

In this section, we will state and prove the sufficient conditions for the existence and -exponential stability of periodic solution of (1).

Theorem 12. Let hold. Suppose further that

there exists an integer such that , where

Then (1) has an -periodic solution, which is -exponentially stable.

Proof. We can rewrite (10) as follows:

By the method of variation parameter, for , , from the first equation of (12), we have the following:

For , denote

Considering expectations, using Lemma 6, for , we have

For , we evaluate the first term of (15) as follows:

For the second term of (15), we have

For the third term of (15), we have

As for the fourth term of (15), for , we have

For the fifth term of (15), by Hölder inequality, we have

As for the last term of (15), using Proposition  1.9 in  [37] and Hölder inequality, for , we have

where . Therefore, for , we have

Similarly, for ,  , from the second equation of (12), we can obtain the following:

Hence, by (22) and (23), we have that

Set . By (24), we have that

where , . By and Corollary 10, the solutions of (10) are -uniformly bounded and it also shows that the family of all solutions of (10) is -point dissipative. Then, it follows from Lemma 5 that (10) has an -periodic solution .

Suppose that is an arbitrary solution of (10). Then it follows from (10) that

Let . Proceeding as the proof of the existence of periodic solution of (10), from (26), we obtain that

By and Corollary 10, the periodic solution of (10) is -exponentially stable; that is, the periodic solution of (1) is -exponentially stable. Therefore, (1) has an -periodic solution, which is -exponentially stable. This completes the proof of Theorem 12.

4. An Example

In this section, we present an example to illustrate the feasibility of our results obtained in the previous section.

Let . Consider the following impulsive stochastic BAM neural network:

where and the coefficients are as follows:

By calculating, we have

Taking , we can verify that all conditions of Theorem 12 are satisfied. Hence, (28) has a -periodic solution, which is -exponentially stable.

By the numerical simulation in Figures 1, 2, and 3 we can show that our results are plausible and effective.

Figure 1: Transient states of stochastic BAM neural networks.

Figure 2: State response of and .

Figure 3: State response of and .

Acknowledgment

This work is supported by the National Natural Science Foundation of People’s Republic of China under Grant 10971183.

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