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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 358362, 14 pages
http://dx.doi.org/10.1155/2012/358362
Research Article

Globally Exponential Stability of Periodic Solutions to Impulsive Neural Networks with Time-Varying Delays

1College of Science, Guilin University of Technology, Guangxi, Guilin 541004, China
2Guizhou Key Laboratory of Economics System, Guizhou College of Finance and Economics, Guizhou, Guiyang 550004, China

Received 12 January 2012; Accepted 26 February 2012

Academic Editor: Josef Diblík

Copyright © 2012 Yuanfu Shao 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

By using Schaeffer's theorem and Lyapunov functional, sufficient conditions of the existence and globally exponential stability of positive periodic solution to an impulsive neural network with time-varying delays are established. Applications, examples, and numerical analysis are given to illustrate the effectiveness of the main results.

1. Introduction

It is well known that in implementation of neural networks, time delays are inevitably encountered because of the finite switching speed of amplifiers. Specially in electronic neural networks, delays are usually time-varying and often become sources of instability. So it is important to investigate the dynamics of neural networks with delays [17]. Recently, the study of the existence of periodic solutions of neural networks has received much attention. The common approaches are based on using Mawhin continuation theorem [1, 2, 810], Banach’s fixed point theorem [1113], fixed point theorem in a cone [14], Schaeffer’s theorem [15, 16], and so on. On the other hand, studies on neural dynamical systems not only involve the existence of periodic solutions, but also involve other dynamical behaviors such as stability of periodic solutions, bifurcations, and chaos. In recent years, the stability of solutions of neural networks has attracted attention of many researchers and many nice results have been obtained [13, 513, 1625]. For example, M. Tan and Y. Tan [1] considered the following neural network with variable coefficients and time-varying delays: By using the Mawhin continuation theorem, they discussed the existence and globally exponential stability of periodic solutions.

However, in real world, many physical systems often undergo abrupt changes at certain moments due to instantaneous perturbations, which lead to impulsive effects. In fact, impulsive differential equation represents a more natural framework for mathematical modelling of many real world phenomena such as population dynamic and neural networks. The theory of impulsive differential equations is now being recognized to be richer than the corresponding theory of differential equations without impulse, and various kinds of impulsive differential equations have been extensively studied, see [816, 18, 19, 2125] and references therein. Then, considering impulsive effects, it is necessary and interesting for us to study further the dynamics of system (1.1). Furthermore, as pointed by Gopalsamy and Sariyasa [4], it would be of great interest to study neural networks in periodic environment. On the other hand, to the best of our knowledge, few authors considered the existence of periodic solutions by using Schaeffer’s theorem. Hence, in this paper, by using Schaeffer’s theorem and Lyapunov functional, we aim to discuss the existence and exponential stability of periodic solutions to a class of impulsive neural networks with periodic coefficients and time-varying delays. The model is as follows: with initial conditions where corresponds to the state of the th unit, represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, denotes the output of the th unit, and denote the strength of the th unit on the th unit, respectively, is the external bias on the th unit, corresponds to the transmission delay along the axon of the th unit, denotes the impulsive moment, and is a strictly increasing sequence such that , denotes the Banach space of continuous mapping from to equipped with the norm for all , where .

Throughout this paper, we always assume the following.() are all continuous -periodic functions for .() is continuous and there exists positive constant such that for any and .()There exists positive integer such that . Then For convenience, we use the following notations: where is continuous and -periodic function.

The rest of this paper is organized as follows. In Section 2, by using Schaeffer’s theorem, sufficient conditions of the existence of -periodic solution to system (1.2) with initial conditions (1.3) are established. In Section 3, by using Lyapunov functional, we derive the conditions under which the periodic solution is globally exponentially stable. In Section 4, applications, illustrative examples, and simulations are given to show the effectiveness of the main results. Finally, some conclusions are drawn in Section 5.

2. Existence of Periodic Solution

First we make some preparations. As usual in the theory of impulsive differential equation, by a solution of model (1.2), it means the following.(i) is piecewise continuous such that exists, and is differentiable on for .(ii) satisfies (1.2) for .

Definition 2.1. The set is said to be quasi-equicontinuous in if for any , there exists such that, if and , then .

Lemma 2.2 (see [26, Compactness criterion]). The set is relatively compact if and only if(i) is bounded, that is, for each and some ,(ii) is quasi-equicontinuous in .

The following lemma is fundamental to our discussion. The method is similar to that of [13, 16], so the proof is omitted here.

Lemma 2.3. is an -periodic solution of system (1.2) which is equivalent to is an -periodic solution of the following equation: where , and
It is easy to show that and where and .

Lemma 2.4 (see [27, Schaeffer’s theorem]). Let be a normed space and be a compact operator. Define Then either(i)set is unbounded, or(ii)operator has a fixed point in .
In order to use Lemma 2.4, let with the norm , then is a Banach space.
Define a mapping by , where and By Lemma 2.3, it is easy to see that the existence of -periodic solution of (1.2) is equivalent to the existence of fixed point of the mapping in .

Theorem 2.5. Suppose that hold. Further,(). Then system (1.2) admits an -periodic solution.

Proof. By Lemma 2.3, it suffices to prove that the mapping admits a fixed point in .
For any constant , let . For , from (2.3) and , we have It implies that is uniformly bounded.
For any , we have If , it is obvious that . Hence, from (2.7) and (2.8), we have Therefore, is a family of uniformly bounded and equicontinuous subset. By Lemma 2.2, the mapping is compact.
Let , and considering the following operator equation: If is a solution of (2.10), then According to , we deduce that It implies that is bounded, which is independent of . By Lemma 2.4, we obtain that the mapping admits a fixed point in . Hence system (1.2) admits an -periodic solution such that . This completes the proof.

3. Globally Exponentially Stable

In this section, the sufficient conditions ensuring that (1.2) admits a unique -periodic solution and all solutions of (1.2) exponentially converge to the unique -periodic solution are to be established.

Definition 3.1. Let be an -periodic solution of system (1.2) with initial value . If there exist constants , for every solution of (1.2) with initial , such that then is said to be globally exponentially stable.

Theorem 3.2. Suppose that hold. Further,()  ,()  ,where is a constant, is a constant determined in (3.5).
Then system (1.2) admits a unique -periodic solution, which is globally exponentially stable.

Proof. By Theorem 2.5, system (1.2) admits an -periodic solution with initial value . Let be an arbitrary solution of (1.2) with initial value . Define and , then we have By , we have for . Let It is clear that is continuous on and . In addition, and , then is strictly monotone increasing. Therefore, there exists a unique such that for . Let then Obviously, for and the above , we have where .
Define by In view of (3.2) and (3.8), for , we have We claim that If not, then there exist and such that for . Then, it follows from (3.9) and (3.11) that Equation (3.12) leads to which contradicts (3.6). Thus (3.10) holds, that is, If , we have for . Similar to the steps of (3.10)–(3.14), we can derive that If , then By repeating the same procedure, then It follows from that , which leads to for any . So the combination (3.18) and (3.19) gives In addition, it is clear that Therefore, from (3.20) and (3.21), for any , we have It implies that the -periodic solution of (1.2) is globally exponentially stable. Hence, (1.2) admits a unique -periodic solution, which is globally exponentially stable. This completes the proof.

Remark 3.3. Theorem 3.2 implies that the impulse affects the existence and exponential stability of the periodic solution of system (1.2). It shows the dynamics of impulsive differential system (1.2) is richer than the corresponding system (1.1) without impulse.

4. Applications and Examples

In (1.2), if , then (1.2) reads: For system (4.1), we have the following result.

Proposition 4.1. Suppose that hold. Further,(),(),(), where is a constant, is determined in Theorem 3.2.
Then system (4.1) admits a unique -periodic solution, which is globally exponentially stable.

If the impulses are absent in system (1.2), that is, , then (1.2) leads to (1.1). Similarly we have the following.

Proposition 4.2. Suppose that hold. Further,()  ,()  ,then system (1.1) admits a unique -periodic solution, which is globally exponentially stable.

Remark 4.3. Proposition 4.2 implies that the sufficient conditions of the existence and globally exponential stability of periodic solution to (1.1) are independent of the time-varying delays, while the corresponding results obtained by authors [5] are dependent on delays. Without effect from time-varying delays, our results are better for people to keep the stability of system (1.1). Although the authors [1] also established similar conditions which are independent of delays, their employed tool and analysis techniques are very different so that their main results are different from ours. Particularly, (1.1) is the special case of (1.2) without impulse. Hence, in this sense, results of this paper complement or improve some previously known results [1, 5].

Finally, two examples and numerical analysis are given to show the usefulness of the main results.

Example 4.4. Let where ,  ,  . Then .
By easy computation, , and , which implies holds. On the other hand, it is easy to verify that holds. By verification, , namely, holds too. From Theorems 2.5 and 3.2, we obtain that (4.2) has a unique 2-periodic solution, which is globally exponentially stable, see Figure 1.

fig1
Figure 1: Dynamics of (4.2)—(a) time series of , (b) time series of , and (c) portrait of .

Example 4.5. Let where for , ,,, , , , . Then .
By computation, , which implies that holds. It is easy to verify that holds too. From Proposition 4.2, system (4.3) has a unique 1-periodic solution, which is globally exponentially stable, see Figure 2. However, by calculation, conditions of the results of [1] fail, then one cannot obtain the existence of periodic solution of system (4.3) by results of reference [1], which further shows that the results complement or improve previously known results.

fig2
Figure 2: Dynamics of (4.3)—(a) time series of , (b) time series of .

5. Conclusions

In this paper, the existence and globally exponential stability of the periodic solution of system (1.2) are studied. Model (1.2) is very general, including such models as continuous bidirectional associative memory networks, cellular neural networks, and Hopfield-type neural networks (see, e.g., [6, 7, 28]). The main methods employed here are Schaeffer’ theorem, differential inequality techniques, and Lyapunov functional, which are very different from [1]. The sufficient conditions obtained here are new and complement or improve the previously known results [1, 57]. Finally, applications, two illustrative examples and simulations, are given to show the effectiveness of the main results.

Acknowledgments

The authors would like to thank the reviewers for their valuable comments and constructive suggestions, which are very useful for improving the quality of this paper. This paper is supported by National Natural Science Foundation of China (11161015) and Doctoral Foundation of Guilin University of Technology (2010).

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