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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 140153, 12 pages
http://dx.doi.org/10.1155/2013/140153
Research Article

Complete Periodic Synchronization of Memristor-Based Neural Networks with Time-Varying Delays

Department of Applied Mathematics, Yanshan University, Qinhuangdao 066001, China

Received 6 April 2013; Revised 4 June 2013; Accepted 8 June 2013

Academic Editor: Zhengqiu Zhang

Copyright © 2013 Huaiqin Wu 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

This paper investigates the complete periodic synchronization of memristor-based neural networks with time-varying delays. Firstly, under the framework of Filippov solutions, by using M-matrix theory and the Mawhin-like coincidence theorem in set-valued analysis, the existence of the periodic solution for the network system is proved. Secondly, complete periodic synchronization is considered for memristor-based neural networks. According to the state-dependent switching feature of the memristor, the error system is divided into four cases. Adaptive controller is designed such that the considered model can realize global asymptotical synchronization. Finally, an illustrative example is given to demonstrate the validity of the theoretical results.

1. Introduction

Memristor, as the fourth fundamental passive circuit, was firstly postulated by Chua [1] in 1971. On May 1, 2008, the Hewlett-Packard (HP) research team announced their realization of a memristor prototype, with an official publication in Nature [2, 3]. This new circuit element of memristor shares many properties of resistors and shares the same unit of measurement. Recently, memristor has received a great deal of attention because of its potential applications in next generation computer and powerful brain-like “neural” computer. The papers [421] have given a detailed introduction on the memristor, so readers can consult [421] to get more explanation. As noted in [10], from a systems-theoretic point of view and a mathematical point of view, memristor dynamics strictly obey Bernoulli's nonlinear differential equation, so the mathematical framework and its usefulness are worth studying. The paper [10] by Wu and Zeng discussed the exponential stabilization of memristive neural networks with time delays. The papers [1114] investigated the synchronization and antisynchronization control of a class of memristor-based recurrent neural networks. A series of results on stability analysis of memristor-based recurrent neural networks were presented in [1518]. The papers [1921] dealt with the existence and stability of periodic solution of almost periodic of a class of memristor-based recurrent neural networks.

Different from the previous works, in this paper, we will study complete periodic synchronization of memristor-based neural networks described by the following differential equation: where in which switching jumps , , , , , , , , are constants; and are feedback functions, is the time delay with and ( and are negative constants). At first glance, one might intuitively believe that the chaotic motion is more complicated compared with the periodic motion, the synchronization of chaotic oscillators is also complicated than those of periodic oscillators [22]. However, this is not always true, just as indicated in [23, 24], where an opposite result was given.

The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced. In Section 3, the proof of the existence of periodic solutions is presented. Complete periodic synchronization is discussed in Section 4. In Section 5, a numerical example is presented to demonstrate the validity of the proposed results. Some conclusions are drawn in Section 6.

Notation 1. denotes the set of real numbers, denotes the -dimensional Euclidean space, and denotes the set of all real matrices. Given the vectors , , , . For , denotes the family of continuous function from to with the norm . represents the interval. denotes the closure of the convex hull of . denotes the identity matrix of size . A vector or matrix means that all entries of are greater than or equal to zero; can be defined similarly. For vectors or matrices and , (or ) means that (or ). denotes the collection of all nonempty compact subsets of with the Hausdorff metric defined by , , and , {}, { is convex}.

2. Preliminaries

In this section, we give some definitions and properties, which are needed later.

Definition 1 (see [25]). Suppose ; then is called a set-valued map from , if for each point , there exists a nonempty set . A set-valued map with nonempty values is said to be upper semicontinuous (USC) at , if for any open set containing , there exists a neighborhood of such that . The map is said to have a closed (convex, compact) image if for each , is closed (convex, compact).

Definition 2 (see [26]). For the system , with discontinuous right-hand sides, a set-valued map is defined as where is the closure of the convex hull of set , and is Lebesgue measure of set . A solution in Filippov's sense of the Cauchy problem for this system with initial condition is an absolutely continuous function , which satisfies and differential inclusion

The initial value associated with system (1) is , . Let , , , , , , , and . We define Clearly, system (1) is a differential equation with discontinuous right-hand side; its solution in the conventional sense does not exist. Inspired by [10, 20, 21, 2730], we adopt the following definition of the solution in the sense of Filippov for system (1).

Definition 3. Suppose that is a continuous function. An absolutely continuous function is said to be a solution with initial data of system (1), if satisfies the differential inclusion for all , or equivalently, there exist , , and , such that

Remark 4. From the theoretical point of view, the above parameters , , and in (7) are measurable functions and depend on the state and time .

Definition 5. We say that real matrix is an -matrix, if and only if we have , , , and all successive principal minors of are positive.

Lemma 6 (see [31]). Let be an matrix with nonpositive off-diagonal elements. Then is an -matrix if and only if one of the following conditions hold:(1)there exists a vector such that ;(2)there exists a vector such that .

Lemma 7 (Mawhin-Like Coincidence Theorem [32]). Suppose that : is USC and -periodic in . If the following conditions hold:(1) there exists a bounded open set , the set of all continuous, -periodic functions: , such that for any , each -periodic function of the inclusion satisfies if it exists;(2) each solution of the inclusion satisfies ;(3), then differential inclusion (4) has at least one -periodic solution with .

To proceed with our analysis, we need the following assumptions for system (1). and are continuous -periodic functions. For , for all , , the neural activation function is bounded and satisfy Lipschitz condition; that is, there exist , and , such that

3. Existence of Periodic Solution

In this section, we will give a sufficient condition which ensures the existence of periodic solution of memristor-based neural network (1).

Theorem 8. Under assumptions and , if is an -matrix, where and then system (1) has at least one -periodic solution.

Proof. Set , . Then is a Banach space with the norm . Let , , where . It is obvious that the set-valued map has nonempty compact convex values. Futhermore, it is USC.
Based on the conditions of Lemma 7, the proof will be divided into three steps.
Step  1. We need to search for appropriate open bounded subset . Assume that is an arbitrary -periodic solution of differential inclusion for a certain . Then, one has or equivalently, there exist measurable functions , , and , such that for . Multiplying both sides of (14) by and integrating over the interval , one has Noting that where is the inverse function of , , from (15) and (16), it yields where . This means Define , , , and . From (18), we have Since is an -matrix, we can choose a vector such that By combining (19) and (20) together, one can derive Thus, we can easily get that where . Then, there exists such that Obviously, for , . It follows from (23) that On the other hand, from (14), one easily obtains that Let ; combining (24) and (25), we can derive Clearly, is independent of . In addition, since is an -matrix, there exists a vector such that . Thus, we can choose a sufficiently large constant such that , , and Taking , then, is an open bounded set of and for any . This proves that condition (1) in Lemma 7 is satisfied.
Step  2. Suppose that there exists a solution of the inclusion ; then is a constant vector on such that for some . Therefore, one has or equivalently, there exist , , and , such that Then, there exists such that It follows from (30) that This means , which contradicts (27).
Step  3. We define a homotopic set-valued map by .
If , then is a constant vector on such that for some . It follows that In fact, we have , . If , that is, or There exists , such that that is,
Therefore, we have This means that , which contradicts (27). Thus, , . It follows that , for any , . Therefore, by the homotopy invariance and the solution properties of the topological degree, one has where denotes the topological degree for USC set-valued maps with compact convex values.
Up to now, we have proved that satisfies all the conditions in Lemma 7, then system (1) has at least one -periodic solution. This completes the proof.

Notice that a constant function can be regarded as a special periodic function with arbitrary period or zero amplitude. Hence, we can obtain the following result.

Corollary 9. Suppose assumption holds and , , if is an -matrix, where and then system (1) exists at least one equilibrium point.

Remark 10. By employing the method based on the -matrix theory, our results can be easily verified and are much different from these in the literature [20, 21]. It is also worth mentioning that the -matrix theory is one of the effective and important methods to deal with the existence of periodic solution and equilibrium point for large-scale dynamical neuron systems.

4. Complete Periodic Synchronization

In this paper, we consider model (1) as the master system, and a slave system for (1) can be described by the following equation: where is the controller to be designed, and Let , ; one can obtain the following result.

Theorem 11. Suppose that all the conditions of Theorem 8 are satisfied; then the salve system (40) can globally synchronize with the master system (1) under the following adaptive controller: where , are arbitrary positive constants and .

Proof. Consider the following Lyapunov functional: where
The master system (1) and the slave system (40) are state-dependent switching systems; then, the four cases may appear in the following at time .
Case  1. If , at time , then the master system (1) and the slave system (40) reduce to the following systems, respectively, Correspondingly, the error system can be written as where , . Under assumption , evaluating the upper right derivation of along the trajectory of (47) gives Considering the definition of , and , , one has Case  2. If , at time , then the master system (1) and the slave system (40) reduce to the following systems, respectively, Correspondingly, the error system can be rewritten as Arguing as in the proof of Case  1, we can obtain Case  3. If , at time , then the master system (1) and the slave system (40) reduce to (50) and (46). Correspondingly, the error system can be rewritten as Similarly, evaluating the upper right derivation of along the trajectory of (54), we have Note that ; by using assumption , one has According to the definition of , , and , one has Case  4. If , at time , then the master system (1) and the slave system (40) reduce to (45) and (51). Correspondingly, the error system can be rewritten as By using , we can also have The above proving procedures clearly imply that one always has at time . Therefore, the salve system (40) globally synchronizes with the master system (1) under the adaptive controller (42). This completes the proof.

Remark 12. In the literature, some results on stability analysis of periodic solution (or equilibrium point) or synchronization (or antisynchronization) control of memristor-based neural network were obtained [1113, 16, 17, 20, 21]. A typical assumption is that However, We can prove that this assumption holds only when and have different sign, or , or . Without this assumption, we divide the error system into four cases in this paper. Under the adaptive controller (42), globally periodic synchronization criterion between system (1) and (40) is derived. The synchronization criterion of this paper which does not solve any inequality or linear matrix inequality is easily verified.

Remark 13. As far as we know, there is no work on the periodic synchronization of memristor-based neural network via adaptive control. Thus, our outcomes are brand new and original compared to the existing results ([1114]). In addition, the obtained results in this paper are also applicable to the common systems without memristor or the memductance of the memristor equals a constant since they are special cases of memristor-based neural networks.

5. Numerical Example

In this section, one example is offered to illustrate the effectiveness of the results obtained in this paper.

Example 1. Consider the second-order memristor-based neural network (1) with the following system parameters: and the activation functions are taken as follows: It can be verified that , , , , , , and .
We take , . A straightforward calculation gives and . Then, we get . Obviously, is an -matrix. Thus, the conditions required in Theorem 8 are satisfied. When is a periodic function, in the view of Theorem 8, this neural network has at least one periodic solution.
For numerical simulations, we choose the external input . The periodic dynamic behavior of the master system (1) and the slave system (40) with is given in Figures 1, 2, and 3, with the initial states chosen as and for .
In order to demonstrate the adaptive controller (42) can realize complete periodic synchronization of memristor-based neural networks, some initial parameters are taken as , for , , ,