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

Huaiqin Wu, Luying Zhang, Sanbo Ding, Xueqing Guo, and Lingling Wang Department of Applied Mathematics, Yanshan University, Qinhuangdao 066001, China Correspondence should be addressed to Huaiqin Wu; huaiqinwu@ysu.edu.cn 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. 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 setvalued 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.

Different from the previous works, in this paper, we will study complete periodic synchronization of memristorbased neural networks described by the following differential equation:  ( in which switching jumps   > 0, d > 0, d > 0, á  , à  , b  , b  , ,  = 1, 2, . . ., , are constants;   and   are feedback functions,   () is the time delay with 0 ≤   () ≤  and τ  ≤   < 1 ( 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.

Preliminaries
In this section, we give some definitions and properties, which are needed later.
Definition 1 (see [25]).Suppose  ⊆ R  ; then  → () is called a set-valued map from  → R  , if for each point  ∈ , there exists a nonempty set () ⊆ R  .A set-valued map  with nonempty values is said to be upper semicontinuous (USC) at  0 ∈ , if for any open set  containing ( 0 ), there exists a neighborhood  of  0 such that () ⊆ .The map () is said to have a closed (convex, compact) image if for each  ∈ , () is closed (convex, compact).
To proceed with our analysis, we need the following assumptions for system (1).

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).
Based on the conditions of Lemma 7, the proof will be divided into three steps.
Up to now, we have proved that Ω 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.
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.

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   () =   () −   (),  = 1, 2, . . ., ; 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   > 1.
Proof.Consider the following Lyapunov functional: where The master system (1) and the slave system (40) are statedependent switching systems; then, the four cases may appear in the following at time .
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 (A 2 ), evaluating the upper right derivation  + () of () along the trajectory of (47) gives ] (51) Correspondingly, the error system can be rewritten as ( Arguing as in the proof of Case 1, we can obtain Similarly, evaluating the upper right derivation  + () of () along the trajectory of (54), we have Correspondingly, the error system can be rewritten as By using |  ()| ≤   , we can also have The above proving procedures clearly imply that one always has  + () ≤ 0 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 [11-13, 16, 17, 20, 21].A typical assumption is that However, We can prove that this assumption holds only when  and  have different sign, or  = 0, or  = 0. 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 ( [11][12][13][14]).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.

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

Conclusion
In this paper, complete periodic synchronization of a class of memristor-based neural networks has been investigated.The master system synchronizes with the slave system by using adaptive control.The obtained results are novel since there are few works about complete periodic synchronization issue of memristor-based neural networks via adaptive control.
In addition, the easily testable condition which ensures the   existence of periodic solution of a class of memristor-based recurrent neural network is also much different from the existing work.The obtained results are also applicable to the continuous systems without switching jumps.Finally, a numerical example has been given to illustrate the validity of the present results.