- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 647561, 13 pages

http://dx.doi.org/10.1155/2012/647561

## Impulsive Control for the Synchronization of Chaotic Systems with Time Delay

^{1}College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China^{2}Department of Mathematics, Southeast University, Nanjing 210096, China

Received 21 August 2012; Accepted 9 October 2012

Academic Editor: Jinde Cao

Copyright © 2012 Ming Han 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 considers impulsive control for the synchronization of chaotic systems with time delays. Based on the Lyapunov functions and the Razumikhin technique, some new synchronization criteria with an exponential convergence rate are derived. Our results show that impulses do contribute to globally exponential synchronization of dynamical systems. Besides, the impulsive moments are independent of the upper bound of time delays. Furthermore, a bigger upper bound of impulsive intervals for the synchronization of chaotic systems can be obtained when compared with many previous studies. Hence, our results are less conservative and more effective for the synchronization analysis. A numerical example is given to show the validity and potential of the developed results.

#### 1. Introduction

In the last two decades, control and synchronization problems of chaotic systems have been extensively studied, due to their potential applications in many areas. For instance, they are used to understand self-organizational behavior in the brain as well as in ecological systems, and they also have been applied to produce secure message communication between a sender and a receiver. So far, different synchronization techniques have been proposed and implemented in practice, such as PC (Pecora and Carroll) method [1], active control [2], adaptive control [3, 4], sliding mode control [5], and impulsive method [6–12].

Impulsive phenomena exist in many biological systems and mechanics fields. Impulsive control has been widely studied and gradually become an interesting and useful synchronization approach [13–16]. Impulsive control can provide an efficient method for some cases in which the systems cannot endure continuous disturbance. The main idea of the impulsive control strategy is to change the states of a system by some sudden jumps instantaneously. Furthermore, using the impulsive control method, the response system needs to receive the information from the drive system only at some discrete instants which means that impulsive control is easier to be implemented to some extent. The asymptotical synchronization of chaotic systems without time delay [17–20] and with time delay [21, 22] has been widely investigated by using impulsive control. However, most of the previous impulsive stability criteria are only valid for some specific systems with small delays due to the restrictive requirement that the time delays are required to be smaller than the length of impulsive interval [23–25].

In this paper, we will investigate globally exponential synchronization of coupled chaotic systems with time delay by using impulsive control. The main contributions of this paper include the following. (i) Our results show that impulses do contribute to globally exponential synchronization of dynamical systems and the time delays in systems not to be smaller than the length of impulsive intervals. Therefore, they can be usually used as an effective control strategy to synchronize the underlying delayed dynamical systems in more practical application. (ii) A new approach for the exponential synchronization of impulsive chaotic systems with any finite delay is given. (iii) In order to deal with uncertainty and/or measurement noise effectively in nominal case for identical chaotic and hyperchaotic systems, the impulse distance should increases which in turn decreases the control cost in accordance with [19]. Thus, having a minimum level of synchronization error with the largest impulsive intervals is generally desired [26]. In our proposed results, a bigger upper bound of impulsive intervals for the synchronization of chaotic systems can be obtained by comparing with the results in [13–16]. It should also be noticed that our paper was inspired in part by the work of Zhou and Wu in [27] for delayed linear differential equations.

The rest of this paper is organized as follows. In Section 2, the problem and some preliminaries are presented. The main result on exponential synchronization is given in Section 3. Section 4 gives an example for illustration, and some conclusions are finally drawn in Section 5.

#### 2. Preliminaries

Denote that is the set of real numbers, is the set of nonnegative real numbers, is the -dimensional real space, and is the set of positive integers. Let and . For with and for , we define the following function: , for all , for all and , for all but at most a finite number of points . For , the norm of is, respectively, defined by are defined by and for .

Now we introduce the coupled chaotic systems that will be studied. It usually consists of two chaotic systems at the transmitter and the receiver ends.

At the transmitter end, we have and, at the receiver end, we have where the matrices , are the synchronization error states between the states of system (2.1) and system (2.2), and the functions are the continuous functions in their respective domain of definition. The time sequence satisfies , , the time delay , and , in which for . A typical form of the function is given as follows: where are constant matrices. The second term is known as the delayed feedback controller, which is applied to the input state and then influences the system function.

Then, we can get the following error dynamical system: where .

Obviously, is a trivial solution of system (2.4). We shall analyze the dynamics of system (2.4) and drive criteria under which its trivial solution is globally exponentially stable. It is clear that the globally and exponential stability of trivial solution of (2.4) implies the global synchronization of systems (2.1) and (2.2).

For a given and , the initial value problem of (2.4) is

We assume that (2.4) has a unique solution with respect to initial conditions. Denote by the solution of (2.4) such that , . Also we assume that the function satisfies the following assumption: for certain positive definite matrix , there exist constant matrices , , such that Now we have the following definitions.

*Definition 2.1. *The trivial solution of (2.4) is said to be globally and exponentially stable if for any initial data and , there exist some constants and such that

*Definition 2.2. *Given a function , the upper right-hand derivative of with respect to system (2.4) is defined by

#### 3. Synchronization of Chaotic Systems

In the following, we shall address the exponential stability problem for impulsive delayed nonlinear differential equation (2.4), which implies the global synchronization of two chaotic systems. Our result shows that impulses play an important role in making the delayed nonlinear differential equations globally and exponentially stable.

Theorem 3.1. *Let be a positive definite matrix and be largest eigenvalue of . Assume that there exist constants , with and , , such that for all , one has*(i)*(ii)**
where ,*(iii)*Then, the trivial solution of system (2.4) is globally and exponentially stable with a convergence rate of for any fixed delay , that is, system (2.1) and system (2.2) are globally and exponentially synchronized. *

*Proof. *Let be any solution of system (2.5), and consider a Lyapunov function as follows:
Let and be the smallest and largest eigenvalues of positive definite matrix , respectively. Then, we can obtain
Now, we are in a position to prove that, for any ,
For , it follows from assumption (2.6) and condition (i) that
It further follows from condition (i) that
Defining in condition (iii), we have
Then, in view of the inequality in (3.9), we can find constant such that
which shows that
Firstly, we prove that
To do this, we only need to prove that
If the inequality (3.13) is not true, it follows from (3.11) that there must exist some such that
which implies that there exists such that
and there exists such that
Hence, for any , we can obtain that
By condition (ii), (3.8), and (3.17), we can obtain
which shows that
It is obviously a contradiction. Hence, (3.12) holds, and then (3.6) is true for . Now, we suppose that (3.6) holds for , that is,
Next, we will prove that (3.6) holds for , that is,
For the sake of contradiction, we suppose that (3.21) is not true. Then, we define
From condition (iii) and (3.20), we can get

and so . By the continuity of in the interval , we obtain
From (3.23), we can derive that there exists such that

As for any , , then either or . These two cases will be discussed in the following.

If , from (3.20), we have
whereas if , from (3.24), then
Above all, from (3.25)–(3.27), we can obtain, for any ,
Hence, by condition (ii), (3.8), and (3.28), we can conclude that
Thus, in view of condition (iii), we have
which is a contradiction. It implies that the supposition is not true. Hence, (3.6) holds for , then, we can conclude by some induction that (3.6) holds for any . It immediately follows from (3.6) that
that is, the trivial solution of the impulsive delayed system (2.5) is globally and exponentially stable with a convergence rate of for any fixed delays . Then, it implies that system (2.1) and system (2.2) are globally synchronized. The proof is thus complete.

*Remark 3.2. *In LMI (3.1) of Theorem 3.1, the constant is used to measure the level of instability for delay-free system and is determined by the matrix , while is decided by the matrix .

*Remark 3.3. *Compared with Theorem 3.1 in [28], a distinct feature of Theorem 3.1 in this paper is to eliminate the restriction that the impulsive interval can not be too large, that is, the additional assumption (iv) in Theorem 3.1 of [28] is indeed deleted here. Moreover, a controlled parameter is introduced to adjust the degree of convergence rate on globally exponential stability of the error system (2.4). It should be mentioned that our results allow us to develop an effective impulse control strategy to exponentially synchronize chaotic systems, and it is particularly meaningful for some practical applications.

Similarly to Theorem 3.1, we can obtain the following result.

Corollary 3.4. *Let be a symmetric and positive definite matrix, , where is given by Theorem 3.1 for each . Assume that there exist constants , with , such that for all the following are satisfied:*(i)*(ii)**Then, the trivial solution of system (2.4) is globally and exponentially stable for any fixed delay , that is, system (2.1) and system (2.2) are globally and exponentially synchronized. *

*Proof. *Condition (ii) in Theorem 3.1 can be reduced to the following form:
Then, it is easy to get the following inequality, which is equivalent to (3.34) and condition (iii) in Theorem 3.1:
Hence, there must exist a small-enough real number such that (3.35) holds. And if is extremely tiny, we can obtain that condition (ii) in Corollary 3.4 holds. From Theorem 3.1, the proof is completed.

*Remark 3.5. *As Corollary 3.4 can be applied to deal with globally exponential stability of impulsive differential equations with any time delays , it is obviously more applicable than those existing in the works in the recent literature [23–25, 28] where the time delays need to be assumed not bigger than the length of impulsive interval. In view of Corollary 3.4, it is clear that we have removed the restriction of time delay indeed.

*Remark 3.6. *Since the amount of transmitted information decreases leading to reduced control cost, the cost of impulsive synchronization of chaotic systems is closely related to the impulse distances and having a minimum level of synchronization error with the largest impulsive intervals as generally a desired [26]. From the proposed result in Corollary 3.4, the upper bound of impulsive intervals for the globally and exponentially synchronization can be given by for each . From the following numerical example, we will show that our upper bound of impulsive intervals is bigger than some existing results.

#### 4. Numerical Simulation

In this section, an example is presented here to illustrate our main results.

*Example 4.1. *Consider the Lorenz system in [13] as follows:
where , , , and . Let , then we can rewrite the Lorenz system in the form of (2.1) as follows:
where
We also choose defined by (2.3) with . Set the delay and the impulsive matrices for all . Then, it can be observed that inequality (2.6) holds with .

Solving (3.1) in Theorem 3.1, we get that , , and . Since , we can get . According to condition (ii) in Corollary 3.4, when
the trivial solution of error system based on impulsive delayed differential equation (4.1) is globally exponentially stable for any fixed delay . With the same parameter, a comparison of the upper bound of impulsive intervals with [13–16] is presented in Table 1, which shows that our upper bound is much bigger. Choose , and the simulation result of error system is shown in Figure 1.

#### 5. Conclusion

In this paper, we have investigated the synchronization of coupled chaotic systems with time delay by using impulsive control. The time delays in systems need not to be smaller than the length of impulsive interval as many works existing in the literature assumed. A numerical example has been given to demonstrate the effectiveness of the theoretical results, and the estimation of the stable region of the impulsive intervals has also been presented. By comparison, the upper bound of impulsive intervals is greater than it was in some existing results. The obtained results can be easily used to control many systems, especially to stabilize and synchronize chaotic systems.

#### Acknowledgments

The authors would like to take this opportunity to thank Professor. Jinde Cao and the reviewers for their constructive comments and useful suggestions. This work was partially supported by the NNSF of China (Grants nos. 61175119, 11101373, 11271333, and 61074011), the NSF of Jiangsu Province of China (Grant no. BK2010408), Huo Ying-Dong Education Foundation (Grant no. 132037), and Zhejiang Innovation Project, China (Grant no. T200905).

#### References

- L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,”
*Physical Review Letters*, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. N. Agiza and M. T. Yassen, “Synchronization of Rossler and Chen chaotic dynamical systems using active control,”
*Physics Letters A*, vol. 278, no. 4, pp. 191–197, 2001. View at Publisher · View at Google Scholar - J. Q. Lu and J. D. Cao, “Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters,”
*Chaos*, vol. 15, no. 4, Article ID 043901, 10 pages, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Cao and J. Lu, “Adaptive synchronization of neural networks with or without time-varying delay,”
*Chaos*, vol. 16, no. 1, Article ID 013133, 6 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Zhang, X. K. Ma, and W. Z. Liu, “Synchronization of chaotic systems with parametric uncertainty using active sliding mode control,”
*Chaos, Solitons and Fractals*, vol. 21, no. 5, pp. 1249–1257, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. Ji, C. Y. Wen, and Z. G. Li, “Impulsive synchronization of chaotic systems via linear matrix inequalities,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 16, no. 1, pp. 221–227, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y.-W. Wang, C. Wen, Y. C. Soh, and Z. H. Guan, “Partial state impulsive synchronization of a class of nonlinear systems,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 19, no. 1, pp. 387–393, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Q. Lu, D. W. C. Ho, and J. D. Cao, “A unified synchronization criterion for impulsive dynamical networks,”
*Automatica*, vol. 46, no. 7, pp. 1215–1221, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Liu, S. W. Zhao, and J. Q. Lu, “A new fuzzy impulsive control of chaotic systems based on T-S fuzzy model,”
*IEEE Transactions on Fuzzy Systems*, vol. 19, no. 2, pp. 393–398, 2011. View at Publisher · View at Google Scholar · View at Scopus - B. Wu, Y. Liu, and J. Q. Lu, “Impulsive control of chaotic systems and its applications in synchronization,”
*Chinese Physics B*, vol. 20, no. 5, Article ID 050508, 2011. View at Publisher · View at Google Scholar - B. Wu, Y. Liu, and J. Q. Lu, “New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 837–843, 2012. View at Publisher · View at Google Scholar - T. Yang and L. O. Chua, “Practical stability of impulsive synchronization between two nonautonomous chaotic systems,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 10, no. 4, pp. 859–867, 2000. View at Scopus - W. X. Xie, C. Y. Wen, and Z. G. Li, “Impulsive control for the stabilization and synchronization of Lorenz systems,”
*Physics Letters A*, vol. 275, no. 1-2, pp. 67–72, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Li, “Some new less conservative criteria for impulsive synchronization of a hyperchaotic Lorenz system based on small impulsive signals,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 2, pp. 713–719, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Run-Zi, “Impulsive control and synchronization of a new chaotic system,”
*Physics Letters A*, vol. 372, no. 5, pp. 648–653, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Q. Wu, J. A. Lu, C. K. Tse, J. J. Wang, and J. Liu, “Impulsive control and synchronization of the Lorenz systems family,”
*Chaos, Solitons and Fractals*, vol. 31, no. 3, pp. 631–638, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. D. Cao, D. W. C. Ho, and Y. Q. Yang, “Projective synchronization of a class of delayed chaotic systems via impulsive control,”
*Physics Letters A*, vol. 373, no. 35, pp. 3128–3133, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Hu, H. J. Jiang, and Z. D. Teng, “Fuzzy impulsive control and synchronization of general chaotic system,”
*Acta Applicandae Mathematicae*, vol. 109, no. 2, pp. 463–485, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Haeri and M. Dehghani, “Modified impulsive synchronization of hyperchaotic systems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 3, pp. 728–740, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Chen, H. Liu, J. A. Lu, and Q. J. Zhang, “Projective and lag synchronization of a novel hyperchaotic system via impulsive control,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 4, pp. 2033–2040, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. M. Liu and W. Ding, “Impulsive synchronization for a chaotic system with channel time-delay,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 2, pp. 958–965, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Z. Liu, “Impulsive synchronization of chaotic systems subject to time delay,”
*Nonlinear Analysis*, vol. 71, no. 12, pp. e1320–e1327, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Z. Liu and Q. Wang, “Impulsive stabilization of high-order Hopfield-type neural networks with time-varying delays,”
*IEEE Transactions on Neural Networks*, vol. 19, no. 1, pp. 71–79, 2008. View at Publisher · View at Google Scholar · View at Scopus - X. Z. Liu, X. S. Shen, Y. Zhang, and Q. Wang, “Stability criteria for impulsive systems with time delay and unstable system matrices,”
*IEEE Transactions on Circuits and Systems. I*, vol. 54, no. 10, pp. 2288–2298, 2007. View at Publisher · View at Google Scholar - A. Khadra, X. Z. Liu, and X. S. Shen, “Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses,”
*IEEE Transactions on Automatic Control*, vol. 54, no. 4, pp. 923–928, 2009. View at Publisher · View at Google Scholar - C. Li, X. Liao, and X. Zhang, “Impulsive synchronization of chaotic systems,”
*Chaos*, vol. 15, no. 2, Article ID 023104, 5 pages, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Zhou and Q. J. Wu, “Exponential stability of impulsive delayed linear differential equations,”
*IEEE Transactions on Circuits and Systems II*, vol. 56, no. 9, pp. 744–748, 2009. View at Publisher · View at Google Scholar - Q. Wang and X. Z. Liu, “Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method,”
*Applied Mathematics Letters*, vol. 20, no. 8, pp. 839–845, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH