Journal of Chaos

Volume 2013 (2013), Article ID 304643, 7 pages

http://dx.doi.org/10.1155/2013/304643

## Finite-Time Combination-Combination Synchronization for Hyperchaotic Systems

^{1}Department of Mathematics, Zhangzhou Normal University, Zhangzhou 363000, China^{2}College of Finance, Fujian Jiangxia University, Fuzhou 350108, China

Received 27 April 2013; Revised 13 July 2013; Accepted 24 July 2013

Academic Editor: Uchechukwu E. Vincent

Copyright © 2013 Huini Lin 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

A new type of finite-time synchronization with two drive systems and two response systems is presented. Based on the finite-time stability theory, step-by-step control and nonlinear control method, a suitable controller is designed to achieve finite-time combination-combination synchronization among four hyperchaotic systems. Numerical simulations are shown to verify the feasibility and effectiveness of the proposed control technique.

#### 1. Introduction

As a new subject in 1980s, chaos almost covers all the fields of science. It is known that chaos is an interesting nonlinear phenomenon which may lead to irregularity and unpredictability in the dynamic system, and it has been intensively studied in the last three decades. Since Pecora and Carroll proposed the PC method to synchronize two chaotic systems in 1990 [1, 2], the study of synchronization of chaotic systems has been widely investigated due to their potential applications in various fields, for instance, in chemical reactions, biological systems, and secure communication. Over the past decades, a variety of control approaches such as adaptive control [3], linear feedback control [4], active control [5], and backstepping control [6] have been proposed for various types of synchronization, which include complete synchronization [7], projective synchronization [8, 9], general synchronization [10], lag synchronization [11], and novel compound synchronization [12].

Most of the aforementioned works are based on the synchronization scheme which consists of one drive system and one response system and can be seen as one-to-one system. However, we found it not secure and flexible enough in many real world applications, for instance, in secure communication. Recently, Runzi et al. presented a new type of synchronization with two drive systems and one response system [13]. Then, Sun et al. extended multi-to-one system to multi-to-two systems and reported a new type of synchronization, namely, combination-combination synchronization, where synchronization is achieved between two drive systems and two response systems [14]. The type of synchronization can improve the security of communication; for instance, we can split the transmitted signals into several parts, then load each part in different drive systems, and then restore it to the original signals by combining the received signals of different response systems correctly.

Notice that the mentioned literatures mainly investigated the asymptotic synchronization of chaotic systems. However, in the view of practical application, optimizing the synchronization time is more important than achieving synchronization asymptotically [15–19]. Recently, based on the step-by-step control method, Wang et al. realized the finite-time synchronization of two chaotic systems by designing a proper controller [15]. The method has the ability to achieve global stability in finite time. In addition, the step-by-step technique has the advantage of reducing controller complexity.

Motivated by the previous discussion, this paper aims to study the finite-time synchronization between a combination of two drive systems and a combination of two response systems in drive-response synchronization scheme. We have applied the finite-time stability theory to our analysis to achieve finite-time combination-combination synchronization. The step-by-step control method and nonlinear control technique are adopted to synchronize four different hyperchaotic systems. Numerical simulations are presented to verify the theoretical findings.

#### 2. The Finite-Time Combination-Combination Synchronization Scheme

Consider the drive systems and response systems as follows:
where , , , and are the state vectors of systems (1), (2), (3), and (4), respectively. * * are four continuous functions, and are two controllers of the response systems (3) and (4) which will be designed, respectively.

*Definition 1. *If there exist four constant matrices and , such that and for , where , , , and , one gets that the drive systems (1) and (2) are realized as finite-time combination-combination synchronization with the response systems (3) and (4), where represents the matrix norm.

Lemma 2 (see [15]). *Assume that a continuous, positive-definite function satisfies the following differential inequality:
**
where and are constants. Then, for any initial time , satisfies
**
with given by
*

Thus, for any initial value , the system (5) has in ; that is, the system can achieve global stability in finite time.

#### 3. Problem Statement and Control Scheme

##### 3.1. Problem Statement

Consider two identity hyperchaotic Chen systems as the drive systems [20]:

Consider two identity hyperchaotic Lorenz systems as the response systems [21]: where and are two controllers of the response systems (10) and (11) which will be designed, respectively.

Without loss of generality, we choose , , , and .

The objective of the synchronization scheme is to design a suitable controller such that and for , where , , , and ; that is, the drive systems (8) and (9) are realized as finite-time combination-combination synchronization with the response systems (10) and (11).

##### 3.2. The Control Scheme

Let , , , and . The controller to be designed is , , , and .

Thus, we can get the error system as follows:

Our aim is to design a suitable controller, such that the drive systems (8) and (9) are realized as combination-combination synchronization with the response systems (10) and (11) in finite time. Then, the problem is changed to design a suitable controller, such that the error system (12) achieves the finite-time stability at the origin.

The design plan and its steps are as follows.

*Step 1. *Choose
where

is a proper rational number, and is a positive odd number,.

Substituting into the fourth equation of (12), we get

Choose a candidate Lyapunov function

Thus, the derivative of along the solution of error equation (15) is

According to Lemma 2, the system (15) is finite-time stability, which implies that there exists , such that for .

*Step 2. *Choose
where

For , substituting into the first equation of (12), we get

Choose a candidate Lyapunov function

Thus, the derivative of along the solution of error equation (20) is

According to Lemma 2, the system (20) is finite-time stability, which implies that there exists , such that for .

*Step 3. *Choose
where

For , substituting into the second equation of (12), we get

Choose a candidate Lyapunov function

Thus, the derivative of along the solution of error equation (25) is

According to Lemma 2, the system (25) is finite-time stability, which implies that there exists , such that for .

*Step 4. *Choose
where

For , substituting into the third equation of (12), we get

Choose a candidate Lyapunov function

Thus, the derivative of along the solution of error equation (30) is

According to Lemma 2, the system (30) is finite-time stability, which implies that there exists , such that for .

The controller is designed as follows:

According to what we discussed previously, we can obtain this conclusion that the error system (12) achieves finite-time stability under the control of the controller (33). Furthermore, the drive systems (8) and (9) are realized as combination-combination synchronization with the response systems (10) and (11) in finite time , where .

#### 4. Numerical Simulation

To verify the effectiveness of the proposed finite-time synchronization method, we consider the hyperchaotic Chen system with the parameters ,, , , and . The hyperchaotic attractor of the system is shown in Figure 1.

Consider the hyperchaotic Lorenz system with the parameters , , , and . The hyperchaotic attractor of the system is shown in Figure 2.

In the following simulation, we assume, , and , and the initial states for the drive systems and response systems are arbitrarily given by , ,, and . Then, and we choose the synchronization controller with ,,,,. The synchronization evolution for this controller is shown in Figure 3.

The error vector is achieved to zero which implies that systems (8), (9) and (10), (11) have achieved finite-time combination-combination synchronization.

#### 5. Conclusion

In this paper, the problem of finite-time combination-combination synchronization with two drive systems and two response systems was investigated. Based on the finite-time stability theory, the step-by-step control and nonlinear control approach, a suitable controller was introduced. The simulation results demonstrated that the proposed controller works well for synchronizing four hyperchaotic systems in finite time.

#### Acknowledgments

The authors would like to thank the Editor and the referees for their valuable comments and suggestions that helped to improve the quality of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 61074012) and the Natural Science Foundation of Fujian Province (Grant no. 2011J01025).

#### 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 Scopus - L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar, and J. F. Heagy, “Fundamentals of synchronization in chaotic systems, concepts, and applications,”
*Chaos*, vol. 7, no. 4, pp. 520–543, 1997. View at Google Scholar · View at Scopus - T. L. Liao and S. H. Tsai, “Adaptive synchronization of chaotic systems and its application to secure communications,”
*Chaos, solitons and fractals*, vol. 11, no. 9, pp. 1387–1396, 2000. View at Publisher · View at Google Scholar · View at Scopus - M. T. Yassen, “Controlling chaos and synchronization for new chaotic system using linear feedback control,”
*Chaos, Solitons and Fractals*, vol. 26, no. 3, pp. 913–920, 2005. View at Publisher · View at Google Scholar · View at Scopus - Y. Lei, W. Xu, and H. Zheng, “Synchronization of two chaotic nonlinear gyros using active control,”
*Physics Letters A*, vol. 343, no. 1–3, pp. 153–158, 2005. View at Publisher · View at Google Scholar · View at Scopus - S. Tong, C. Li, and Y. Li, “Fuzzy adaptive observer backstepping control for MIMO nonlinear systems,”
*Fuzzy Sets and Systems*, vol. 160, no. 19, pp. 2755–2775, 2009. View at Publisher · View at Google Scholar - C. Yao, Q. Zhao, and J. Yu, “Complete synchronization induced by disorder in coupled chaotic lattices,”
*Physics Letters A*, vol. 377, no. 5, pp. 370–377, 2013. View at Publisher · View at Google Scholar - J. Sun, Y. Shen, G. Zhang, et al., “General hybrid projective complete dislocated synchronization with non-derivative and derivative coupling based on parameter identification in several chaotic and hyperchaotic systems,”
*Chinese Physics B*, vol. 22, no. 4, pp. 040508–040518, 2013. View at Publisher · View at Google Scholar - J. Sun, Y. Shen, and G. Zhang, “Transmission projective synchronization of multi-systems with non-delayed and delayed coupling via impulsive control,”
*Chaos*, vol. 22, no. 4, pp. 043107–043116, 2012. View at Google Scholar - H. G. Enjieu Kadji and R. Yamapi, “General synchronization dynamics of coupled Van der Pol-Duffing oscillators,”
*Physica A*, vol. 370, no. 2, pp. 316–328, 2006. View at Publisher · View at Google Scholar · View at Scopus - Y. Shen and J. Wang, “Robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 23, no. 1, pp. 87–96, 2012. View at Publisher · View at Google Scholar - J. Sun, Y. Shen, Q. Yin, et al., “Compound synchronization of four memristor chaotic oscillator systems and secure communication,”
*Chaos*, vol. 23, no. 1, pp. 013140–013149, 2013. View at Publisher · View at Google Scholar - R. Luo, Y. Wang, and S. Deng, “Combination synchronization of three classic chaotic systems using active backstepping design,”
*Chaos*, vol. 21, no. 4, pp. 043114–043119, 2011. View at Publisher · View at Google Scholar - J. Sun, Y. Shen, G. Zhang, et al., “Combination-combination synchronization among four identical or different chaotic systems,”
*Nonlinear Dynamics*, vol. 73, no. 3, pp. 1211–1222, 2013. View at Publisher · View at Google Scholar - H. Wang, Z. Han, Q. Xie, and W. Zhang, “Finite-time chaos synchronization of unified chaotic system with uncertain parameters,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 2239–2247, 2009. View at Publisher · View at Google Scholar · View at Scopus - Y. Yang and X. Wu, “Global finite-time synchronization of a class of the non-autonomous chaotic systems,”
*Nonlinear Dynamics*, vol. 70, no. 1, pp. 197–208, 2012. View at Google Scholar - U. E. Vincent and R. Guo, “Finite-time synchronization for a class of chaotic and hyperchaotic systems via adaptive feedback controller,”
*Physics Letters A*, vol. 375, no. 24, pp. 2322–2326, 2011. View at Publisher · View at Google Scholar · View at Scopus - R. Luo and Y. Wang, “Finite-time stochastic combination synchronization of three different chaotic systems and its application in secure communication,”
*Chaos*, vol. 22, no. 2, pp. 023109–023118, 2012. View at Publisher · View at Google Scholar - P. He, S. Ma, and T. Fan, “Finite-time mixed outer synchronization of complex networks with coupling time-varying delay,”
*Chaos*, vol. 22, no. 4, pp. 043151–043161, 2012. View at Publisher · View at Google Scholar - Y. X. Li, W. K. S. Tang, and G. R. Chen, “Generating hyperchaos via state feedback control,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 15, no. 10, pp. 3367–3375, 2005. View at Publisher · View at Google Scholar · View at Scopus - X. Wang and M. Wang, “Hyperchaotic Lorenz system,”
*Acta Physica Sinica*, vol. 56, no. 9, pp. 5136–5141, 2007. View at Google Scholar · View at Scopus