The Scientific World Journal

Volume 2013 (2013), Article ID 159194, 6 pages

http://dx.doi.org/10.1155/2013/159194

Research Article

Dual Synchronization of Fractional-Order Chaotic Systems via a Linear Controller

Jian Xiao, Zhen-zhen Ma, and Ye-hong Yang

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received 11 July 2013; Accepted 13 August 2013

Academic Editors: N. Herisanu, T. Li, Y. Xia, and Q. Xie

Copyright © 2013 Jian Xiao 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.

The problem of the dual synchronization of two different fractional-order chaotic systems is studied. By a linear controller, we realize the dual synchronization of fractional-order chaotic systems. Finally, the proposed method is applied for dual synchronization of Van der Pol-Willis systems and Van der Pol-Duffing systems. The numerical simulation shows the accuracy of the theory.

1. Introduction

In recent years, the topic of chaos synchronization has attracted increasing attention in many fields. The result of synchronization of chaotic oscillators is used in nonlinear oscillators [1], circuit experiment [2], secret communication [3], and some other fields. In 1990, the first concept of synchronization was presented by Carroll and Perora [4]. And there are many methods about chaos synchronization such as Lyapunov equation [5], Perora-Carroll (PC) [4] and backstepping control [6]. All of these methods are amid of the synchronization between one master and one slave system do not consist of the synchronization of multimaster systems and multislave systems.

Dual synchronization is a special circumstance in synchronization of chaotic oscillators. The first idea of multiplexing chaos using synchronization was investigated in a small map and an electronic circuit model by Tsimring and Sushchik in 1996 in [7]; then the concept of dual synchronization was raised by Liu and Davids in 2000 in [8], which concentrates on using a scalar signal to simultaneously synchronize two different pairs chaotic oscillators, that is, the synchronization between two master systems and two slave systems.

Nowadays, there are many dual synchronization methods, such as in 2000 Liu and Davids introduce the dual synchronization of 1-D discrete chaotic systems via specific classes of piecewise-linear maps with conditional linear coupling in [8]. The dual synchronization between the Lorenz and Rossler systems by the Lyapunov stabilization theory is investigated in [9]. The output feedback strategy is used to study the dual synchronization of two different 3-D continuous chaotic systems in [10]. Then the dual synchronization in modulated time-delayed systems is investigated by designing a delay feedback controller in [11]. All of these works are amid of the dual synchronization of integer-order chaotic systems and do not consist of the dual synchronization of fractional-order chaotic systems. In this paper, a new method of dual synchronization of fractional-order chaotic systems is proposed, by a linear controller; the dual synchronization of chaos is obtained.

The rest of this paper is organized as follows: in Section 2, we construct a theory frame about the dual synchronization of two different fractional-order chaotic systems. By a linear controller, we obtain dual synchronization between two different fractional-order chaotic systems in Section 3. In Section 4, the proposed method is applied to dual synchronization of Van der Pol-Willis systems and Van der Pol-Duffing systems for evaluating the performance of the method and by numerical simulation; the result shows that the controller designed by the application of this method is effective. Finally, conclusions are drawn in Section 5.

2. Problem Analysis

We define the following two systems as two master systems.

Master 1:

Master 2:

where and are the state vectors of the two master systems.    and are two known functions. is the order of the two master systems. By a linear combination of the two master systems states, a signal is give as

where and are two known matrices, and , , , cannot be zero at the same time. So is a known matrix. is a combination of the two master systems states. The corresponding two slave systems are as follows:

Slave 1:

Slave 2:

where and are the state vectors of the two slave systems, and are vectors of manipulated variables, and is the order of the two slave systems. Similarly, by a linear combination of two slave systems states, a signal is generated as follows:

where is a combination of the two slave systems states.

The error signal for dual synchronization is

The main goal is to synchronize the master systems and the slave systems is equivalent to

where is the Euclidian norm.

3. Dual Synchronization Strategy

Lemma 1. Considering the fractional-order system

where , , and ; then system (9) is stable if and only if , , where denotes the argument of the eigenvalue of .

Theorem 2. The dual synchronization of fractional-order chaotic systems between the master systems and the slave systems is achieved if and only if the following condition satisfies

where is the coefficient matrix of master systems and is a control gain vector.

Proof. We can rewrite (1) and (2) in the following form by defining :

Similarly, (4) and (5) can be rewritten as

where , one defines and .

Equation (12) is transformed into

so the error system is transformed into

The error system is obtained as

Using the first-order Taylor expansion, the function is rewritten as

where denotes the higher order terms of the series. We substitute (16) into (15) and yield

We can transfer the (17) into

according to Lemma 1, we can know that the error system is asymptotically stable at zero if and only if the following condition is satisfied

4. The Example Analysis and Numerical Simulations

Example 3 (dual synchronization of Van der Pol-Willis systems). In the first example, we can use the proposed method to achieve the dual synchronization of the Van der Pol system and the Willis system.

Master 1: Van der Pol system

Master 2: Willis system

So the corresponding slave systems are as follows:

Slave 1:

Slave 2:

where ,  ,  ,  , and .

The matrix of the master systems is achieved as

so the corresponding error matrix are as follows:

We should choose the appropriate parameters so that all the eigenvalues of the Jacobian matrix of (25) satisfy Matignon condition; that is, the eigenvalues evaluated at the equilibrium point are satisfied:

The eigenvalue equation of the equilibrium point is locally asymptotically stable. From what we have discussed above, we can know that and are two known matrices; the parameter can be appropriately selected for satisfying the Matignon condition.

Dual synchronization of the Van der Pol system and the Willis system is simulated. The system parameters are set to be , , , , , , , , , , , , ,  and , so

If , , , and , which satisfy (18), the eigenvalue equation of the equilibrium point is locally asymptotically stable. We choose  , , , and . The initial conditions of the master system 1 and the master system 2 are taken as , and ,  ; the initial conditions of the slave system 1 and the slave system 2 are taken as ,   and , , so the initial conditions of the error system are set to be , , , and . In Figures 1 and 2, we can see that all error variables have converged to zero; that is, we achieve the dual synchronization between the Van der Pol and the Willis systems.

Figure 1: Error signals between the pair of Van der Pol system.

Figure 2: Error signals between the pair of Willis system.

Example 4 (dual synchronization of Van der Pol and Duffing systems). For Example 4, the dual synchronization of Van der Pol and Duffing systems is investigated.

Master 1: Van der Pol system

Master 2: Duffing system

So the corresponding slave systems are

Slave 1:

Slave 2:

where ,   ,  , , and .

The matrix of the master systems is achieved as

So the corresponding error matrix are as follows:

We should choose the appropriate parameters so that all the eigenvalues of the Jacobian matrix of (33) satisfy Matignon condition; that is, the eigenvalues evaluated at the equilibrium point are satisfied:

The eigenvalue equation of the equilibrium point is locally asymptotically stable. Because and are two known matrices, the parameter can be appropriately selected for satisfying the Matignon condition.

According to what we have studied above, parameters are set to , ,  , , , , , , , , , , and , so

If , , , and , which satisfy (34), the eigenvalue equation of the equilibrium point is locally asymptotically stable. We choose  , , , and . The initial conditions of the master system 1 and the master system 2 are taken as , and , , the initial conditions of the slave system 1 and the slave system 2 are taken as ,     and , , so the initial conditions of the error system are set to be , , , and . In Figures 3 and 4, we can see that all error variables have converged to zero; that is, we achieve the dual synchronization between the Van der Pol and the Duffing systems.

Figure 3: Error signals between the pair of Van der Pol system.

Figure 4: Error signals between the pair of Duffing system.

5. Conclusions

In this work, we construct a theory frame about dual synchronization of two different fractional-order chaotic systems and propose a method of dual synchronization. In addition, this method is used for designing a synchronization controller to achieve the dual synchronization of two different fractional-order chaotic systems. Finally, the proposed method is applied for dual synchronization of the Van der Pol-Willis systems and the Van der Pol-Duffing systems. The numerical simulations proves the accuracy of the theory.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities of China under Grant no. CQDXWL-2012-007.

References

G. M. Mahmoud, E. E. Mahmoud, A. A. Farghaly, and S. A. Aly, “Chaotic synchronization of two complex nonlinear oscillators,” Chaos, Solitons & Fractals, vol. 42, no. 5, pp. 2858–2864, 2009.

Z. Xu, C.-X. Liu, and T. Yao, “Study on a new chaotic system with analysis and circuit experiment,” Acta Physica Sinica, vol. 59, no. 1, pp. 131–139, 2010.

D. Matignon, “Stability results for fractional differential equations with application to control processing,” in Proceedings of the International IMACS IEEE-SMC Multiconference on Computational Engineering in Systems Applications, vol. 2, pp. 963–968, Lille, France, 1996.

T. Carroll and L. M. Perora, “Synchronization of chaotic systems,” Physical Review Letters, vol. 64, pp. 821–824, 1990.

J.-B. Hu, Y. Han, and L.-D. Zhao, “Synchronizing fractional chaotic systems based on Lyapunov equation,” Acta Physica Sinica, vol. 57, no. 12, pp. 7522–7526, 2008.

J.-B. Hu, Y. Han, and L.-D. Zhao, “A novel stablility theorem for fractional systems and its applying in synchronizing fractional chaotic system based on back-stepping approach,” Acta Physica Sinica, vol. 58, no. 4, pp. 2235–2239, 2009.

L. S. Tsimring and M. M. Sushchik, “Multiplexing chaotic signals using synchronization,” Physics Letters A, vol. 213, no. 3-4, pp. 155–166, 1996.

Y. Liu and P. Davids, “Dual synchronization of chaos,” Physical Review E, vol. 61, pp. 2176–2184, 2000.

D. Ning, J.-A. Lu, and X. Han, “Dual synchronization based on two different chaotic systems: Lorenz systems and Rössler systems,” Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 1046–1050, 2007.

H. Salarieh and M. Shahrokhi, “Dual synchronization of chaotic systems via time-varying gain proportional feedback,” Chaos, Solitons & Fractals, vol. 38, no. 5, pp. 1342–1348, 2008.

D. Ghosh and A. R. Chowdhury, “Dual-anticipating, dual and dual-lag synchronization in modulated time-delayed systems,” Physics Letters A, vol. 374, no. 34, pp. 3425–3436, 2010.