About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 151025, 5 pages
http://dx.doi.org/10.1155/2013/151025
Research Article

Synchronization between Fractional-Order and Integer-Order Hyperchaotic Systems via Sliding Mode Controller

College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China

Received 16 September 2012; Accepted 19 January 2013

Academic Editor: Magdy A. Ezzat

Copyright © 2013 Yan-Ping Wu and Guo-Dong Wang. 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 synchronization between fractional-order hyperchaotic systems and integer-order hyperchaotic systems via sliding mode controller is investigated. By designing an active sliding mode controller and choosing proper control parameters, the drive and response systems are synchronized. Synchronization between the fractional-order Chen chaotic system and the integer-order Chen chaotic system and between integer-order hyperchaotic Chen system and fractional-order hyperchaotic Rössler system is used to illustrate the effectiveness of the proposed synchronization approach. Numerical simulations coincide with the theoretical analysis.

1. Introduction

During the past decades, fractional calculus has become a powerful tool to describe the dynamics of complex systems such as power systems, mathematics, biology, medicine, secure communication, and chemical reactors [16]. Chaos synchronization has attracted lots of attention in a variety of research fields [713] over the last two decades, because it can be applied in vast areas of physics and engineering and secure communication [14, 15]. Moreover, many theoretical analysis and numerical simulation results about the synchronization of chaotic systems are obtained. Wang et al. [16] deal with the finite-time chaos synchronization of the unified chaotic system with uncertain parameters. Chen and Liu [17] propose a simple linear state feedback controller to realize the stability control of a unified chaotic system. The problem of chaos synchronization between two different chaotic systems with fully unknown parameters is investigated in [18]. Moreover, Chen and his partners [19] investigate the chaos control of a class of fractional-order chaotic systems via sliding mode.

All of above articles mainly focus on integer-order chaotic systems or fractional-order chaotic systems. There is little information about the synchronization between fractional-order chaotic systems and integer-order chaotic systems [20, 21]. The study of synchronization between fractional-order hyperchaotic systems and integer-order hyperchaotic systems is also limited.

Motivated by the above discussion, this paper investigates a sliding mode method for synchronization between a class of fractional-order hyperchaotic systems and integer-order hyperchaotic systems. And the integer-order hyperchaotic systems are regarded as response system in the proposed synchronous technique which is simple and theoretically rigorous.

2. System Description and Problem Formulation

Consider the following fractional-order hyperchaotic system as a drive system where denotes four-dimensional state vector. represents the linear part of the system, and is the nonlinear part of the system.

And the response system can be described as where is four-dimensional state vector, and imply the same roles as and in the drive system, respectively.

Remark 1. and in the drive system can be same as and in the response system, respectively.
One adds the controller into the response system, which is given by Define the synchronous errors as . The aim is to choose a suitable controller , so that the drive system and response system can achieve chaotic synchronization (i.e., , where is the Euclidean norm).

3. Design of Sliding Mode Controller

Let the controller be where is a compensation controller and . Here, in the response system belongs to hyperchaotic fractional-order drive system. is a vector function, and it will be designed later.

From (4), the system (3) can be rewritten as

In accordance with the active control design procedure, the nonlinear part of the error dynamics is eliminated by the following choice of the input vector [22]

The error system (5) is rewritten as where is a constant gain vector and is the control input which satisfies

To design a sliding mode controller, one has two steps. First, one constructs a sliding surface that represents a desired system dynamics. Next, one develops a switching control law such that a sliding mode exists on every point of the sliding surface, and any states outside the surface are driven to reach the surface in a finite time [23]. As a choice for the sliding surface, one has which can also be easily given by

In the sliding mode, the sliding surface and its derivative must satisfy

Consider

One can get that

Replacing for in (7) from of (13), the error dynamics on the sliding surface are determined by the following relation:

To satisfy the sliding condition, the discontinuous reaching law is chosen as follows: where and , are the gains of the controller.

In the sliding phase, it implies that . Considering (12) and (15), one gets

Now, the total control law can be defined as follows:

Replacing in (7) by (17), the error dynamics are determined by

Theorem 2 (see [24]). The following system is as follows: where , and . System (20) is asymptotically stable if  , where are the eigenvalues of matrix . Also, this system is stable if   and those critical eigenvalues that satisfy have geometric multiplicity one.

Theorem 3 (see [24]). Consider a system given by the following linear state space form with inner dimension as follows: where , , , and . Assuming that the triplet is minimal, then system (21) is stable if .

According to Theorem 2, the error dynamics on the sliding surface defined by (14) is asymptotically stable, as long as all eigenvalues of satisfy the condition . In the sliding phase, as a linear fractional-order system with bounded inputs ( for and for ), the error system (19) is stable if . It can be shown that choosing appropriate , , and can make the error dynamics stable; hence, the synchronization is realized.

4. Numerical Simulation

This section presents two illustrative examples to verify and demonstrate the effectiveness of the proposed control scheme. Case 1 is the synchronization between the same structure hyperchaotic systems. Case 2 is the synchronization between the different structure hyperchaotic systems.

Case 1. Synchronization between fractional-order and integer-order hyperchaotic Chen systems.
Consider Chen hyperchaotic system which is written as [25]
When , the system is integer-order system; otherwise we call the system (22) a fractional-order system.
Take the fractional-order system with fractional-order as a drive system, and the integer-order Chen hyperchaotic system as a response system with the following initial conditions: and , and the system parameters are .
The controller parameters are chosen as , , , and . This selection of parameters results in eigenvalues , which are located in the stable region. According to (18), the control inputs are taken as follows:
The simulation results are given in Figure 1. As we can see, the errors converge to zero which implies that synchronization between the two systems is realized.

151025.fig.001
Figure 1: Synchronization errors between Chen systems.

Case 2. Synchronization between integer-order hyperchaotic Chen system and fractional-order hyperchaotic Rössler system.
Consider hyperchaotic Rössler system which is written as [26]
Similarly, take the fractional-order Rössler hyperchaotic system with fractional-order as a drive system, and take the integer-order Chen hyperchaotic system as a response system with the following initial conditions: and , and the system parameters are .
We choose the design parameters in the simulations as , , , and . This selection of parameters results in eigenvalues, , which are located in the stable region. According to (19), we can yield the response system easily, that is,
The synchronization errors are shown in Figure 2, which show that the proposed method is succeeded in synchronizing the two different structure systems.

151025.fig.002
Figure 2: Synchronization between integer-order Chen system and fractional-order Rössler system.

5. Conclusion

In this paper, the problem of synchronization between fractional-order hyperchaotic systems and integer-order hyperchaotic systems is investigated. The integer-order hyperchaotic system is regarded as the response system. A sliding mode controller is designed to synchronize two systems with different orders successfully. It is rigorously proven that the proposed synchronization approach can be achieved between two different order hyperchaotic systems. Some numerical simulations are presented to show the applicability and feasibility of the proposed scheme.

Acknowledgments

This wok was supported by the “111” Project from the Ministry of Education of People’s Republic of China and the State Administration of Foreign Experts Affairs of People’s Republic of China (B12007) and the National Science and Technology Supporting Plan from the Ministry of Science and Technology of People’s Republic of China (2011BAD29B08).

References

  1. A. M. Harb and N. Abdel-Jabbar, “Controlling Hopf bifurcation and chaos in a small power system,” Chaos, Solitons & Fractals, vol. 18, no. 5, pp. 1055–1063, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 22, no. 1, pp. 64–69, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. Ma, Y. Jia, M. Yi, J. Tang, and Y. F. Xia, “Suppression of spiral wave and turbulence by using amplitude restriction of variable in a local square area,” Chaos, Solitons & Fractals, vol. 41, no. 3, pp. 1331–1339, 2009. View at Publisher · View at Google Scholar · View at Scopus
  4. W. X. Wang, L. Huang, Y. C. Lai, and G. Chen, “Onset of synchronization in weighted scale-free networks,” Chaos, vol. 19, no. 1, Article ID 013134, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. Z. Wei and Q. Yang, “Controlling the diffusionless Lorenz equations with periodic parametric perturbation,” Computers & Mathematics with Applications, vol. 58, no. 10, pp. 1979–1987, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. Liu and Q. Yang, “Dynamics of a new Lorenz-like chaotic system,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2563–2572, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D.-Y. Chen, W.-L. Zhao, X.-Y. Ma, and R.-F. Zhang, “Control and synchronization of chaos in RCL-shunted Josephson junction with noise disturbance using only one controller term,” Abstract and Applied Analysis, vol. 2012, Article ID 378457, 14 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. S. Hegazi and A. E. Matouk, “Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1938–1944, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. D. Y. Chen, Z. T. Sun, X. Y. Ma, and L. Chen, “Circuit implementation and model of a new multi-scroll chaotic system,” International Journal of Circuit Theory and Applications. In press. View at Publisher · View at Google Scholar
  10. J. Zhang, K. Zhang, J. Feng, and M. Small, “Rhythmic dynamics and synchronization via dimensionality reduction: application to human gait,” PLoS Computational Biology, vol. 6, no. 12, Article ID e1001033, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. D. Y. Chen, C. Wu, C. F. Liu, X.-Y. Ma, Y.-J. You, and R.-F. Zhang, “Synchronization and circuit simulation of a new double-wing chaos,” Nonlinear Dynamics, vol. 67, no. 2, pp. 1481–1504, 2012. View at Publisher · View at Google Scholar
  12. D.-Y. Chen, L. Shi, H.-T. Chen, and X.-Y. Ma, “Analysis and control of a hyperchaotic system with only one nonlinear term,” Nonlinear Dynamics, vol. 67, no. 3, pp. 1745–1752, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  13. D. Chen, R. Zhang, X. Ma, and S. Liu, “Chaotic synchronization and anti-synchronization for a novel class of multiple chaotic systems via a sliding mode control scheme,” Nonlinear Dynamics, vol. 69, no. 1-2, pp. 35–55, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. D.-Y. Chen, W.-L. Zhao, X.-Y. Ma, and R.-F. Zhang, “No-chattering sliding mode control chaos in Hindmarsh-Rose neurons with uncertain parameters,” Computers & Mathematics with Applications, vol. 61, no. 10, pp. 3161–3171, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. L. Pan, W. Zhou, L. Zhou, and K. Sun, “Chaos synchronization between two different fractional-order hyperchaotic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 6, pp. 2628–2640, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. H. Wang, Z. Z. Han, Q. Y. 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
  17. X. Chen and C. Liu, “Passive control on a unified chaotic system,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 683–687, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. P. Aghababa, S. Khanmohammadi, and G. Alizadeh, “Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique,” Applied Mathematical Modelling, vol. 35, no. 6, pp. 3080–3091, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. D.-Y. Chen, Y.-X. Liu, X.-Y. Ma, and R.-F. Zhang, “Control of a class of fractional-order chaotic systems via sliding mode,” Nonlinear Dynamics, vol. 67, no. 1, pp. 893–901, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Z. Ping, C. Yuan-Ming, and K. Fei, “Synchronization between fractional-order chaotic systems and integer orders chaotic systems (fractional-order chaotic systems),” Chinese Physics B, vol. 19, no. 9, Article ID 090503, 2010. View at Publisher · View at Google Scholar · View at Scopus
  21. D. Y. Chen, R. Zhang, J. C. Sprott, H. T. Chen, and X. Y. Ma, “Synchronization between integer-order chaotic systems and a class of fractional-order chaotic systems via sliding mode control,” Chaos, vol. 22, no. 2, Article ID 023130, 2012. View at Publisher · View at Google Scholar
  22. M. S. Tavazoei and M. Haeri, “Synchronization of chaotic fractional-order systems via active sliding mode controller,” Physica A, vol. 387, no. 1, pp. 57–70, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. S. Dadras and H. R. Momeni, “Control of a fractional-order economical system via sliding mode,” Physica A, vol. 389, no. 12, pp. 2434–2442, 2010. View at Publisher · View at Google Scholar · View at Scopus
  24. D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Proceedings of the IMACS-IEEE/SMC Multiconference on Computational Engineering in Systems and Applications, vol. 2, pp. 963–968, Lille, France, 1996.
  25. C. Li and G. Chen, “Chaos in the fractional order Chen system and its control,” Chaos, Solitons & Fractals, vol. 22, no. 3, pp. 549–554, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. K. Moaddy, I. Hashim, and S. Momani, “Non-standard finite difference schemes for solving fractional-order Rössler chaotic and hyperchaotic systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1068–1074, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet