Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013, Article ID 179428, 8 pages
http://dx.doi.org/10.1155/2013/179428
Research Article

A New Four-Dimensional Autonomous Hyperchaotic System and the Synchronization of Different Chaotic Systems by Using Fast Terminal Sliding Mode Control

1Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China
2School of Mathematics and Information Science, Qujing Normal University, Qujing 655011, China

Received 24 September 2013; Revised 28 November 2013; Accepted 28 November 2013

Academic Editor: Tomasz Kapitaniak

Copyright © 2013 Baojie Zhang and Hongxing Li. 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 presents a new continuous-time four-dimensional autonomous system based on Lorenz system. We analyze the dissipation, equilibrium, and Lyapunov exponents of the system. Lyapunov exponent spectrum demonstrates that the system possesses rich dynamic behaviors if the parameters of the system vary. In a large range of parameters, the system is hyperchaotic. By using fast terminal sliding mode control method, the synchronization of two different chaotic systems is studied. Synchronization between the new system and hyperchaotic Chen system with noise perturbation is illustrated. Simulation results verify the effectiveness of the proposed method.

1. Introduction

Since Lorenz first discovered the chaotic attractor [1] in 1963, many new chaotic systems have been found, and there is a lot of interesting work on the study of chaotic systems, such as Rossler system [2], Genesio system [3], Chen system [4], L system [5, 6], Liu system [7], and Qi system [8]. These chaotic systems have unique positive Lyapunov exponent. Positive Lyapunov exponent is an important character of chaotic system, and it is a value that describes the sensitive dependence on initial conditions. More positive Lyapunov exponents mean that the system is of much more chaos. Rossler defined that a system with at least two positive Lyapunov exponents is hyperchaotic in 1979. Meanwhile, he presented the first hyperchaotic system—hyperchaotic Rossler system [9]. The dynamic behavior of hyperchaotic system is more complicated and unpredictable. With these characters, it is important to use hyperchaotic system in engineering, especially in secure communication. Over the past decades, many hyperchaotic systems have been introduced, such as hyperchaotic Chen system [10] by Chen and Dong, hyperchaotic Chen system [11] by Li et al., hyperchaotic L system by Chen et al. [12], hyperchaotic Nikolov system [13] by Nikolov and Clodong, hyperchaotic Lorenz system [14] by X. Wang and M. Wang, and hyperchaotic Lorenz system [15] by Barboza. This paper presents a new hyperchaotic system by adding a state variable and analyze the new system’s dissipation, the existence of the attractor, the stability of the equilibriums, and the dynamic behavior as a parameter varies.

Since Pecora and Carroll observed the synchronization of chaotic systems [16], there have been many efforts on the scheme of synchronization and a variety of approaches have been proposed for synchronization of chaotic systems, such as linear and nonlinear feedback synchronization [17, 18], adaptive synchronization [1921], active synchronization [22] and sliding mode synchronization [23, 24]. For the synchronization of different chaotic systems that exists extensively, it is more challenging to study the synchronization of two different chaotic systems than to research the identical ones. It is very useful in secure communication. Here we utilize terminal sliding mode method to synchronize two different chaotic systems. Synchronization of the new hyperchaotic system and the hyperchaotic Chen system illustrated the presented approach.

The structure of this paper is organized as follows. In Section 2, the design of the new hyperchaotic system is introduced. The dynamic behaviors are analyzed. In Section 3, the scheme of terminal sliding mode synchronization between two different chaotic systems with disturbances is proposed. In Section 4, numerical simulations of the synchronization between the new hyperchaotic system and the hyperchaotic Chen system verify the effectiveness of the scheme. Finally, conclusions are given in Section 5.

2. The Design of a New Hyperchaotic System

Lorenz system is a paradigm of chaos given by

where , , and are state variables and , , and are parameters. Here , , and . Adding a new state variable to the second equation with the change rate of being and adding to the third equation of Lorenz system, we get a system as follows:

When , , , and , the Lyapunov exponents are , , and . is close to . Obviously, with these parameters, the system is hyperchaotic. The phase portraits are shown in Figure 1.

fig1
Figure 1: The phase portrait of hyperchaotic system.
2.1. Dissipativity and Existence of Attractor

For the system (2), we have

If , the system is dissipative. When , , , , and , the system (2) is dissipative and it converges with the rate . That is to say, when , each volume containing a trajectory contracts to vacancy and the trajectory is confined in an attractor.

2.2. Equilibriums and Stability

By solving the equilibrium equation of system (2), we know that the system has three equilibria , , . By linearizing system (2), we obtain the Jacobian

From the Jacobian (4), we know that the eigenvalues at are , , and ; the eigenvalues at are , , , and ; the eigenvalues at are , , and . Because each equilibrium has at least an eigenvalue with positive real part, the three equilibriums are all unstable.

2.3. Bifurcation and Lyapunov Exponent Spectrum

Without loss of generality, we fix the parameters , , and . Let , , and . The bifurcation diagram in -direction during varies in the interval is shown in Figure 2.

179428.fig.002
Figure 2: Bifurcation diagram in -direction with varies.

The bifurcation diagrams in -direction, -direction, and -direction are almost the same as Figure 2. The system dynamic behavior can be defined by one component.

The Lyapunov exponents are shown in Figures 3 and 4.

179428.fig.003
Figure 3: Lyapunov exponents , , .
179428.fig.004
Figure 4: Lyapunov exponent .

In Figures 3 and 4, we observe that there are two positive Lyapunov exponents over a large range of parameters . Two positive Lyapunov exponents imply that system (2) is hyperchaotic. As we all know, the signs of Lyapunov exponents characterize different dynamic behaviors of the system. Specifically, when , , , , and , one Lyapunov exponent is zero and the other three are negative, and system (2) is periodic; when , , , two Lyapunov exponents are zero and the other two are negative, it is quasiperiodic; when , , , and , only one Lyapunov exponent is positive, it is chaotic; when , , and , two Lyapunov exponents are positive, it is hyperchaotic. To observe the phase portraits of system (2), we get four typical attractors of the system by selecting , , and . Their phase portraits are shown in Figures 5, 6, 7, and 8, respectively.

fig5
Figure 5: , periodic attractor.
fig6
Figure 6: , quasi-periodic attractor.
fig7
Figure 7: , chaotic attractor.
fig8
Figure 8: , hyperchaotic attractor.

3. Fast Terminal Sliding Synchronization between Two Different Chaotic Systems

Terminal sliding mode control is an effective approach for chaotic synchronization [25, 26]. In this section, we present the fast terminal sliding synchronization method to synchronize two different chaoses with disturbance. Let and be drive system and response system, respectively. are state variables. is the disturbance and is the controller. The state error , we choose a terminal sliding mode surface , where where , are diagonal matrixes. are parameters and .

Theorem 1. If and are bounded, with the above terminal sliding mode surface, one can synchronize two different chaoses with disturbance by using the terminal sliding mode controller , provided by
where , are diagonal matrices, is parameter, and .

Proof. First, we all know that our design target is to make . For the given terminal sliding mode surface ,
If we use the controller above, then,
And then we have
So with the given controller two different chaoses can be synchronized.

From the proof above, we know that switching item is two-valued; that is, it equals or and it always work. If we let it be a step function, that is,

or use saturation function. These schemes can attenuate the chattering of the sliding mode surface. Also we know the item guarantees that the state trajectory reaches the sliding mode surface in a short time.

4. Numerical Simulation

Here, we assume that the new system introduced in this paper is the drive system and hyperchaotic Chen system, is the response system: where , , , and are variables of hyperchaotic Chen system and , , , , and are parameters. Adding controllers and noise perturbations to the response system, we get where , , , and are controllers, , , , and are disturbances. The state errors are , , , and .

The sliding surface is where , , , , , , .

To demonstrate the effectiveness of the controller, we give the parameters and initial value of the drive system as , , , and . And the parameters and initial value of the response system are given as , , , , and while the disturbaces are chosen as , , . But the cotroller parameters are the following , , , , , , , , , , , , , , , , , , , , , , , , and . By the way, the diagrams of the synchronized states, the state errors and the controllers of the new system and hyperchaotic Chen system are presented as Figures 9, 10, and 11.

179428.fig.009
Figure 9: The synchronization phase portrait of the new system and hyperchaotic Chen system with disturbance.
179428.fig.0010
Figure 10: State errors , , , and .
179428.fig.0011
Figure 11: The controllers , , , .

From the above diagrams, we know that the precision synchronization of the two different systems with disturbance can be realized by less than seconds. Synchronization time can change by tuning parameters.

5. Conclusion

In this paper, we have introduced a new hyperchaotic system by adding a state variable and investigated the characters of the system. By using fast terminal sliding mode control scheme, the synchronization of two different chaotic systems is studied. Synchronization of the new hyperchaotic system and hyperchaotic Chen system with disturbance has illustrated effectiveness of the scheme.

Acknowledgments

This work is supported by National Natural Science Foundation (NNSF) of China under Grant no. 61374118 and Youth Foundation of Qujing Normal University (2008QN034).

References

  1. E. N. Lorenz, “Deterministic non-periodic flow,” Journal of the Atmospheric Sciences, vol. 20, pp. 130–141, 1963. View at Google Scholar
  2. O. E. Rossler, “An equation for continuous chaos,” Physics Letters A, vol. 57, no. 5, pp. 397–398, 1976. View at Publisher · View at Google Scholar
  3. R. Genesio and A. Tesi, “Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems,” Automatica, vol. 28, no. 3, pp. 531–548, 1992. View at Publisher · View at Google Scholar · View at Scopus
  4. G. Chen and T. Ueta, “Yet another chaotic attractor,” International Journal of Bifurcation and Chaos, vol. 9, no. 7, pp. 1465–1466, 1999. View at Publisher · View at Google Scholar · View at Scopus
  5. J. Lü and G. Chen, “A new chaotic attractor coined,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 3, pp. 659–661, 2002. View at Publisher · View at Google Scholar · View at Scopus
  6. G. R. Chen and J. H. Lü, Dynamics of the Lorenz System Family: Analysis, Control and Synchronization, Science Press, Beijing, China, 2003, Chinese.
  7. C. Liu, T. Liu, L. Liu, and K. Liu, “A new chaotic attractor,” Chaos, Solitons & Fractals, vol. 22, no. 5, pp. 1031–1038, 2004. View at Publisher · View at Google Scholar · View at Scopus
  8. G. Qi, S. Du, G. Chen, Z. Chen, and Z. Yuan, “On a four-dimensional chaotic system,” Chaos, Solitons & Fractals, vol. 23, no. 5, pp. 1671–1682, 2005. View at Publisher · View at Google Scholar · View at Scopus
  9. O. E. Rossler, “An equation for hyperchaos,” Physics Letters A, vol. 71, no. 2-3, pp. 155–157, 1979. View at Google Scholar · View at Scopus
  10. G. R. Chen and X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, Singapore, 1998.
  11. Y. Li, W. K. S. Tang, and G. 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
  12. A. Chen, J. Lu, J. Lü, and S. Yu, “Generating hyperchaotic Lü attractor via state feedback control,” Physica A, vol. 364, pp. 103–110, 2006. View at Publisher · View at Google Scholar · View at Scopus
  13. S. Nikolov and S. Clodong, “Occurrence of regular, chaotic and hyperchaotic behavior in a family of modified Rossler hyperchaotic systems,” Chaos, Solitons & Fractals, vol. 22, no. 2, pp. 407–431, 2004. View at Publisher · View at Google Scholar · View at Scopus
  14. X. Wang and M. Wang, “A hyperchaos generated from Lorenz system,” Physica A, vol. 387, no. 14, pp. 3751–3758, 2008. View at Publisher · View at Google Scholar · View at Scopus
  15. R. Barboza, “Dynamics of a hyperchaotic Lorenz system,” International Journal of Bifurcation and Chaos, vol. 17, no. 12, pp. 4285–4294, 2007. View at Publisher · View at Google Scholar · View at Scopus
  16. 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
  17. M.-C. Ho and Y.-C. Hung, “Synchronization of two different systems by using generalized active control,” Physics Letters A, vol. 301, no. 5-6, pp. 424–428, 2002. View at Publisher · View at Google Scholar · View at Scopus
  18. L. Huang, R. Feng, and M. Wang, “Synchronization of chaotic systems via nonlinear control,” Physics Letters A, vol. 320, no. 4, pp. 271–275, 2004. View at Publisher · View at Google Scholar · View at Scopus
  19. T.-L. Liao, “Adaptive synchronization of two Lorenz systems,” Chaos, Solitons & Fractals, vol. 9, no. 9, pp. 1555–1561, 1998. View at Google Scholar · View at Scopus
  20. S. Chen and J. Lü, “Synchronization of an uncertain unified chaotic system via adaptive control,” Chaos, Solitons & Fractals, vol. 14, no. 4, pp. 643–647, 2002. View at Google Scholar · View at Scopus
  21. Z. Li, C. Han, and S. Shi, “Modification for synchronization of Rossler and Chen chaotic systems,” Physics Letters A, vol. 301, no. 3-4, pp. 224–230, 2002. View at Publisher · View at Google Scholar · View at Scopus
  22. E.-W. Bai and K. E. Lonngren, “Sequential synchronization of two Lorenz systems using active control,” Chaos, Solitons & Fractals, vol. 11, no. 7, pp. 1041–1044, 2000. View at Publisher · View at Google Scholar · View at Scopus
  23. M. Yahyazadeh, A. Ranjbar Noei, and R. Ghaderi, “Synchronization of chaotic systems with known and unknown parameters using a modified active sliding mode control,” ISA Transactions, vol. 50, no. 2, pp. 262–267, 2011. View at Publisher · View at Google Scholar · View at Scopus
  24. H.-T. Yau, “Chaos synchronization of two uncertain chaotic nonlinear gyros using fuzzy sliding mode control,” Mechanical Systems and Signal Processing, vol. 22, no. 2, pp. 408–418, 2008. View at Publisher · View at Google Scholar · View at Scopus
  25. X. Yu and Y. Wu, “Adaptive terminal sliding mode control of uncertain nonlinear systems—backstepping approach,” Control Theory and Applications, vol. 15, no. 6, pp. 900–907, 1998. View at Google Scholar · View at Scopus
  26. N. N. Zhang, The Theory and Application of Terminal Sliding Mode Control, Science Press, Beijing, China, 2011.