Mathematical Problems in Engineering

Volume 2013 (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

Baojie Zhang1,2 and Hongxing Li¹

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 [19–21], 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.

(a)

(b)

(c)

(d)

(e)

(f)

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.

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.

Figure 3: Lyapunov exponents , , .

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.

(a)

(b)

Figure 5: , periodic attractor.

(a)

(b)

Figure 6: , quasi-periodic attractor.

(a)

(b)

Figure 7: , chaotic attractor.

(a)

(b)

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.

Figure 9: The synchronization phase portrait of the new system and hyperchaotic Chen system with disturbance.

Figure 10: State errors , , , and .

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).

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