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

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], Lü 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 Lü 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 𝑤 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 ̇,̇̇𝑥=𝑎−𝑥+𝑦𝑦=𝑏𝑥−𝑦−𝑥𝑧,𝑧=−𝑐𝑧+𝑥𝑦, (1)

where 𝑥, 𝑦, and 𝑧 are state variables and 𝑎, 𝑏, and 𝑐 are parameters. Here 𝑎=10, 𝑏=28, and 𝑐=8/3. Adding a new state variable 𝑤 to the second equation with the change rate of 𝑤 being 𝑑𝑤−𝑥𝑧 and adding 0.67𝑤 to the third equation of Lorenz system, we get a system as follows: ̇,̇̇̇𝑥=𝑎−𝑥+𝑦𝑦=𝑏𝑥−𝑦+𝑤−𝑥𝑧,𝑧=−𝑐𝑧+0.67𝑤+𝑥𝑦,𝑤=𝑑𝑤−𝑥𝑧. (2)

When 𝑎=31, 𝑏=54, 𝑐=13, and 𝑑=1.8, the Lyapunov exponents are LE₁=1.156884, LE₂=0.720394, LE₃=0.026702 and LE₄=−44.999849. LE₃=0.026702 is close to 0. 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 𝜕𝑥̇∇𝑉=𝑥+𝜕𝑦̇𝑦+𝜕𝑧̇𝑧+𝜕𝑤̇𝑤=−𝑎−1−𝑐+𝑑. (3)

If ∇𝑉=−𝑎−1−𝑐+𝑑<0, the system is dissipative. When 𝑎=31, 𝑏=54, 𝑐=13, 𝑑=1.8, and ∇𝑉=−43.2, the system (2) is dissipative and it converges with the rate 𝑒^−43.2𝑡. That is to say, when 𝑡→∞, 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 𝑂₁(0,0,0,0), 𝑂₂(−67.3912,−67.3912,119.25,−4464.6692), 𝑂₃(23.0037,23.0037,119.25,1523.9974). By linearizing system (2), we obtain the Jacobian 𝐴=−𝑎𝑎00𝑏−𝑧−1−𝑥1𝑦𝑥−𝑐0.67−𝑧0−𝑥𝑑. (4)

From the Jacobian (4), we know that the eigenvalues at 𝑂₁ are 𝜆_𝑂11=−59.5775,𝜆_𝑂12=27.5775, 𝜆_𝑂13=−13.0000, and 𝜆_𝑂14=1.8000; the eigenvalues at 𝑂₂ are 𝜆_𝑂21=0.2302+83.2288𝑖, 𝜆_𝑂22=0.2302−83.2288𝑖, 𝜆_𝑂23=−44.1544, and 𝜆_𝑂24=0.4939; the eigenvalues at 𝑂₃ are 𝜆_𝑂31=−9.7166+48.9668𝑖, 𝜆_𝑂32=−9.7166−48.9668𝑖, 𝜆_𝑂33=−24.6077 and 𝜆_𝑂34=0.8409. 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 𝑎, 𝑏, and 𝑐. Let 𝑎=31, 𝑏=54, and 𝑑=1.8. The bifurcation diagram in 𝑥-direction during 𝑐 varies in the interval [1,20] is shown in Figure 2.

Figure 2 

Bifurcation diagram in 𝑥-direction with 𝑐 varies.

The bifurcation diagrams in 𝑦-direction, 𝑧-direction, and 𝑤-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 LE1, LE2, LE3.

Figure 4 

Lyapunov exponent LE4.

In Figures 3 and 4, we observe that there are two positive Lyapunov exponents over a large range of parameters 𝑐. 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 𝑐∈[1,1.2864], 𝑐∈(4.0553,4.4372], 𝑐∈(6.5377,7.6834], 𝑐∈(8.4472,8.6382], and 𝑐∈(9.7839,10.6432], one Lyapunov exponent is zero and the other three are negative, and system (2) is periodic; when 𝑐∈(1.2864,1.7638], 𝑐∈(4.4372,4.6281], 𝑐∈(7.6834,7.8744], two Lyapunov exponents are zero and the other two are negative, it is quasiperiodic; when 𝑐∈(1.7638,2.2412], 𝑐∈(5.1055,6.5377], 𝑐∈(7.8744,8.4472], 𝑐∈(8.6382,9.2111] and 𝑐∈(15.8945,20], only one Lyapunov exponent is positive, it is chaotic; when 𝑐∈(2.2412,4.0553], 𝑐∈(4.6281,5.1055], 𝑐∈(9.2111,9.7839] and 𝑐∈(10.6432,15.8945], 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 𝑐=1.1, 𝑐=7.87, 𝑐=6 and 𝑐=2.8. Their phase portraits are shown in Figures 5, 6, 7, and 8, respectively.

(a)

(b)

Figure 5 

𝑐=1.1, periodic attractor.

(a)

(b)

Figure 6 

𝑐=7.87, quasi-periodic attractor.

(a)

(b)

Figure 7 

𝑐=6, chaotic attractor.

(a)

(b)

Figure 8 

𝑐=2.8, 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 𝑥̇=𝑓(𝑥) and 𝑦̇=𝑔(𝑦)+𝛿(𝑡)+𝑢 be drive system and response system, respectively. 𝑥,𝑦∈𝑅^𝑛 are state variables. 𝛿(𝑡)∈𝑅^𝑛 is the disturbance and 𝑢∈𝑅^𝑛 is the controller. The state error 𝑒=𝑦−𝑥, we choose a terminal sliding mode surface 𝑠=0, where 𝑠=𝑒̇+𝛼𝑒+𝛽𝑒^𝑞/𝑝, (5) where 𝑠∈𝑅^𝑛, 𝛼,𝛽 are diagonal matrixes. 𝑒^𝑞/𝑝=[𝑒^𝑞1/𝑝1₁,𝑒^𝑞2/𝑝2₂,…,𝑒^{𝑞𝑛/𝑝𝑛}_𝑛]^𝑇(𝑞_𝑖,𝑝_𝑖) are parameters and 𝑞_𝑖<𝑝_𝑖<2𝑞_𝑖.

Theorem 1. If 𝛿(𝑡) and 𝛿̇(𝑡) are bounded, with the above terminal sliding mode surface, one can synchronize two different chaoses with disturbance by using the terminal sliding mode controller 𝑢, provided by 𝑢̇+𝜕𝑔𝑞𝜕𝑒+𝛼+𝛽𝑝𝑒^{𝑞/𝑝−1}𝑢=−𝜕𝑔−𝜕𝑥𝜕𝑓̇𝜕𝑥𝑥−𝜕𝑔𝑞𝜕𝑒+𝛼+𝛽𝑝𝑒^{𝑞/𝑝−1}𝑔−𝑓−𝜖𝑠−2^−𝜂𝜔𝑠^2𝜂−1−𝐷sign(𝑠), (6)
where 𝜖, 𝜔 are diagonal matrices, 𝜂∈𝑅^𝑛 is parameter, and 𝐷>|(𝜕𝑔/𝜕𝑒+𝛼+𝛽(𝑞/𝑝)𝑒^{𝑞/𝑝−1})𝛿(𝑡)+𝑑𝛿(𝑡)/𝑑𝑡|.

Proof. First, we all know that our design target is to make 𝑠𝑠̇<0. For the given terminal sliding mode surface 𝑠, 𝑠̇=𝜕𝑔̇𝜕𝑥𝑥+𝜕𝑔̇(𝑡)𝜕𝑒𝑒+𝑑𝛿+̇̇𝑞𝑑𝑡𝑢+𝛼𝑒+𝛽𝑝𝑒^{𝑞/𝑝−1}𝑒̇−𝜕𝑓𝑥̇=𝜕𝑥𝜕𝑔−𝜕𝑥𝜕𝑓̇𝜕𝑥𝑥+𝜕𝑔𝑞𝜕𝑒+𝛼+𝛽𝑝𝑒^{𝑞/𝑝−1}+𝑔−𝑓𝜕𝑔𝑞𝜕𝑒+𝛼+𝛽𝑝𝑒^{𝑞/𝑝−1}𝛿(𝑡)+𝜕𝑔𝑞𝜕𝑒+𝛼+𝛽𝑝𝑒^{𝑞/𝑝−1}𝑢+𝑑𝛿(𝑡)+̇𝑑𝑡𝑢. (7)
If we use the controller above, then, 𝑠̇=𝜕𝑔𝑞𝜕𝑒+𝛼+𝛽𝑝𝑒^{𝑞/𝑝−1}𝛿(𝑡)+𝑑𝛿(𝑡)𝑑𝑡−𝜖𝑠−2^−𝜂𝜔𝑠^2𝜂−1−𝐷sign(𝑠). (8)
And then we have 𝑠𝑠̇=𝜕𝑔𝑞𝜕𝑒+𝛼+𝛽𝑝𝑒^{𝑞/𝑝−1}𝛿(𝑡)+𝑑𝛿(𝑡)𝑑𝑡𝑠−𝜖𝑠²−2^−𝜂𝜔𝑠^2𝜂−𝐷|𝑠|<−𝜖𝑠²−2^−𝜂𝜔𝑠^2𝜂−𝐷|𝑠|. (9)
So with the given controller two different chaoses can be synchronized.

From the proof above, we know that switching item 𝐷sign(𝑠) is two-valued; that is, it equals 𝐷 or −𝐷 and it always work. If we let it be a step function, that is, 𝐷^∗=𝐷apositiveconstant,if𝜕𝑔𝑞𝜕𝑒+𝛼+𝛽𝑝𝑒^{𝑞/𝑝−1}𝛿(𝑡)+𝑑𝛿(𝑡)𝑑𝑡𝑠−𝜖𝑠²−2^−𝜂𝜔𝑠^2𝜂>00,otherwise, (10)

or use saturation function. These schemes can attenuate the chattering of the sliding mode surface. Also we know the item −𝜖𝑠−2^−𝜂𝜔𝑠^2𝜂−1 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: 𝑥̇₁=𝑎₁−𝑥₁+𝑦₁+𝑤₁,𝑦̇₁=𝑏₁𝑥₁−𝑥₁𝑧₁+𝑐₁𝑦₁,𝑧̇₁=𝑥₁𝑦₁−𝑑₁𝑧₁,𝑤̇₁=𝑦₁𝑧₁+𝑛₁𝑤₁, (11) where 𝑥₁, 𝑦₁, 𝑧₁, and 𝑤₁ are variables of hyperchaotic Chen system and 𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁, and 𝑛₁ are parameters. Adding controllers and noise perturbations to the response system, we get 𝑥̇₁=𝑎₁−𝑥₁+𝑦₁+𝑤₁+𝑑₁+𝑢₁,𝑦̇₁=𝑏₁𝑥₁−𝑥₁𝑧₁+𝑐₁𝑦₁+𝑑₂+𝑢₂,𝑧̇₁=𝑥₁𝑦₁−𝑑₁𝑧₁+𝑑₃+𝑢₃,𝑤̇₁=𝑦₁𝑧₁+𝑛₁𝑤₁+𝑑₄+𝑢₄, (12) where 𝑢₁, 𝑢₂, 𝑢₃, and 𝑢₄ are controllers, 𝑑₁, 𝑑₂, 𝑑₃, and 𝑑₄ are disturbances. The state errors are 𝑒₁=𝑥₁−𝑥, 𝑒₂=𝑦₁−𝑦, 𝑒₃=𝑧₁−𝑧, and 𝑒₄=𝑤₁−𝑤.

The sliding surface is 𝑠=𝑒̇+𝛼𝑒+𝛽𝑒^𝑞/𝑝, (13) where 𝑠=[𝑠₁,𝑠₂,𝑠₃,𝑠₄]^𝑇, 𝑒̇=[𝑒₁̇,𝑒₂̇,𝑒₃̇,𝑒₄̇]^𝑇𝑒=[𝑒₁,𝑒₂,𝑒₃,𝑒₄]^𝑇, 𝛼=diag(𝛼₁,𝛼₂,𝛼₃,𝛼₄), 𝛽=diag(𝛽₁,𝛽₂,𝛽₃,𝛽₄), 𝑒^𝑞/𝑝=[𝑒^𝑞1/𝑝1₁,𝑒^𝑞2/𝑝2₂,𝑒^𝑞3/𝑝3₃,𝑒^𝑞4/𝑝4₄]^𝑇, 𝑝_𝑖<𝑞_𝑖<2𝑝_𝑖, 1≤𝑖≤4.

To demonstrate the effectiveness of the controller, we give the parameters and initial value of the drive system as 𝑎=31, 𝑏=54, 𝑐=13, 𝑑=1.8 and 𝑥₀=[−60,40,8,10]. And the parameters and initial value of the response system are given as 𝑎₁=35, 𝑏₁=7, 𝑐₁=12, 𝑑₁=3, 𝑛₁=0.7 and 𝑥^∗₀=[20,−80,3,2] while the disturbaces are chosen as 𝛿₁=0.025sin(𝑡), 𝛿₂=0.05cos(𝑡), 𝛿₃=𝛿₄=0.5. But the cotroller parameters are the following 𝛼₁=50, 𝛼₂=48, 𝛼₃=50, 𝛼₄=51, 𝛽₁=50, 𝛽₂=52, 𝛽₃=53, 𝛽₄=62, 𝑞₁=3, 𝑞₂=3, 𝑝₂=5, 𝑝₄=5, 𝑝₃=5, 𝑝₁=5, 𝑞₄=3, 𝑞₃=3, 𝜀₁=30, 𝜀₂=30, 𝜀₃=30, 𝜀₄=15, 𝜔₁=10, 𝜔₂=10, 𝜔₃=10, 𝜔₄=7, 𝜂=2 and 𝐷=1. 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 𝑒1, 𝑒2, 𝑒3, and 𝑒4.

Figure 11 

The controllers 𝑢1, 𝑢2, 𝑢3, 𝑢4.

From the above diagrams, we know that the precision synchronization of the two different systems with disturbance can be realized by less than 0.1 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 𝑤 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).