This paper presents a new hyperchaotic system by introducing an additional state variable into Lorenz system. The system’s characteristics, including the dissipativity, equilibrium, and Lyapunov exponents, are studied. A controller is developed which consists of an active control term and a variable universe adaptive fuzzy system. By using this controller, the synchronization of the new hyperchaotic systems with uncertain linear part is accomplished according to Lyapunov’s direct method. Simulation results illustrate the effectiveness of the proposed method.

1. Introduction

Rössler first proposed the hyperchaotic system for a model of a chemical reaction in 1979 [1]. Hyperchaotic system is usually defined as a system with at least two positive Lyapunov exponents. The positive Lyapunov exponent means that a system is complex and unpredictable. It is believed that the adoption of hyperchaotic system improves the security of the communication scheme. Up to now, there are many references focusing on hyperchaotic systems, for example, hyperchaotic Chen system [2], hyperchaotic Lü system [3], and the hyperchaotic system proposed by X. Wang and M. Wang [4].

This paper presents a new hyperchaotic system by modifying the second equation of Lorenz system with a nonlinear feedback. The dissipativity, equilibrium, and Lyapunov exponents spectrum are studied. Simulation results show that the system has two positive Lyapunov exponents if proper parameters are given.

There are many efforts focused on the synchronization of hyperchaotic systems, and many various control methods have been proposed, such as optimal control, adaptive control, active control, and fuzzy control [511]. Fuzzy logic is a universal approximator and it has advantages in the aspect of handling uncertain problems. The idea of variable universe adaptive fuzzy control has been proposed by Li [1215] since 1995. Needing a few fuzzy rules with getting high control precision, variable universe fuzzy control method has been extensively used in the inverted pendulum [16, 17], aerospace vehicle [18], near space vehicle [19], and so forth.

In practical situations, the parameters of chaotic system are unknown and time-varying [7]. Here, we try to solve the synchronization of the new system with uncertain linear part (regarding as the generalization of uncertain parameters) by using a new controller, which is called an active variable universe adaptive fuzzy controller (AVUAFC). The controller consists of an active control term and a variable universe adaptive fuzzy system.

The structure of this paper is organized as follows. In Section 2, the design of the new hyperchaotic system is introduced and the characteristics of the new system is studied. In Section 3, the scheme of synchronization of nearly identical hyperchaotic systems is proposed with AVUAFC. In Section 4, the specific design of AVUAFC is given and the simulation results verify the effectiveness of the controller. Finally, the paper is concluded in Section 5.

2. The Design of the New Hyperchaotic System

Lorenz system is one of the paradigms of chaos capturing many features of chaotic systems. It is given by where , , and are state variables and , , and are parameters. In order to get hyperchaotic systems, there are three important requisites: the system has dissipative structure; the minimal dimension of the phase space that embeds a hyperchaotic attractor should be at least four; the number of terms in the coupled equations giving rise to instability should be at least two, of which at least one should have a nonlinear function.

In [4], X. Wang and M. Wang discovered a hyperchaotic system by adding a nonlinear controller to the first equation of Lorenz system. Here, by adding a state feedback to the second equation and changing the term into , a new four-dimensional system is constructed as follows: where the change rate of is ; , and are state variables; , , and are the parameters. Given proper parameters, one can get a hyperchaotic attractor. For example, when , , and , the Lyapunov exponents are , , , and , obviously, the system is a hyperchaotic system (see Figure 3(e)).

2.1. The Dissipativity and Equilibrium

The dissipativity of system (2) is described as .

So, when the parameters satisfy , the system is dissipative.

By solving the equilibrium equation of system (2), it can be observed that the system has unique equilibrium point .

By linearizing system (2) at , we get the Jacobian matrix

For is an upper triangular matrix, obviously, the eigenvlues are , , , and . When , , and , equilibrium is unstable. It is possible to generate chaos or hyperchaos in system (2).

2.2. Lyapunov Exponents and Bifurcation

Fixing parameters , , let vary in the interval . First of all, we know the system is dissipative with the parameters’ assumption. The bifurcation diagram in -direction and Lyapunov exponent spectrum is as follows (Figures 1 and 2).

Three Lyapunov exponents are no less than , while the smallest Lyapunov exponent is always less than . For clarity, we enlarge Figure 2(a) into Figure 2(b) by neglecting the smallest Lyapunov exponent.

From Figure 2, we observe that the system shows rich dynamical behaviors as varies. We can obtain the following.

When or or or or or or or , system (2) is periodic; when or , it is quasiperiodic; when or or or or , it is a chaotic attractor; when or , it is a hyperchaotic attractor.

To observe the orbits of system (2), we give some typical attractors of the system by selecting , , , , and . The phase portraits are shown in Figure 3.

3. Synchronization of the Hyperchaotic System with Linear Uncertainty

We investigate the synchronization of the hyperchaotic system. Suppose the drive system and the response system are written as follows:

In practical situations, these parameters are uncertain. Moreover, these parameters change from time to time. Suppose the linear parts are uncertain. In matrix manner, consider where , are the coefficient matrix of the known linear part of the system. Assume are the coefficient matrix of the unknown linear part of the system. Suppose , , where is 1-norm and , and are positive constants. are the nonlinear part of the system, is the controller added to the response system.

Our purpose is to make the state of system (5) and the state of (6) identical when , that is, ().

Suppose , , (), (6)–(5),

According to active control design strategy [6]; let , where is the variable universe adaptive fuzzy cotroller to be designed. Then, (9) becomes

For the purpose of stability, we add a compensating controller to (10), that is, where . , . is designed as follows: where , and are the contraction-expansion factors of and , respectively, and is the jth membership function of . Consider

In [12], Li demonstrated that fuzzy reasoning is equivalent to an interpolation. For the length of no longer than 1 and needing a few fuzzy roles, the design of is very simple.

Suppose the maximal eigenvalue of is , then let

We have where is a stable matrix. Obviously, the ideal controller of (16) is . By using (11), we have

Theorem 1. Choosing suitable , system (16) can be asymptotically stable.

Proof. Because is stable, for any positive definite symmetric matrix , there exists a positive definite symmetric matrix which satisfies the Lyapunov equations . Given energy function , our purpose is to demonstrate it by using the Lyapunov function. Its derivative with respect to is
Suppose , where () is the th column of . If then .
Otherwise, let
If (19) is satisfied, then Since , obviously, .
If (20), satisfied, let , , , where , , () are matrix elements of , , and . Then
If (20) can be satisfied, we have so .

The stability of (16) is obtained.

So, we get the conclusion that if , , are defined as above, system (5) can synchronize (6).

4. Simulation

Let , , , , , . The initial value of the drive system is and the initial value of the response system is .

From Figure 3(d), we know that , , , and . So , , , and . The universe of () is . For simplicity, we define the contraction-expression factors of () as .

We define 6 membership functions about which are taken as triangle waves (see Figure 4). Expressions are as follows: From (12), we know where , .

The membership functions about are the same as those of except for the subscripts. and take the place of and , respectively.

From (12), we know where , .

We also define 6 membership functions about which are taken as triangle waves (see Figure 5). Expressions are as follows:

From (12), we know where , .

We also define 6 membership functions about which are taken as triangle waves (see Figure 6). Expressions are as follows:

From (12), we know where , .

The compensating controller is defined by (19) and (20). With the controller , we get the simulation results in Figures 7, 8, 9, and 10.

From the simulation results, we find that the systems can be synchronized only less than 0.07 second.

5. Conclusion

We developed a new hyperchaotic system and a new controller (AVUAFC) which achieve the synchronization of the new system with uncertainty. By using Lyapunov direct method, the stability of the controlled system is demonstrated. Simulation results of complete synchronization verify the effectiveness of the proposed controller.


This work is supported by the National Natural Science Foundation of China (no. 61374118; no. 11101066) and the Youth Foundation of Qujing Normal University (2008QN034).