A New Hyperchaotic System and the Synchronization Using Active Variable Universe Adaptive Fuzzy Controller

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.


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 [5][6][7][8][9][10][11]. 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 [12][13][14][15] 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.

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  = 10,  = 28, and  = 8/3 are parameters.In order to get hyperchaotic systems, there are three important requisites: (1) the system has dissipative structure; (2) the minimal dimension of the phase space that embeds a hyperchaotic attractor should be at least four; (3) 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  = 26,  = 14, and  = 3.36, the Lyapunov exponents are  1 = 0.2406,  2 = 0.1243,  3 = 0, and  4 = −14.7253,obviously, the system is a hyperchaotic system (see Figure 3(e)).
By solving the equilibrium equation of system (2), it can be observed that the system has unique equilibrium point  = (0, 0, 0, 0).
By linearizing system (2) at , we get the Jacobian matrix For  is an upper triangular matrix, obviously, the eigenvlues are −, , −, and 1.When  > 0,  > 0, and  > 0, equilibrium  is unstable.It is possible to generate chaos or hyperchaos in system (2).

Lyapunov Exponents and Bifurcation.
Fixing parameters  = 26,  = 14, let  vary in the interval [1,10].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 −3, while the smallest Lyapunov exponent is always less than −10.For clarity, we enlarge Figure 2 From Figure 2, we observe that the system shows rich dynamical behaviors as  varies.We can obtain the following.
When To observe the orbits of system (2), we give some typical attractors of the system by selecting  = 1,  = 5.5,  = 1.15,  = 2.3, and  = 3.36.The phase portraits are shown in Figure 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 Δ 1 , Δ 2 ∈  4 × 4 are the coefficient matrix of the unknown linear part of the system.Suppose where ‖ ⋅ ‖ is 1-norm and  1 , and  2 are positive constants.ℎ(), ℎ() ∈  3 are the nonlinear part of the system, = ( 1 ,  2 ,  3 ,  4 )  is the controller added to the response system.
Our purpose is to make the state  of system (5) and the state  of (6 According to active control design strategy [6]; let  = −ℎ()+ℎ()+V fuzz , where V fuzz is the variable universe adaptive fuzzy cotroller to be designed.Then, (9) becomes For the purpose of stability, we add a compensating controller V  to (10), that is, where V fuzz is designed as follows: where   () and   (  ) are the contraction-expansion factors of   and   , respectively, and   (⋅) is the jth membership function of V fuzzi .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 V fuzz 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 V * =  − Δ 2  + Δ 1 .By using (11), we have   Theorem 1. Choosing suitable V  , system (16) can be asymptotically stable.
From  We define 6 membership functions about  1 which are taken as triangle waves (see Figure 4).Expressions are as follows: From ( 12), we know where β 1 () =  1  1 (),  1 (0) = 1.The membership functions about  2 are the same as those of  1 except for the subscripts. 2 and  2 take the place of  1 and  1 , respectively.
The tracking curve of the third state The tracking curve of the 4th state The compensating controller V  is defined by ( 19) and (20).With the controller  = −ℎ() + ℎ() + V fuzz + V  , 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.

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.
(a) into Figure 2(b) by neglecting the smallest Lyapunov exponent.

4
are the coefficient matrix of the known linear part of the system.Assume c = 5.5, quasiperiodic attractor (a) c = 1, periodic orbit