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

Abstract

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 [5–11]. 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 [12–15] 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.

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 ̇,̇̇𝑥=𝑎𝑦−𝑥𝑦=𝑏𝑥−𝑦−𝑥𝑧,𝑧=−𝑐𝑧+𝑥𝑦, (1) where 𝑥, 𝑦, and 𝑧 are state variables and 𝑎=10, 𝑏=28, and 𝑐=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 𝑤 to the second equation and changing the term 𝑏𝑥−𝑦 into 𝑏𝑦, a new four-dimensional system is constructed as follows: ̇,̇̇̇𝑥=𝑎𝑦−𝑥𝑦=−𝑥𝑧+𝑏𝑦+𝑤,𝑧=𝑥𝑦−𝑐𝑧,𝑤=−𝑥𝑧+𝑤, (2) where the change rate of 𝑤 is −𝑥𝑧+𝑤; 𝑥,𝑦,𝑧, and 𝑤 are state variables; 𝑎, 𝑏, and 𝑐 are the parameters. Given proper parameters, one can get a hyperchaotic attractor. For example, when 𝑎=26, 𝑏=14, and 𝑐=3.36, the Lyapunov exponents are 𝜆₁=0.2406, 𝜆₂=0.1243, 𝜆₃=0, and 𝜆₄=−14.7253, 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 ∇𝑉=𝜕𝑥̇/𝜕𝑥+𝜕𝑦/̇𝜕𝑦+𝜕𝑧/̇𝜕𝑧+𝜕𝑤̇/𝜕𝑤=−𝑎+𝑏−𝑐+1.

So, when the parameters satisfy −𝑎+𝑏−𝑐+1<0, the system is dissipative.

By solving the equilibrium equation of system (2), it can be observed that the system has unique equilibrium point 𝑀=(0,0,0,0).

By linearizing system (2) at 𝑀, we get the Jacobian matrix 𝐽=−𝑎𝑎000𝑏0100−𝑐00001. (3)

For 𝐽 is an upper triangular matrix, obviously, the eigenvlues are −𝑎, 𝑏, −𝑐, and 1. When 𝑎>0, 𝑏>0, and 𝑐>0, equilibrium 𝑀 is unstable. It is possible to generate chaos or hyperchaos in system (2).

2.2. Lyapunov Exponents and Bifurcation

Fixing parameters 𝑎=26, 𝑏=14, let 𝑐 vary in the interval [1,10]. First of all, we know the system is dissipative with the parameters’ assumption. The bifurcation diagram in 𝑥-direction and Lyapunov exponent spectrum is as follows (Figures 1 and 2).

Figure 1 

Bifurcation diagram in 𝑥-direction as 𝑐 varies.

(a)

(b)

Figure 2 

(a) Lyapunov exponent spectrum as 𝑐 varies. (b) Lyapunov exponent spectrum as 𝑐 varies.

 

Figure 3 

Phase portraits of system (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(a) into Figure 2(b) by neglecting the smallest Lyapunov exponent.

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

When 1≤𝑐<1.08 or 1.29≤𝑐<1.43 or 1.48≤𝑐<1.55 or 1.68≤𝑐<1.71 or 1.81≤𝑐<1.84 or 3.78≤𝑐<4.08 or 4.76≤𝑐<4.867 or 6.08≤𝑐<10, system (2) is periodic; when 4.08≤𝑐<4.76 or 4.867≤𝑐<6.08, it is quasiperiodic; when 1.08≤𝑐<1.29 or 1.55≤𝑐<1.59 or 1.84≤𝑐<2.20 or 2.35≤𝑐<3.29 or 3.52≤𝑐<3.78, it is a chaotic attractor; when 2.20≤𝑐<2.35 or 3.29≤𝑐<3.52, it is a hyperchaotic attractor.

To observe the orbits of system (2), we give some typical attractors of the system by selecting 𝑐=1, 𝑐=5.5, 𝑐=1.15, 𝑐=2.3, and 𝑐=3.36. 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: 𝑥̇₁=𝑎𝑥₂−𝑥₁,𝑥̇₂=−𝑥₁𝑥₃+𝑏𝑥₂+𝑥₄,𝑥̇₃=𝑥₁𝑥₂−𝑐𝑥₃,𝑥̇₄=−𝑥₁𝑥₃+𝑥₄,𝑦̇₁=𝑎𝑦₂−𝑦₁+𝑢₁,𝑦̇₂=−𝑦₁𝑦₃+𝑏𝑦₂+𝑦₄+𝑢₂,𝑦̇₃=𝑦₁𝑦₂−𝑐𝑦₃+𝑢₃,𝑦̇₄=−𝑦₁𝑦₃+𝑦₄+𝑢₄. (4)

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 𝑥̇=𝐴+Δ𝐴₁(𝑥),̇𝑥+ℎ𝑦=𝐴+Δ𝐴₂𝑦+ℎ𝑦+𝑢, (5)(6) where 𝑥=(𝑥₁,𝑥₂,𝑥₃,𝑥₄)^𝑇, 𝑦=(𝑦₁,𝑦₂,𝑦₃,𝑦₄)^𝑇∈𝑅^𝑛𝐴∈𝑅^4×4 are the coefficient matrix of the known linear part of the system. Assume 𝐴=−𝑎𝑎000𝑏0100−𝑐0000𝑑. (7)Δ𝐴₁,Δ𝐴₂∈𝑅^4×4 are the coefficient matrix of the unknown linear part of the system. Suppose ‖Δ𝐴₁‖≤𝐿₁, ‖Δ𝐴₂‖≤𝐿₂, where ‖⋅‖ is 1-norm and 𝐿₁, and 𝐿₂ are positive constants. ℎ(𝑥),ℎ(𝑦)∈𝑅³ are the nonlinear part of the system, ℎ(𝑥)=0−𝑥₁𝑥₃𝑥₁𝑥₂−𝑥₁𝑥₃𝑦=0,ℎ−𝑦₁𝑦₃𝑦₁𝑦₂−𝑦₁𝑦₃. (8)𝑢=(𝑢₁,𝑢₂,𝑢₃,𝑢₄)^𝑇 is the controller added to the response system.

Our purpose is to make the state 𝑥 of system (5) and the state 𝑦 of (6) identical when 𝑡→∞, that is, lim_𝑡→∞|𝑦_𝑖−𝑥_𝑖|=0 (𝑖=1,2,3,4).

Suppose 𝑒=(𝑒₁,𝑒₂,𝑒₃,𝑒₄)^𝑇, 𝑒_𝑖=𝑦_𝑖−𝑥_𝑖, (𝑖=1,2,3,4), (6)–(5), 𝑒̇=𝐴𝑒+Δ𝐴₂𝑦−Δ𝐴₁𝑥+ℎ𝑦−ℎ(𝑥)+𝑢. (9)

According to active control design strategy [6]; let 𝑢=−ℎ(𝑦)+ℎ(𝑥)+𝑣_fuzz, where 𝑣_fuzz is the variable universe adaptive fuzzy cotroller to be designed. Then, (9) becomes 𝑒̇=𝐴𝑒+Δ𝐴₂𝑦−Δ𝐴₁𝑥+𝑣_fuzz. (10)

For the purpose of stability, we add a compensating controller 𝑣_𝑠 to (10), that is, 𝑒̇=𝐴𝑒+Δ𝐴₂𝑦−Δ𝐴₁𝑥+𝑣, (11) where 𝑣=𝑣_fuzz+𝑣_𝑠. 𝑣_fuzz=(𝑣_fuzz1,𝑣_fuzz2,𝑣_fuzz3,𝑣_fuzz4)^𝑇, 𝑣_𝑠=(𝑣_𝑠1,𝑣_𝑠2,𝑣_𝑠3,𝑣_𝑠4)^𝑇. 𝑣_fuzz is designed as follows:𝑣_fuzz≜𝑈_𝑖𝛽_𝑖(𝑡)^𝑚_𝑗=1𝐴_𝑖𝑗𝑒_𝑖𝛼_𝑖𝑒_𝑖𝑦_𝑖𝑗,(𝑖=1,2,3,4), (12) where 𝛽_𝑖(𝑡), and 𝛼_𝑖(𝑒_𝑖) are the contraction-expansion factors of 𝑦_𝑖𝑗 and 𝑒_𝑖, respectively, and 𝐴_𝑖𝑗(⋅) is the jth membership function of 𝑣_fuzz𝑖. Consider 𝛽̇_𝑖(𝑡)=𝑘_𝑖𝑒_𝑖(𝑖=1,2,3,4),𝑘_𝑖𝛼isconstant,_𝑖(𝑡)=1−𝑙_𝑖exp−𝛿_𝑖𝑒²_𝑖(𝑖=1,2,3,4). (13)(14)

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

Suppose the maximal eigenvalue of 𝐴 is 𝜆, then let ()()𝐾=diag0,0,0,0,𝜆<0diag−𝜆−1,−𝜆−1,−𝜆−1,−𝜆−1,𝜆≥0. (15)

We have 𝑒̇=𝐵𝑒−𝐾𝑒+Δ𝐴₂𝑦−Δ𝐴₁𝑥+𝑣, (16) where 𝐵=𝐴+𝐾 is a stable matrix. Obviously, the ideal controller of (16) is 𝑣^∗=𝐾𝑒−Δ𝐴₂𝑦+Δ𝐴₁𝑥. By using (11), we have 𝑒̇=𝐵𝑒−𝑣^∗+𝑣. (17)

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

Proof. Because 𝐵 is stable, for any positive definite symmetric matrix 𝑄, there exists a positive definite symmetric matrix 𝑃 which satisfies the Lyapunov equations 𝐵^𝑇𝑃+𝑃𝐵=−𝑄. Given energy function 𝑉(𝑒)=(1/2)𝑒^𝑇𝐵𝑒, our purpose is to demonstrate it by using the Lyapunov function. Its derivative with respect to 𝑡 is 𝑉̇=−12𝑒^𝑇𝑄𝑒+𝑒^𝑇𝑃−𝑣^∗+𝑣_fuzz+𝑣_𝑠1=−2𝑒^𝑇𝑄𝑒+𝑒^𝑇𝑃−𝐾𝑒+Δ𝐴₂𝑦−Δ𝐴₁𝑥+𝑣_fuzz+𝑣_𝑠. (18)
Suppose 𝑃=(𝑃₁,𝑃₂,𝑃₃,𝑃₄), where 𝑃_𝑖 (𝑖=1,2,3,4) is the 𝑖th column of 𝑃. If −𝑒^𝑇𝑃𝐾𝑒+𝐿₂𝑒^𝑇𝑃𝑦+𝐿₁𝑒^𝑇𝑃𝑥+𝑒^𝑇𝑃𝑣_fuzz≤12𝑒^𝑇𝑄𝑒 (19) then 𝑣_𝑠=0.
Otherwise, let 𝑣_𝑠𝑖=−sgn𝑒^𝑇𝑃_𝑖𝐿₂𝑦_𝑖+𝐿₁⋅𝑥_𝑖−𝑣_fuzz𝑖+𝑘_𝑖𝑖𝑒_𝑖. (20)
If (19) is satisfied, then 𝑉̇(𝑒)=−12𝑒^𝑇𝑄𝑒+𝑒^𝑇𝑃−𝐾𝑒+Δ𝐴₂𝑦−Δ𝐴₁𝑥+𝑣_fuzz+𝑣_𝑠1≤−2𝑒^𝑇𝑄𝑒−𝑒^𝑇𝑃𝐾𝑒+𝐿₂𝑒^𝑇𝑃𝑦+𝐿₁𝑒^𝑇𝑃𝑥+𝑒^𝑇𝑃𝑣_fuzz+𝑒^𝑇𝑃𝑣_𝑠. (21) Since 𝑣_𝑠=0, obviously, 𝑉̇(𝑒)≤0.
If (20), satisfied, let 𝐾=(𝑘_𝑖𝑗), Δ𝐴₁=(Δ𝑎_𝑖𝑗), Δ𝐴₂=(Δ𝑏_𝑖𝑗), where 𝑘_𝑖𝑗, Δ𝑎_𝑖𝑗, Δ𝑏_𝑖𝑗 (𝑖,𝑗=1,2,…,𝑛) are matrix elements of 𝐾, Δ𝐴, and Δ𝐵. Then 𝑒^𝑇𝑃−𝐾𝑒+Δ𝐴₂𝑦−Δ𝐴₁𝑥+𝑣_fuzz+𝑣_𝑠=𝑒^𝑇𝑃₁,𝑒^𝑇𝑃₂,𝑒^𝑇𝑃₃,𝑒^𝑇𝑃₄×−𝑘₁₁𝑒₁+⁴_𝑖=1Δ𝑏_𝑖1𝑦₁+⁴_𝑖=1Δ𝑎_𝑖1𝑥₁+𝑣_fuzz1+𝑣_𝑠1−𝑘₂₂𝑒₂+⁴_𝑖=1Δ𝑏_𝑖2𝑦₂+⁴_𝑖=1Δ𝑎_𝑖2𝑥₂+𝑣_fuzz2+𝑣_𝑠2−𝑘₃₃𝑒₃+⁴_𝑖=1Δ𝑏_𝑖3𝑦₃+⁴_𝑖=1Δ𝑎_𝑖3𝑥₃+𝑣_fuzz3+𝑣_𝑠3−𝑘₄₄𝑒₄+⁴_𝑖=1Δ𝑏_𝑖4𝑦₄+⁴_𝑖=1Δ𝑎_𝑖4𝑥₄+𝑣_fuzz4+𝑣_𝑠4≤𝑒^𝑇𝑃₁,𝑒^𝑇𝑃₂,𝑒^𝑇𝑃₃,𝑒^𝑇𝑃₄×𝐿₂𝑦₁+𝐿₁𝑥₁𝐿₂𝑦₂+𝐿₁𝑥₂𝐿₂𝑦₃+𝐿₁𝑥₃𝐿₂𝑦₄+𝐿₁𝑥₄+𝑒^𝑇𝑃₁,𝑒^𝑇𝑃₂,𝑒^𝑇𝑃₃,𝑒^𝑇𝑃₄×−𝑘₁₁𝑒₁+𝑣_fuzz1+𝑣_𝑠1−𝑘₂₂𝑒₂+𝑣_fuzz2+𝑣_𝑠2−𝑘₃₃𝑒₃+𝑣_fuzz3+𝑣_𝑠3−𝑘₄₄𝑒₄+𝑣_fuzz4+𝑣_𝑠4. (22)
If (20) can be satisfied, we have 𝑒^𝑇𝑃−𝐾𝑒+Δ𝐵𝑦−𝛼Δ𝐴𝑥+𝑣_fuzz+𝑣_𝑠≤0, (23) so 𝑉̇(𝑒)≤−(1/2)𝑒^𝑇𝑄𝑒≤0.

The stability of (16) is obtained.

So, we get the conclusion that if 𝑢=−ℎ(𝑦)+ℎ(𝑥)+𝑣_fuzz+𝑣_𝑠, 𝑣_fuzz, 𝑣_𝑠 are defined as above, system (5) can synchronize (6).

4. Simulation

Let 𝑎=26, 𝑏=14, 𝑐=2.3, 𝐿1=𝐿2=0.3, Δ𝐴₁=,0.090.010.210.020.070.060.04−0.05−0.050.02−0.110.030.060.050.030.07Δ𝐴₂=,−0.020.050.110.070.070.050.060.02−0.080.030.010.040.070.11−0.05−0.03 (24)𝐾=diag(0,−15,0,−2), 𝑄=diag(10,10,10,10). The initial value of the drive system is (1,2,3,1)^𝑇 and the initial value of the response system is (12,13,14,15)^𝑇.

From Figure 3(d), we know that 𝑥∈[−20,20], 𝑦∈[−20,20], 𝑧∈[0,30], and 𝑤∈[−60,60]. So 𝑒₁∈[−40,40], 𝑒₂∈[−40,40], 𝑒₃∈[−60,60], and 𝑒₄∈[−120,120]. The universe of 𝑣_fuzz𝑖 (𝑖=1,2,3,4) is [−1,1]. For simplicity, we define the contraction-expression factors of 𝑒_𝑖 (𝑖=1,2,3,4) as 𝛼(𝑒_𝑖(𝑡))=1−0.995exp(−0.08𝑒_𝑖²(𝑡)).

We define 6 membership functions about 𝑒₁ which are taken as triangle waves (see Figure 4). Expressions are as follows: 𝐴₁₁𝑒₁=1,𝑒₁𝑒≤−40,₁+20−20,−40≤𝑒₁𝐴≤−20,0,else;₁₂𝑒₁=𝑒₁+4020,−40≤𝑒₁𝑒≤−20,₁+10−10,−20≤𝑒₁𝐴≤−10,0,else;₁₃𝑒₁=𝑒₁+2010,−20≤𝑒₁𝑒≤−10,₁−10,−10≤𝑒₁𝐴≤0,0,else;₁₄𝑒₁=𝑒₁−20−10,10≤𝑒₁𝑒≤20,₁10,0≤𝑒₁𝐴≤10,0,else;₁₅𝑒₁=𝑒₁−40−20,20≤𝑒₁𝑒≤40,₁−1010,10≤𝑒₁𝐴≤20,0,else;₁₆𝑒₁=1,𝑒₁𝑒≥40,₁−2020,20≤𝑒₁≤40,0,else. (25) From (12), we know 𝑣_fuzz1=𝑈₁𝛽₁(𝑡)−𝐴₁₁𝑒₁(𝑡)𝛼₁𝑒₁(𝑡)−0.5𝐴₁₂𝑒₁(𝑡)𝛼₁𝑒₁(𝑡)+0.25×𝐴₁₄𝑒₁(𝑡)𝛼₁𝑒₁(𝑡)+0.5𝐴₁₅𝑒₁(𝑡)𝛼₁𝑒₁(𝑡)+𝐴₁₆𝑒₁(𝑡)𝛼₁𝑒₁(𝑡), (26) where 𝛽̇₁(𝑡)=𝑘₁𝑒₁(𝑡), 𝛽₁(0)=1.

Figure 4 

The membership functions about 𝑒₂ are the same as those of 𝑒₁ except for the subscripts. 𝐴_2𝑖 and 𝑒₂ take the place of 𝐴_1𝑖 and 𝑒₁, respectively.

From (12), we know 𝑣_fuzz2=𝑈₂𝛽₂(𝑡)−𝐴₂₁𝑒₂(𝑡)𝛼₂𝑒₂(𝑡)−0.5𝐴₂₂𝑒₂(𝑡)𝛼₂𝑒₂(𝑡)−0.25𝐴₂₃𝑒₂(𝑡)𝛼₂𝑒₂(𝑡)+0.25𝐴₂₄𝑒₂(𝑡)𝛼₂𝑒₂(𝑡)+0.5𝐴₂₅𝑒₂(𝑡)𝛼₂𝑒₂(𝑡)+𝐴₂₆𝑒₂(𝑡)𝛼₂𝑒₂(𝑡), (27) where 𝛽̇₂(𝑡)=𝑘₂𝑒₂(𝑡), 𝛽₂(0)=1.

We also define 6 membership functions about 𝑒₃ which are taken as triangle waves (see Figure 5). Expressions are as follows: 𝐴₃₁𝑒₃=1,𝑒₃𝑒≤−60,₃+30−30,−60≤𝑒₃𝐴≤−30,0,else;₃₂𝑒₁=𝑒₃+6030,−60≤𝑒₃𝑒≤−30,₃+15−15,−30≤𝑒₃𝐴≤−15,0,else;₃₃𝑒₃=𝑒₃+3015,−30≤𝑒₃𝑒≤−15,₃−15,−15≤𝑒₃𝐴≤0,0,else;₃₄𝑒₃=𝑒₃−30−15,15≤𝑒₃𝑒≤30,₃15,0≤𝑒₃𝐴≤15,0,else;₃₅𝑒₃=𝑒₃−60−30,30≤𝑒₃𝑒≤60,₃−1515,15≤𝑒₃𝐴≤30,0,else;₃₆𝑒₃=1,𝑒₃𝑒≥60,₃−3030,30≤𝑒₃≤60,0,else. (28)

Figure 5 

From (12), we know 𝑣_fuzz3=𝑈₃𝛽₃(𝑡)−𝐴₃₁𝑒₃(𝑡)𝛼₃𝑒₃(𝑡)−0.5𝐴₃₂𝑒₃(𝑡)𝛼₃𝑒₃(𝑡)−0.25𝐴₃₃𝑒₃(𝑡)𝛼₃𝑒₃(𝑡)+0.25𝐴₃₄𝑒₃(𝑡)𝛼₃𝑒₃(𝑡)+0.5𝐴₃₅𝑒₃(𝑡)𝛼₃𝑒₃(𝑡)+𝐴₃₆𝑒₃(𝑡)𝛼₃𝑒₃(𝑡), (29) where 𝛽̇₃(𝑡)=𝑘₃𝑒₃(𝑡), 𝛽₃(0)=1.

We also define 6 membership functions about 𝑒₄ which are taken as triangle waves (see Figure 6). Expressions are as follows: 𝐴₄₁𝑒₄=1,𝑒₄𝑒≤−120,₄+60−60,−120≤𝑒₄𝐴≤−60,0,else;₄₂𝑒₄=𝑒₄+12060,−120≤𝑒₄𝑒≤−60,₄+30−30,−60≤𝑒₄𝐴≤−30,0,else;₄₃𝑒₄=𝑒₄+6030,−60≤𝑒₄𝑒≤−30,₄−30,−30≤𝑒₄𝐴≤0,0,else;₄₄𝑒₄=𝑒₄−60−30,30≤𝑒₄𝑒≤60,₃30,0≤𝑒₄𝐴≤30,0,else;₄₅𝑒₄=𝑒₄−120−60,60≤𝑒₄𝑒≤120,₄−3030,30≤𝑒₄𝐴≤60,0,else;₄₆𝑒₄=1,𝑒₄𝑒≥120,₄−6060,60≤𝑒₄≤120,0,else. (30)

Figure 6 

From (12), we know 𝑣_fuzz4=𝑈₄𝛽₄(𝑡)−𝐴₄₁𝑒₄(𝑡)𝛼₄𝑒₄(𝑡)−0.5𝐴₄₂𝑒₄(𝑡)𝛼₄𝑒₄(𝑡)−0.25𝐴₄₃𝑒₄(𝑡)𝛼₄𝑒₄(𝑡)+0.25𝐴₄₄𝑒₄(𝑡)𝛼₄𝑒₄(𝑡)+0.5𝐴₄₅𝑒₄(𝑡)𝛼₄𝑒₄(𝑡)+𝐴₄₆𝑒₄(𝑡)𝛼₄𝑒₄(𝑡), (31) where 𝛽̇₄(𝑡)=𝑘₄𝑒₄(𝑡), 𝛽₄(0)=1.

The compensating controller 𝑣_𝑠 is defined by (19) and (20). With the controller 𝑢=−ℎ(𝑦)+ℎ(𝑥)+𝑣_fuzz+𝑣_𝑠, we get the simulation results in Figures 7, 8, 9, and 10.

Figure 7 

Figure 8 

Figure 9 

Figure 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.

Acknowledgments

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).