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

Authors: Baojie Zhang; Hongxing Li; Haigang Guo
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 Section2, 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(1)x˙=a(y-x),y˙=bx-y-xz,z˙=-cz+xy,
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:
(2)x˙=a(y-x),y˙=-xz+by+w,z˙=xy-cz,w˙=-xz+w,
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=∂x˙/∂x+∂y˙/∂y+∂z˙/∂z+∂w˙/∂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
(3)J=[-aa000b0100-c00001].ForJ 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 parametersa=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 inx-direction as c varies.(a) Lyapunov exponent spectrum asc varies. (b) Lyapunov exponent spectrum as c varies.
(a)
(b)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 Figure2, we observe that the system shows rich dynamical behaviors as c varies. We can obtain the following.When1≤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.


## 2.1. The Dissipativity and Equilibrium

The dissipativity of system (2) is described as ∇V=∂x˙/∂x+∂y˙/∂y+∂z˙/∂z+∂w˙/∂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
(3)J=[-aa000b0100-c00001].ForJ 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 parametersa=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 inx-direction as c varies.(a) Lyapunov exponent spectrum asc varies. (b) Lyapunov exponent spectrum as c varies.
(a)
(b)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 Figure2, we observe that the system shows rich dynamical behaviors as c varies. We can obtain the following.When1≤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:(4)x˙1=a(x2-x1),x˙2=-x1x3+bx2+x4,x˙3=x1x2-cx3,x˙4=-x1x3+x4,y˙1=a(y2-y1)+u1,y˙2=-y1y3+by2+y4+u2,y˙3=y1y2-cy3+u3,y˙4=-y1y3+y4+u4.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(5)x˙=(A+ΔA1)x+h(x),(6)y˙=(A+ΔA2)y+h(y)+u,
where x=(x1,x2,x3,x4)T, y=(y1,y2,y3,y4)T∈RnA∈R4×4 are the coefficient matrix of the known linear part of the system. Assume
(7)A=(-aa000b0100-c0000d).ΔA1,ΔA2∈R4×4 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,
(8)h(x)=(0-x1x3x1x2-x1x3),h(y)=(0-y1y3y1y2-y1y3).u=(u1,u2,u3,u4)T is the controller added to the response system.Our purpose is to make the statex of system (5) and the state y of (6) identical when t→∞, that is,  limt→∞|yi-xi|=0 (i=1,2,3,4).Supposee=(e1,e2,e3,e4)T, ei=yi-xi, (i=1,2,3,4), (6)–(5),
(9)e˙=Ae+ΔA2y-ΔA1x+h(y)-h(x)+u.According to active control design strategy [6]; let u=-h(y)+h(x)+vfuzz, where vfuzz is the variable universe adaptive fuzzy cotroller to be designed. Then, (9) becomes
(10)e˙=Ae+ΔA2y-ΔA1x+vfuzz.For the purpose of stability, we add a compensating controllervs to (10), that is,
(11)e˙=Ae+ΔA2y-ΔA1x+v,
where v=vfuzz+vs. vfuzz=(vfuzz1,vfuzz2,vfuzz3,vfuzz4)T, vs=(vs1,vs2,vs3,vs4)T. vfuzz is designed as follows:(12)vfuzz≜Uiβi(t)∑j=1mAij(eiαi(ei))yij,(i=1,2,3,4),
where βi(t) and αi(ei) are the contraction-expansion factors of yij and ei, respectively,  and Aij(·) is the jth membership function of vfuzzi. Consider
(13)β˙i(t)=kiei(i=1,2,3,4),kiis constant,(14)αi(t)=1-liexp(-δiei2)(i=1,2,3,4).In [12], Li demonstrated that fuzzy reasoning is equivalent to an interpolation. For the length of yij no longer than 1 and needing a few fuzzy roles, the design of vfuzz is very simple.Suppose the maximal eigenvalue ofA is λ, then let
(15)K={diag(0,0,0,0),λ<0diag(-λ-1,-λ-1,-λ-1,-λ-1),λ≥0.We have(16)e˙=Be-Ke+ΔA2y-ΔA1x+v,
where B=A+K is a stable matrix. Obviously, the ideal controller of (16) is v*=Ke-ΔA2y+ΔA1x. By using (11), we have
(17)e˙=Be-v*+v.Theorem 1.
Choosing suitablevs, system (16) can be asymptotically stable.Proof.
BecauseB is stable, for any positive definite symmetric matrix Q, there exists a positive definite symmetric matrix P which satisfies the Lyapunov equations BTP+PB=-Q. Given energy function V(e)=(1/2)eTBe, our purpose is to demonstrate it by using the Lyapunov function. Its derivative with respect to t is
(18)V˙=-12eTQe+eTP(-v*+vfuzz+vs)=-12eTQe+eTP(-Ke+ΔA2y-ΔA1x+vfuzz+vs).
SupposeP=(P1,P2,P3,P4), where Pi (i=1,2,3,4) is the ith column of P. If
(19)-eTPKe+L2|eTPy|+L1|eTPx|+eTPvfuzz≤12eTQe
then vs=0.
Otherwise, let(20)vsi=-sgn(eTPi)(L2|yi|+L1·|xi|)-vfuzzi+kiiei.
If (19) is satisfied, then
(21)V˙(e)=-12eTQe+eTP(-Ke+ΔA2y-ΔA1x+vfuzz+vs)≤-12eTQe-eTPKe+L2|eTPy|+L1|eTPx|+eTPvfuzz+eTPvs.
Since vs=0, obviously, V˙(e)≤0.
If (20), satisfied, let K=(kij), ΔA1=(Δaij), ΔA2=(Δbij), where kij, Δaij, Δbij (i,j=1,2,…,n) are matrix elements of K, ΔA, and ΔB. Then
(22)eTP(-Ke+ΔA2y-ΔA1x+vfuzz+vs)=(eTP1,eTP2,eTP3,eTP4)×(-k11e1+∑i=14Δbi1y1+∑i=14Δai1x1+vfuzz1+vs1-k22e2+∑i=14Δbi2y2+∑i=14Δai2x2+vfuzz2+vs2-k33e3+∑i=14Δbi3y3+∑i=14Δai3x3+vfuzz3+vs3-k44e4+∑i=14Δbi4y4+∑i=14Δai4x4+vfuzz4+vs4)≤(|eTP1|,|eTP2|,|eTP3|,|eTP4|)×(L2|y1|+L1|x1|L2|y2|+L1|x2|L2|y3|+L1|x3|L2|y4|+L1|x4|)+(eTP1,eTP2,eTP3,eTP4)×(-k11e1+vfuzz1+vs1-k22e2+vfuzz2+vs2-k33e3+vfuzz3+vs3-k44e4+vfuzz4+vs4).
If (20) can be satisfied, we have
(23)eTP(-Ke+ΔBy-αΔAx+vfuzz+vs)≤0,
so V˙(e)≤-(1/2)eTQe≤0.The stability of (16) is obtained.So, we get the conclusion that ifu=-h(y)+h(x)+vfuzz+vs, vfuzz, vs are defined as above, system (5) can synchronize (6).


## 4. Simulation

Leta=26, b=14, c=2.3, L1=L2=0.3,
(24)ΔA1=(0.090.010.210.020.070.060.04-0.05-0.050.02-0.110.030.060.050.030.07),ΔA2=(-0.020.050.110.070.070.050.060.02-0.080.030.010.040.070.11-0.05-0.03),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 Figure3(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 vfuzzi (i=1,2,3,4) is [-1,1]. For simplicity, we define the contraction-expression factors of ei (i=1,2,3,4) as α(ei(t))=1-0.995exp(-0.08ei2(t)).We define 6 membership functions aboute1 which are taken as triangle waves (see Figure 4). Expressions are as follows:
(25)A11(e1)={1,e1≤-40,e1+20-20,-40≤e1≤-20,0,else;A12(e1)={e1+4020,-40≤e1≤-20,e1+10-10,-20≤e1≤-10,0,else;A13(e1)={e1+2010,-20≤e1≤-10,e1-10,-10≤e1≤0,0,else;A14(e1)={e1-20-10,10≤e1≤20,e110,0≤e1≤10,0,else;A15(e1)={e1-40-20,20≤e1≤40,e1-1010,10≤e1≤20,0,else;A16(e1)={1,e1≥40,e1-2020,20≤e1≤40,0,else.
From (12), we know
(26)vfuzz1=U1β1(t)[-A11(e1(t)α1(e1(t)))-0.5A12(e1(t)α1(e1(t)))+0.25×A14(e1(t)α1(e1(t)))+0.5A15(e1(t)α1(e1(t)))+A16(e1(t)α1(e1(t)))],
where β˙1(t)=k1e1(t), β1(0)=1.Figure 4The membership functions aboute2 are the same as those of e1 except for the subscripts. A2i and e2 take the place of A1i and e1, respectively.From (12), we know
(27)vfuzz2=U2β2(t)[-A21(e2(t)α2(e2(t)))-0.5A22(e2(t)α2(e2(t)))-0.25A23(e2(t)α2(e2(t)))+0.25A24(e2(t)α2(e2(t)))+0.5A25(e2(t)α2(e2(t)))+A26(e2(t)α2(e2(t)))],
where β˙2(t)=k2e2(t), β2(0)=1.We also define 6 membership functions aboute3 which are taken as triangle waves (see Figure 5). Expressions are as follows:
(28)A31(e3)={1,e3≤-60,e3+30-30,-60≤e3≤-30,0,else;A32(e1)={e3+6030,-60≤e3≤-30,e3+15-15,-30≤e3≤-15,0,else;A33(e3)={e3+3015,-30≤e3≤-15,e3-15,-15≤e3≤0,0,else;A34(e3)={e3-30-15,15≤e3≤30,e315,0≤e3≤15,0,else;A35(e3)={e3-60-30,30≤e3≤60,e3-1515,15≤e3≤30,0,else;A36(e3)={1,e3≥60,e3-3030,30≤e3≤60,0,else.Figure 5From (12), we know
(29)vfuzz3=U3β3(t)[-A31(e3(t)α3(e3(t)))-0.5A32(e3(t)α3(e3(t)))-0.25A33(e3(t)α3(e3(t)))+0.25A34(e3(t)α3(e3(t)))+0.5A35(e3(t)α3(e3(t)))+A36(e3(t)α3(e3(t)))],
where β˙3(t)=k3e3(t), β3(0)=1.We also define 6 membership functions aboute4 which are taken as triangle waves (see Figure 6). Expressions are as follows:
(30)A41(e4)={1,e4≤-120,e4+60-60,-120≤e4≤-60,0,else;A42(e4)={e4+12060,-120≤e4≤-60,e4+30-30,-60≤e4≤-30,0,else;A43(e4)={e4+6030,-60≤e4≤-30,e4-30,-30≤e4≤0,0,else;A44(e4)={e4-60-30,30≤e4≤60,e330,0≤e4≤30,0,else;A45(e4)={e4-120-60,60≤e4≤120,e4-3030,30≤e4≤60,0,else;A46(e4)={1,e4≥120,e4-6060,60≤e4≤120,0,else.Figure 6From (12), we know
(31)vfuzz4=U4β4(t)[-A41(e4(t)α4(e4(t)))-0.5A42(e4(t)α4(e4(t)))-0.25A43(e4(t)α4(e4(t)))+0.25A44(e4(t)α4(e4(t)))+0.5A45(e4(t)α4(e4(t)))+A46(e4(t)α4(e4(t)))],
where β˙4(t)=k4e4(t), β4(0)=1.The compensating controllervs is defined by (19) and (20). With the controller u=-h(y)+h(x)+vfuzz+vs, we get the simulation results in Figures 7, 8, 9, and 10.Figure 7Figure 8Figure 9Figure 10From 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.



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