# Drive-Response Synchronization of a Fractional-Order Hyperchaotic System and Its Circuit Implementation

Authors: Darui Zhu; Chongxin Liu; Bingnan Yan
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A novel fractional-order hyperchaotic system is proposed; the theoretical analysis and numerical simulation of this system are studied. Based on the stability theory of fractional calculus, we propose a novel drive-response synchronization scheme. In order to achieve this synchronization control, the Adams-Bashforth-Moulton algorithm is studied. And then, a drive-response synchronization controller is designed to realize the synchronization of the drive and response system, and the simulation results are given. At last, the fractional oscillator circuit of the new fractional-order hyperchaotic system is designed based on the EWB software, and it is verified that the simulation results of the fractional-order oscillator circuit are consistent with the numerical simulation results through  circuit simulation.


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## Body

## 1. Introduction

Fractional-order calculus is a mathematical theory which studies the characteristic and application of the arbitrary-order differential and integral operator. It has the same history as the integer-order calculus and can be dated back to the 17th century [1, 2]. It has attracted more researchers’ interest and has more broad application prospects due to its unique advantages. But until the last 20 years, the fractional-order calculus theory has been applied to the practical projects; it has been applied to the chaos system, electromagnetism, signal processing, mechanical engineering, robot control, and so on.Since hyperchaos was firstly reported by Rossler [3], there have been considerable achievements in the study of hyperchaos. Hyperchaotic systems have more than one positive Lyapunov exponent, and the strange attractor is usually unstable in more than one direction. Hyperchaotic system is a high-dimensional chaotic system. The fractional-order differential operator is introduced into hyperchaotic system which can reflect the hyperchaotic system with complicated nonlinear dynamic characteristics. Fractional-order hyperchaotic system implementation and application have been attracted more researchers’ interest and in-depth study. In recent years, the fractional-order chaotic dynamical systems began to attract more researchers’ attention, such as the fractional-order Chua system, the fractional-order Lorenz system, Chen’s system, and Liu’s system and so on [4–9].In this paper, a novel fractional-order hyperchaotic system is presented. Based on the stability theory of fractional calculus, a novel drive-response synchronization scheme is proposed. Section4 studied the Adams-Bashforth-Moulton algorithm to achieve this synchronization scheme. The fractional oscillator circuit of the new hyperchaotic system is designed based on the EWB software in Section 5, and the simulation results are given to demonstrate that  the fractional-order oscillator is a hyperchaotic system. Finally, conclusions end this paper.


## 2. A Novel Fractional-Order Hyperchaotic System

A novel four-dimensional fractional-order hyperchaotic system is given as follows:(1)dαxdtα=y-x-y2-w,dαydtα=ay+xz+w,dαzdtα=-bxy-cz+w,dαwdtα=-y,where(a,b,c)=(2.5,4,4) and α is the fractional order satisfying 0<α<1.By calculating, this system has two positive Lyapunov exponents:LE1=0.7980 and LE1=0.1841, and the negative Lyapunov exponents are LE3=-0.9656 and  LE4=-2.5164. According to chaos theory, there are two positive Lyapunov exponents implying that its dynamics is expanded in more than one direction simultaneously and that the system is a hyperchaotic system.Based on the Lyapunov exponents, we can calculate the Hausdorff dimension (Lyapunov dimension) of the nonlinear autonomous system.(2)DL=j+1|LEj+1|∑i=1jLEi=3+LE1+LE2+LE3|LE4|=3.007.In this paper, the following simulations are all performed by usingα=0.9, (a,b,c)=(2.5,4,4). By simulations, we have obtained the 2D phase portraits of the fractional-order system as shown in Figure 1 and the time domain waveform of the x(t) as shown in Figure 2. These figures clearly show that the fractional-order hyperchaotic system exhibits chaotic behaviors.2D strange attractors of the fractional-order hyperchaotic system: (a)x-y phase plane; (b) y-z phase plane; (c) x-z phase plane; (d) x-w phase plane.
(a)
x
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y phase plane
(b)
y-z phase plane
(c)
x-z phase plane
(d)
x-w phase planeFigure 2
The time domain waveform of thex(t).


## 3. The Theory of the Fractional Drive-Response Synchronization

In this section, the drive-response method is used to realize synchronization control of the fractional-order system, assuming that the fractional order chaotic system is studied in the form of the driving equation(3)dαxdtα=Ax+Bf(Cx)+D,where0<α<1, x∈Rn is the column vector, A∈Rn×n, B∈Rn×m, C∈R1×n, and D∈Rn×1 are the continuous matrixes, and Cx∈R, f:R1→Rm is the nonlinear function.It is worth noting that we must select the appropriate driving signal to drive the response system; generally we select the linear term which can be written as follows:(4)y=Cx.The fractional-order response system is defined(5)dαx~dtα=Ax~+Bf(y)+D+K(y-y~),wherey~=Cx~, x~∈Rn are the column vectors and  K=[k1,k2,…,kn]T are the synchronization control parameters. The error system is defined as e(t)=x(t)-x~(t). By (3), (4), and (5), the error equation of the two fractional order chaotic systems can be obtained as follows:
(6)dαedtα=(A-KC)e.Obviously, the system (6) and the following system (7) have the same stability:
(7)dαe^dtα=(AT-CTKT)e^,whereu(t)=-KTe^(t) is the feedback control signal; it determines the synchronization effect of the fractional-order system. Based on the stability theory of the fractional-order system, we propose a novel synchronization theorem [10, 11].Theorem 1.
If the fractional order linear system (7) is asymptotically stable, the necessary and sufficient condition is that all the eigenvalues λncj of the coefficient matrix (AT-CTKT) are satisfied:
(8)απ2<|arg(λncj)|<3απ2,
wherearg(λncj) is the explement of the eigenvalues λncj.Proof.
If the coefficient matrixM=(AT-CTKT) of the system (7) has n different eigenvalues λi(i=1,…,n), there exists a nonsingular transformation matrix, which makes the system (7) convert into
(9)dαe-(t)dtα=Λe-(t),
wheree-(t)=Te^(t), Λ=TMT-1=diag{λ1,λ2,…,λn}, σmax is the maximum singular value of T; and σmin is the minimum singular value of T, hence
(10)σmin∥e^(t)∥≤∥e-(t)∥≤σmax∥e^(t)∥.
The analytical solution of the system (9) can be expressed as the function Eα,β(z) as follows:
(11)e-(t)=Eα,1(Λtα)e-(0).
If∥e-(t)∥→0 is established, it is necessary that Eα,1(λitα)→0, and this is equivalent to that all the eigenvalues λncj are satisfied: απ/2<|arg(λncj)|<3απ/2.
Then, the proof is completed.


## 4. Drive-Response Synchronization of the Fractional-Order System

The proposed fractional-order hyperchaotic system (1) as the drive system, the controlled fractional-order response system, is described:
(12)dαx~dtα=y~-x~-y2-w~-k1(x~-x),dαy~dtα=ay~+xz+w~-k2(y~-y),dαz~dtα=-bxy-cz~+w~-k3(z~-z),dαw~dtα=-y~-k4(w~-w),wherek1, k2, k3, and k4 are the control parameters of the response system. The error system is defined: e1(t)=x~(t)-x(t), e2(t)=y~(t)-y(t), e3(t)=z~(t)-z(t), and e4(t)=w~(t)-w(t); based on the systems (1) and (12), we can obtain the error system:
(13)dαe1dtα=e2-e1-e4-k1e1,dαe2dtα=ae2+e4-k2e2,dαe3dtα=-ce3+e4-k3e3,dαe4dtα=-e2-k4e4,G=[-1-k110-10a-k20100-c-k310-10-k4],whereα=0.9,  (a,b,c)=(2.5,4,4).In order to achieve the synchronization control of the above system, the Adams-Bashforth-Moulton algorithm is applied to the systems (1) and (12). Consider the following differential equations:
(14)dαy(t)dtα=f(t,y(t)),0≤t≤T,y(k)(0)=y0(k),k=0,1,…,[α]-1.They are equivalent to the Volterra integral equation as follows [12, 13]:
(15)y(t)=∑k=0[α]-1tkk!y0(k)+1Γ(α)∫0t(t-w)α-1f(w,y(w))dw.Here,h=T/N, tn=nh, n=0,1,…,N∈Z, and (15) can be discretized into [6]:
(16)yh(tn+1)=∑k=0[α]-1tn+1kk!y0(k)+hαΓ(α+2)(f(tn+1,yh*(tn+1)))yh(tn+1)=+∑j=0naj,n+1f(tj,yh(tj)),(17)aj,n+1={nα+1-(n-α)(n+1)α,j=0,(n-j+2)α+1+(n-j)α+11≤j≤n,-2(n-j+1)α+1,1,j=n+1,yh*(tn+1)=∑k=0[α]-1tn+1kk!y0(k)+1Γ(α)∑j=0nbj,n+1f(tj,yh(tj)),bj,n+1=hαα((n+1-j)α-(n-j)α).The error equation is,maxj=0,1,…,N|y(tj)-yh(tj)|=O(hp), where p=min(2,1+α).By (14), (15), and (16), we can obtain the discrete form of the drive system (1) and response system (12):
(18)xn+1=x0+hαΓ(α+2)xn+1=·(∑j=0naj,n+1·(yj-xj-yj2-wj)(yn+1*-xn+1*-yn+1*2-wn+1*)xn+1=+∑j=0naj,n+1·(yj-xj-yj2-wj)),yn+1=y0+hαΓ(α+2)yn+1=·(∑j=0naj,n+1·(ayj+xjzj+wj)(ayn+1*+xn+1*zn+1*+wn+1*)xn+1=+∑j=0naj,n+1·(ayj+xjzj+wj)),zn+1=z0+hαΓ(α+2)zn+1=·(+∑j=0naj,n+1·(-bxjyj-czj+wj)(-bxn+1*yn+1*-czn+1*+wn+1*)xn+1=+∑j=0naj,n+1·(-bxjyj-czj+wj)),wn+1=w0+hαΓ(α+2)(-yn+1*+∑j=0naj,n+1·(-yj)),x~n+1=x~0+hαΓ(α+2)x~n+1=·(∑j=0naj,n+1·(y~j-x~j-yj2-w~j-k1(x~j-xj))y~n+1*-x~n+1*-yn+1*2-w~n+1*-k1(x~n+1*-xn+1*)x~n+1=+∑j=0naj,n+1·(y~j-x~j-yj2-w~j-k1(x~j-xj))),y~n+1=y~0+hαΓ(α+2)y~n+1=·(∑j=0naj,n+1·(ay~j+xjzj+w~j-k2(y~j-yj))ay~n+1*+xn+1*zn+1*+w~n+1*-k2(y~n+1*-yn+1*)y~n+1=+∑j=0naj,n+1·(ay~j+xjzj+w~j-k2(y~j-yj))),z~n+1=z~0+hαΓ(α+2)z~n+1=·(((z~j-zj))-bxn+1*yn+1*-cz~n+1*+w~n+1*z~n+1=-k3(z~n+1*-zn+1*)z~n+1=+∑j=0naj,n+1·(-bxjyj-cz~jz~n+1=+aj,n+1·(-b+w~j-k3(z~j-zj))),w~n+1=w~0+hαΓ(α+2)w~n+1=·(∑j=0naj,n+1·(-yj-k4(w~j-wj))-y~n+1*-k4(w~n+1*-wn+1*)w~n+1=+∑j=0naj,n+1·(-yj-k4(w~j-wj))),xn+1*=x0+1Γ(α)xn+1*=·∑j=0nbj,n+1·(yj-xj-yj2-wj),yn+1*=y0+1Γ(α)yn+1*=·∑j=0nbj,n+1·(ayj+xjzj+wj),zn+1*=z0+1Γ(α)zn+1*=·∑j=0nbj,n+1·(-bxjyj-czj+wj),wn+1*=w0+1Γ(α)∑j=0nbj,n+1·(-yj),x~n+1*=x~0+1Γ(α)x~n+1*=·∑j=0nbj,n+1·(y~j-x~j-yj2-w~j-k1(x~j-xj)),y~n+1*=y~0+1Γ(α)y~n+1*=·∑j=0nbj,n+1·(ay~j+xjzj+w~j-k2(y~j-yj)),z~n+1*=z~0+1Γ(α)z~n+1*=·∑j=0nbj,n+1·(-bxjyj-cz~j+w~j-k3(z~j-zj)),w~n+1*=w~0+1Γ(α)∑j=0nbj,n+1·(-y~j-k4(w~j-wj)),aj,n+1={nα+1-(n-α)(n+1)α,j=0,(n-j+2)α+1+(n-j)α+11≤j≤n,-2(n-j+1)α+1,1,j=n+1,bj,n+1=hαα[(n-j+1)α-(n-j)α],0≤j≤n+1.Whenk1=10, k2=20, k3=10, k4=40 and the values of a,b,c and k1,k2,k3,k4 are substituted into the system (13), and we can obtain the following: arg(λ1(G))=π, arg(λ2(G))=π, arg(λ3(G))=π, and arg(λ4(G))=π, which are satisfied 0.45π<|arg(λi(G))|<1.35π. The drive system (1) and response system (12) realized the synchronization. The initial values of the system (1) and the system (12) are x(0)=0.1, y(0)=0.2, z(0)=0.3, w(0)=0.4 and x~(0)=0.2, y~(0)=0.3, z~(0)=0.4, w~(0)=0.5, respectively. Based on the above discrete form of the two systems, we can get the synchronization phase diagram by using MATLAB when k1=10, k2=20, k3=10, and k4=40 as shown in Figure 3.Figure 3
The synchronization phase diagram of the systems (1) and (12) when k1=10, k2=20, k3=10, and k4=40.


## 5. Circuit Implementation of the Fractional-Order Hyperchaotic System

In the circuit design, we use linear resistor, capacitor, linear operational amplifier (LM741), analog multiplier (AD633), and the fractional unit circuit structure in the fractional-order hyperchaotic circuits. The linear operational amplifier is used for addition and subtraction, and the analog multiplier is used to realize the nonlinear term. Then, we designed the circuit of fractional-order system (6) with α=0.9, as shown in Figure 4.Figure 4
The circuit realization of the fractional-order system.The resistance values of this circuit are(R1,R2,R3,R4,R7,R8,R13,R15)=1MΩ, (R5,R6,R10,R11,R16,R17)=10kΩ, R9=400kΩ, and (R12,R14)=250kΩ. The component parameter values of the fractional order system with α=0.9 are as follows [14, 15]:
(19)Ra=62.84MΩ,Rb=250kΩ,Rc=2.5kΩ,C1=1.232uF,C2=1.835uF,C3=1.1uF.From Figure5, the fractional-order hyperchaotic system oscillator circuit simulation results are consistent with the MATLAB numerical simulation results. Figure 6 shows phase diagrams of the integer-order hyperchaotic system based on the circuit experimental.2D strange attractors of the fractional-order hyperchaotic system by EWB: (a)x-y phase plane; (b) y-z phase plane; (c) x-z phase plane; (d) x-w phase plane.
(a)
x-y phase plane
(b)
y-z phase plane
(c)
x-z phase plane
(d)
x-w phase planeThe circuit experimental phase diagram of the hyperchaotic system.
(a)
x-y phase plane
(b)
y-z phase plane
(c)
x-z phase plane
(d)
x-w phase plane


## 6. Conclusions

In this paper, we proposed a new fractional-order hyperchaotic system and analyzed the basic properties of this new system. Based on the stability theory of fractional calculus, we proposed a novel drive-response synchronization scheme and studied the Adams-Bashforth-Moulton algorithm. The drive-response synchronization controller is designed to realize synchronization based on the discrete form of the two systems by MATLAB software. And based on the fractional-order unit circuit, we designed the fractional-order oscillator circuit of this system by EWB software. The simulation results of this circuit are consistent with the numerical simulation results. This fractional-order hyperchaotic circuit can be used for the other electronic oscillator and the controller can be used for application in the chaos control due to its simple construction.



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*Source: 815765-2013-09-02.xml*