# Synchronization of Different Fractional Order Time-Delay Chaotic Systems Using Active Control

Authors: Jianeng Tang
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

Chaos synchronization of different fractional order time-delay chaotic systems is considered. Based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method. At last, effects of the fraction order and the time delay on synchronization are further researched.


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## Body

## 1. Introduction

Since the pioneering work of Pecora and Carroll [1], there has been a significant interest in developing powerful techniques for chaos synchronization in the past 20 years. Some different regimes of chaos synchronization have been studied. They contain partial synchronization [2], complete synchronization [3, 4], phase synchronization [5, 6], generalized synchronization [7, 8], projective synchronization [9], and lag synchronization [10]. Of course, Robust synchronization is an important aspect of chaos synchronization. Based on the Lyapunov stability theory and LMI technique, a new sufficient criterion is established for chaos robust synchronization [11]. By use of the sliding mode control technique, a robust control scheme is established even when the parameter uncertainty and external perturbation are present [12]. A robust antisynchronization scheme is proposed according to multiple-kernel least squares support vector machine modeling for two uncertain chaotic systems [13]. Many potential applications of chaos synchronization, especially in secure communications of analog and digital signals and for developing safe and reliable cryptographic systems, chemical oscillators, and electronic systems, have been researched.It is that delayed differential equations have been largely investigated in [14, 15] and references cited therein. Time delays and time-varying delays are recently introduced to chaotic systems; for example, see [16–19]. However, most of these publications are for integer-order or typical differential equations. Although fractional calculus is a 300-year-old mathematical topic, for many years it was not used in physics and engineering. During the last 10 years or so, fractional calculus starts to become focus of attention of physicists and engineers [20, 21]. There are many known systems to display fractional-order dynamics, such as dielectric polarization, electromagnetic waves, and quantum evolution of complex systems. Thus it has been a new trend, that is, the fractional power of the control system dynamics [22–35].According to the Poincare-Bendixon theorem [22], an integer order chaotic nonlinear system must have a minimum order of 3 for chaos to appear. However, it is not the case in the fractional order nonlinear systems. Some examples in the respect are Wu et al. [23] (unified system), Lu and Chen [24] (Chen system), and Lu [25] (Ikeda system). By using Lambert function, the analytical stability bound is obtained for delayed second-order systems with repeatable poles and the bound is obtained delayed linear time-invariant fractional-order dynamic systems [26, 27]. Based on the Laplace transform theory, synchronization scheme to chaotic fractional-order Chen systems is derived in [28]; Deng et al. [29] introduced a characteristic equation for the n-dimensional linear fractional differential system with multiple time delays. In line with the stability theorem of linear fractional systems, a necessary condition is given to check the chaos synchronization of fractional systems with incommensurate order [30]. In [31], Shao et al. proposed a method to achieve projective synchronization of the fractional order chaotic Rossler system. In [32], an analytical justification was proposed for phase synchronization of fractional differential equations. A stability test procedure was proposed for linear nonhomogeneous fractional order systems with a pure time delay [33]. Taghvafard and Erjaee [34] studied the phase and antiphase synchronization between two identical and nonidentical fractional order chaotic systems using techniques from active control theory. The effect of delay on the chaotic behaviour has been investigated for the first time in the literature [35].However, there are few results about chaos synchronization of the fractional order time-delay chaotic systems. In our work, we give an improved version of Adams-Bashforth-Moulton method. By use of active control technique, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed based on the Laplace transform theory. Then effects of the fraction order and the time delay on synchronization are further analyzed.The organization of this paper is as follows. In Section2, some definitions and systems are given and then an improved version of Adams-Bashforth-Moulton method is introduced. In the following three sections, we synchronize the following fractional systems using active control method: (i) Liu with Liu, (ii) Lü with Chen, and (iii) Lorenz with Chen. The effectiveness of our work is verified through numerical simulations. In Section 6, effects of the fraction order and the time delay on synchronization are further researched. Finally concluding remark is given.


## 2. Definitions and Systems

### 2.1. Fractional Calculus

There are several definitions of a fractional-order differential system. In the following, we introduce the most common one of them:(1)D*ax(t)=dαxdtα=Jm-αx(m)(t),
where m=[α]; that is, m is the first integer which is not less than α(0<α<1), x(m) is the m-order derivative in the usual sense, and Jβ(β>0) is the β-order Reimann-Liouville integral operator with expression:
(2)Jβy(t)=1Γ(β)∫0t(t-τ)β-1y(τ)dτ.
Here Γ stands for Gamma function, and the operator D*α is generally called “α-order Caputo differential operator” [33].


### 2.2. System Description

The fractional order Lorenz system has recently been studied in many references [36, 37]. Here we give a fractional order time-delay Lorenz system. The system is described by
(3)dαxdtα=σ(y-x),dαydtα=rx-y-xz,dαzdtα=xy-μz(t-τ),
where σ=10 is the Prandtl number, r=28 is the Rayleigh number over the critical Rayleigh number, μ=8/3 gives the size of the region approximated by the system, and τ>0 is time delay. System (3) displays a chaotic attractor, as shown in Figure 1.Figure 1
The fractional-order time-delay Lorenz chaotic attractors withσ=10,μ=8/3,r=28,α=0.95, and τ=0.4.In 1999, Chen and Ueta [38] introduced a new chaotic system, which is similar but not topologically equivalent to the Lorenz system. Without changing the model structure of the system, we consider the fractional order time-delay Chen system in the following form:
(4)dαxdtα=a(y-x),dαydtα=(c-a)x-xz+cy,dαzdtα=xy-bz(t-τ),
where a,  b, and c are real parameters. When a = 35, b = 3, and c = 28, system (4) displays a chaotic attractor through the suitable selection of time delay, as shown in Figure 2. The fractional order time-delay Lü system is given by
(5)dαxdtα=a1(y-x),dαydtα=c1x-xz,dαzdtα=xy-b1z(t-τ),
where a1=35, b1=3, and c1=28. As shown in Figure 3, system (5) displays a chaotic attractor through the suitable selection of time delay.Figure 2
The fractional-order time-delay Chen chaotic attractors witha=35,b=3,c=28,α=0.98, and τ=0.2.Figure 3
The fractional-order time-delay Lü chaotic attractors witha1=35,b1=3,c1=28,α=0.78, and τ=1.In 2009, Liu et al. proposed a novel three-dimensional autonomous chaos system (called Liu system) [39]. Daftardar-Gejji and Bhalekar [40] studied a fractional version of the chaotic system. Here we investigate a fractional order time-delay Liu system as follows:
(6)dαxdtα=-a2x-ey2,dαydtα=b2x-kxz,dαzdtα=mxy-c2z(t-τ),
where a2=1,b2=2.5,c2=5,e=1, k = 4, and m = 4. The system can be in the chaotic state as shown in Figure 4.Figure 4
The fractional-order time-delay Liu chaotic attractors witha2=1,b2=2.5,c2=5, e=1, k=4, m=5, α=0.985, and τ=0.002.


### 2.3. Numerical Method

According to, [41–43] the predictor-corrector scheme for system (4) is derived. Here we give an improved version of Adams-Bashforth-Moulton method [43]. Based on the fractional order time-delay Chen system, let us illustrate this scheme.The following differential equation:(7)dαxdtα=f(t,x),0≤t≤T,x(k)(0)=x0(k),k=0,1,2,…,n-1,
is equivalent to the Volterra integral equation [42]
(8)x(t)=∑k=0n-1x0(k)tkk!+1Γ(α)∫0tf(τ1,x)(t-τ)1-αdτ1.
Let h=T/N,tn=nh(n=0,1,2,…,N), andn1=τ/h. Then (8) can be discretized as follows:
(9)xh(tn+1)=∑k=0n-1x0(k)tnk+1k!+hαΓ(α+2)f(tn+1,xhρ(tn+1))+hαΓ(α+2)∑aj,n+1f(tj,xh(tj)),
where
(10)aj,n+1={nα+1-(n-α)(n+1)α+1,j=0(n-j+2)α+1+(n-j)α+1-2(n-j+1)α+1,1≤j≤n1,j=n+1,xhρ(tn+1)=∑k=0n-1x0(k)tnk+1k!+1Γ(α)∑j=0nbj,n+1f(tj,xh(tj)),bj,n+1=hαα((n-j+1)α-(n-j)α).
The error estimate is e=Max⁡|x(tj)-xh(tj)|=O(hρ)(j=0,1,…,N) where ρ=Min⁡(2,1+α).Applying the above method, (4) can be discretized as follows:
(11)xn+1=x0+hαΓ(α+2)a(yn+1ρ-xn+1ρ)+hαΓ(α+2)∑j=0nγ1,j,n+1a(yj-xj),yn+1=y0+hαΓ(α+2)[(c-a)xn+1ρ-xn+1ρzn+1ρ+cyn+1ρ]+hαΓ(α+2)∑j=0nγ2,j,n+1[(c-a)xj-xjzj+cyj],zn+1=z0+hαΓ(α+2)(xn+1ρyn+1ρ-bzn+1-n1ρ)+hαΓ(α+2)∑j=0nγ3,j,n+1(xjyj-bzj-n1),
where
(12)xn+1ρ=x0+1Γ(α+2)∑j=0nω1,j,n+1a(yj-xj),yn+1ρ=y0+1Γ(α+2)∑j=0nω2,j,n+1[(c-a)xj-xjzj+cyj],zn+1ρ=z0+1Γ(α+2)∑j=0nω3,j,n+1(xjyj-bzj-n1),γi,j,n+1={nα+1-(n-α)(n+1)α+1,j=0(n-j+2)α+1+(n-j)α+1-2(n-j+1)α+1,1≤j≤n1,j=n+1,ωi,j,n+1=hαα((n-j+1)α-(n-j)α),hhhhhhh0≤j≤n,i=1,2,3.


## 2.1. Fractional Calculus

There are several definitions of a fractional-order differential system. In the following, we introduce the most common one of them:(1)D*ax(t)=dαxdtα=Jm-αx(m)(t),
where m=[α]; that is, m is the first integer which is not less than α(0<α<1), x(m) is the m-order derivative in the usual sense, and Jβ(β>0) is the β-order Reimann-Liouville integral operator with expression:
(2)Jβy(t)=1Γ(β)∫0t(t-τ)β-1y(τ)dτ.
Here Γ stands for Gamma function, and the operator D*α is generally called “α-order Caputo differential operator” [33].


## 2.2. System Description

The fractional order Lorenz system has recently been studied in many references [36, 37]. Here we give a fractional order time-delay Lorenz system. The system is described by
(3)dαxdtα=σ(y-x),dαydtα=rx-y-xz,dαzdtα=xy-μz(t-τ),
where σ=10 is the Prandtl number, r=28 is the Rayleigh number over the critical Rayleigh number, μ=8/3 gives the size of the region approximated by the system, and τ>0 is time delay. System (3) displays a chaotic attractor, as shown in Figure 1.Figure 1
The fractional-order time-delay Lorenz chaotic attractors withσ=10,μ=8/3,r=28,α=0.95, and τ=0.4.In 1999, Chen and Ueta [38] introduced a new chaotic system, which is similar but not topologically equivalent to the Lorenz system. Without changing the model structure of the system, we consider the fractional order time-delay Chen system in the following form:
(4)dαxdtα=a(y-x),dαydtα=(c-a)x-xz+cy,dαzdtα=xy-bz(t-τ),
where a,  b, and c are real parameters. When a = 35, b = 3, and c = 28, system (4) displays a chaotic attractor through the suitable selection of time delay, as shown in Figure 2. The fractional order time-delay Lü system is given by
(5)dαxdtα=a1(y-x),dαydtα=c1x-xz,dαzdtα=xy-b1z(t-τ),
where a1=35, b1=3, and c1=28. As shown in Figure 3, system (5) displays a chaotic attractor through the suitable selection of time delay.Figure 2
The fractional-order time-delay Chen chaotic attractors witha=35,b=3,c=28,α=0.98, and τ=0.2.Figure 3
The fractional-order time-delay Lü chaotic attractors witha1=35,b1=3,c1=28,α=0.78, and τ=1.In 2009, Liu et al. proposed a novel three-dimensional autonomous chaos system (called Liu system) [39]. Daftardar-Gejji and Bhalekar [40] studied a fractional version of the chaotic system. Here we investigate a fractional order time-delay Liu system as follows:
(6)dαxdtα=-a2x-ey2,dαydtα=b2x-kxz,dαzdtα=mxy-c2z(t-τ),
where a2=1,b2=2.5,c2=5,e=1, k = 4, and m = 4. The system can be in the chaotic state as shown in Figure 4.Figure 4
The fractional-order time-delay Liu chaotic attractors witha2=1,b2=2.5,c2=5, e=1, k=4, m=5, α=0.985, and τ=0.002.


## 2.3. Numerical Method

According to, [41–43] the predictor-corrector scheme for system (4) is derived. Here we give an improved version of Adams-Bashforth-Moulton method [43]. Based on the fractional order time-delay Chen system, let us illustrate this scheme.The following differential equation:(7)dαxdtα=f(t,x),0≤t≤T,x(k)(0)=x0(k),k=0,1,2,…,n-1,
is equivalent to the Volterra integral equation [42]
(8)x(t)=∑k=0n-1x0(k)tkk!+1Γ(α)∫0tf(τ1,x)(t-τ)1-αdτ1.
Let h=T/N,tn=nh(n=0,1,2,…,N), andn1=τ/h. Then (8) can be discretized as follows:
(9)xh(tn+1)=∑k=0n-1x0(k)tnk+1k!+hαΓ(α+2)f(tn+1,xhρ(tn+1))+hαΓ(α+2)∑aj,n+1f(tj,xh(tj)),
where
(10)aj,n+1={nα+1-(n-α)(n+1)α+1,j=0(n-j+2)α+1+(n-j)α+1-2(n-j+1)α+1,1≤j≤n1,j=n+1,xhρ(tn+1)=∑k=0n-1x0(k)tnk+1k!+1Γ(α)∑j=0nbj,n+1f(tj,xh(tj)),bj,n+1=hαα((n-j+1)α-(n-j)α).
The error estimate is e=Max⁡|x(tj)-xh(tj)|=O(hρ)(j=0,1,…,N) where ρ=Min⁡(2,1+α).Applying the above method, (4) can be discretized as follows:
(11)xn+1=x0+hαΓ(α+2)a(yn+1ρ-xn+1ρ)+hαΓ(α+2)∑j=0nγ1,j,n+1a(yj-xj),yn+1=y0+hαΓ(α+2)[(c-a)xn+1ρ-xn+1ρzn+1ρ+cyn+1ρ]+hαΓ(α+2)∑j=0nγ2,j,n+1[(c-a)xj-xjzj+cyj],zn+1=z0+hαΓ(α+2)(xn+1ρyn+1ρ-bzn+1-n1ρ)+hαΓ(α+2)∑j=0nγ3,j,n+1(xjyj-bzj-n1),
where
(12)xn+1ρ=x0+1Γ(α+2)∑j=0nω1,j,n+1a(yj-xj),yn+1ρ=y0+1Γ(α+2)∑j=0nω2,j,n+1[(c-a)xj-xjzj+cyj],zn+1ρ=z0+1Γ(α+2)∑j=0nω3,j,n+1(xjyj-bzj-n1),γi,j,n+1={nα+1-(n-α)(n+1)α+1,j=0(n-j+2)α+1+(n-j)α+1-2(n-j+1)α+1,1≤j≤n1,j=n+1,ωi,j,n+1=hαα((n-j+1)α-(n-j)α),hhhhhhh0≤j≤n,i=1,2,3.


## 3. Chaos Synchronization between Fractional Order Time-Delay Liu and Chen System

In this section we study the synchronization between Liu and Chen systems. Assuming that the Liu system drives the Chen system, we define the drive (master) and response (slave) systems as follows:(13)dαx1dtα=-a1x1-ey12,dαy1dtα=b1x1-kx1z1,dαz1dtα=mx1y1-c1z1(t-τ),(14)dαx2dtα=a2(y2-x2)+u1(t),dαy2dtα=(c2-a2)x2-x2z2+c2y2+u2(t),dαz2dtα=x2y2-b2z2(t-τ)+u3(t).The unknown termsu1, u2, and u3 in (14) are active control functions to be determined. Define the error functions as
(15)e1=x2-x1,e2=y2-y1,e3=z2-z1.
Equation (15) together with (13) and (14) yields the error system
(16)dαe1dtα=a2(y2-x2)+u1(t)+a1x1+ey12,dαe2dtα=(c2-a2)x2-x2z2+c2y2+u2(t)-b1x1+kx1z1,dαe3dtα=x2y2-b2z2(t-τ)+u3(t)-mx1y1+c1z1(t-τ).By using Matignon’s theorem [44], so we here define active control function ui(i=1,2,3) as
(17)u1(t)=-3e1-2e1(t-τ)-4e2-3e2(t-τ)-a2(y2-x2)-a1x1-ey12,u2(t)=-6e2-5e2(t-τ)-(c2-a2)x2+x2z2-c2y2+b1x1-kx1z1,u3(t)=-2e1-e1(t-τ)-8e2-3e2(t-τ)-10e3-7e3(t-τ)-x2y2+b2z2(t-τ)+mx1y1-c1z1(t-τ).
Substituting (17) into (16), we achieve
(18)dαe1dtα=-3e1-2e1(t-τ)-4e2-3e2(t-τ),dαe2dtα=-6e2-5e2(t-τ),dαe3dtα=-2e1-e1(t-τ)-8e2-3e2(t-τ)-10e3-7e3(t-τ).Then we start to prove that the error system (18) is asymptotically stable; that is, the system (13) and the system (14) with the active control law (17) are synchronized.Taking the Laplace transformation in two sides of (18), letting Ei(s)=L(ei(t))(i=1,2,3), and utilizing L(dαei/dtα)=sαEi(s)-sα-1ei(0) (see [21]), we obtain
(19)sαE1(s)-sα-1e1(0)=-3E1(s)-2E1(s)e-sτ-4E2(s)-3E2(s)e-sτ,sαE2(s)-sα-1e2(0)=-6E2(s)-5E2(s)e-sτ,sαE3(s)-sα-1e3(0)=-2E1(s)-E1(s)e-sτ-8E2(s)-3E2(s)e-sτ-10E3(s)-7E3(s)e-sτ.
Equation (19) can be rewritten as follows:
(20)E1(s)=sα-1e1(0)+(-4-3e-sτ)E2(s)sα+3+2e-sτ,E2(s)=sα-1e2(0)sα+6+5e-sτ,E3(s)=sα-1e3(0)+(-2-e-sτ)E1(s)+(-8-3e-sτ)E2(s)sα+10+7e-sτ.
By use of the final-value theorem of the Laplace transformation, we have
(21)lim⁡t→+∞e2(t)=lim⁡s→0+sE2(s)=sαe2(0)sα+6+5e-sτ=0.
At the same time, from (20), we can get
(22)lim⁡t→+∞e1(t)=lim⁡s→0+sE1(s)=sαe1(0)+s(-4-3e-sτ)E2(s)sα+3+2e-sτ=0,lim⁡t→+∞e3(t)=lim⁡s→0+sE3(s)=sαe3(0)+s(-2-e-sτ)E1(s)+s(-8-3e-sτ)E2(s)sα+10+7e-sτ=0.
The above analysis implies that the fractional-order drive system (13) and the fractional-order response system (14) with the active control law (17) are synchronized.


### 3.1. Simulation Research

Here, leta1=1, b1=2.5, c1=5, e=1, k=4, m=5 in the Liu system and parameters of the Chen system are taken as a2=35,  b2=3, c2=28. The fractional order α is taken to be 0.985 and assume τ=0.002. The initial conditions for drive and response system are x1(0)=-8, y1(0)=-2,  z1(0)=-0.5 and x2(0)=0.5, y2(0)=1, z2(0)=10, respectively. Thus initial conditions for the error system are e1(0)=8.5, e2(0)=3, e3(0)=10.5. Figures 5(a)–5(c) shows the synchronization between Chen and Liu system. The errors ei(t) for the drive and response system are shown in Figure 5(d).Chaos synchronization between (13) and (14). (a) Signals x1, x2. (b) Signals y1, y2. (c) Signals z1, z2. (d) Error system.
(a)
(b)
(c)
(d)


## 3.1. Simulation Research

Here, leta1=1, b1=2.5, c1=5, e=1, k=4, m=5 in the Liu system and parameters of the Chen system are taken as a2=35,  b2=3, c2=28. The fractional order α is taken to be 0.985 and assume τ=0.002. The initial conditions for drive and response system are x1(0)=-8, y1(0)=-2,  z1(0)=-0.5 and x2(0)=0.5, y2(0)=1, z2(0)=10, respectively. Thus initial conditions for the error system are e1(0)=8.5, e2(0)=3, e3(0)=10.5. Figures 5(a)–5(c) shows the synchronization between Chen and Liu system. The errors ei(t) for the drive and response system are shown in Figure 5(d).Chaos synchronization between (13) and (14). (a) Signals x1, x2. (b) Signals y1, y2. (c) Signals z1, z2. (d) Error system.
(a)
(b)
(c)
(d)


## 4. Chaos Synchronization between Fractional Order Time-Delay Lü and Chen System

In this section we consider Lü system as the drive systems(23)dαx1dtα=a1(y1-x1),dαy1dtα=c1x1-x1z1,dαz1dtα=x1y1-b1z1(t-τ),
and the Chen system as the response system
(24)dαx2dtα=a2(y2-x2)+u1(t),dαy2dtα=(c2-a2)x2-x2z2+c2y2+u2(t),dαz2dtα=x2y2-b2z2(t-τ)+u3(t).The unknown termsu1, u2, u3 in (24) are active control functions to be determined. Define the error functions as
(25)e1=x2-x1,e2=y2-y1,e3=z2-z1,
Equation (25) together with (23) and (24) yields the error system
(26)dαe1dtα=a2(y2-x2)+u1(t)-a1(y1-x1),dαe2dtα=(c2-a2)x2-x2z2+c2y2+u2(t)-c1y1+x1z1,dαe3dtα=x2y2-b2z2(t-τ)+u3(t)-x1y1+b1z1(t-τ).By using of the Matignon’s theorem [44], so we here define active control function ui(i=1,2,3) as
(27)u1(t)=-4e1-3e1(t-τ)-a2(y2-x2)+a1(y1-x1),u2(t)=-6e1-3e1(t-τ)-7.5e2-5.3e2(t-τ)-3e3-e3(t-τ)-(c2-a2)x2+x2z2-c2y2+c1y1-x1z1,u3(t)=-8e1-4e1(t-τ)-9e3-2e3(t-τ)-x2y2+b2z2(t-τ)+x1y1-b1z1(t-τ).
Substituting (27) into (26), we achieve
(28)dαe1dtα=-4e1-3e1(t-τ),dαe2dtα=-6e1-3e1(t-τ)-7.5e2-5.3e2(t-τ)-3e3-e3(t-τ),dαe3dtα=-8e1-4e1(t-τ)-9e3-2e3(t-τ).Then we start to prove that the error system (28) is asymptotically stable, that is, the system (23) and the system (24) with the active control law (27) are synchronized.Taking the Laplace transformation in two sides of (28), letting Ei(s)=L(ei(t))(i=1,2,3), and utilizing L(dαei/dtα)=sαEi(s)-sα-1ei(0) (see [18]), we obtain
(29)sαE1(s)-sα-1e1(0)=-4E1(s)-3E1(s)e-sτ,sαE2(s)-sα-1e2(0)=-6E1(s)-3E1(s)e-sτ-7.5E2(s)-5.3E2(s)e-sτ-3E3(s)-E3(s)e-sτ,sαE3(s)-sα-1e3(0)=-8E1(s)-4E1(s)e-sτ-9E3(s)-2E3(s)e-sτ.
Equation (29) can be rewritten as follows:
(30)E1(s)=sα-1e1(0)sα+4+3e-sτ,E2(s)=sα-1e2(0)+(-6-3e-sτ)E1(s)+(-3-e-sτ)E3(s)sα+7.5+5.3e-sτ,E3(s)=sα-1e3(0)+(-8-4e-sτ)E1(s)sα+9+2e-sτ.
By use of the final-value theorem of the Laplace transformation, we have
(31)lim⁡t→+∞e1(t)=lim⁡s→0+sE1(s)=sαe1(0)sα+4+3e-sτ=0.
At the same time, from (30), we can get
(32)lim⁡t→+∞e3(t)=lim⁡s→0+sE3(s)=sαe3(0)+s(-8-4e-sτ)E1(s)sα+9+2e-sτ=0lim⁡t→+∞e2(t)=lim⁡s→0+sE2(s)=(-3-e-sτ)E3(s)sαe2(0)+s(-6-3e-sτ)E1(s)+s(-3-e-sτ)E3(s)sα+7.5+5.3e-sτ=0.
The above analysis implies that the fractional-order drive system (23) and the fractional-order response system (24) with the active control law (27) are synchronized.


### 4.1. Simulation Research

Here, leta1=35, b1=3, andc1=28 in the Lü system and parameters of the Chen system are the same as in Section 3.1. The fractional order α is taken to be 0.78 and assumeτ=1. The initial conditions for drive and response system are x1(0)=0.2, y1(0)=0, z1(0)=0.5, and x2(0)=-2, y2(0)=2, z2(0)=-4, respectively. Thus initial conditions for the error system (28) are e1(0)=-2.2, e2(0)=2, and e3(0)=-4.5. Figures 6(a)–6(c) show the synchronization between Chen and Lü system. The errors ei(t) for the drive and response system are shown in Figure 6(d).Chaos synchronization between (23) and (24). (a) Signals x1, x2. (b) Signals y1, y2. (c) Signals z1, z2. (d) Error system.
(a)
(b)
(c)
(d)


## 4.1. Simulation Research

Here, leta1=35, b1=3, andc1=28 in the Lü system and parameters of the Chen system are the same as in Section 3.1. The fractional order α is taken to be 0.78 and assumeτ=1. The initial conditions for drive and response system are x1(0)=0.2, y1(0)=0, z1(0)=0.5, and x2(0)=-2, y2(0)=2, z2(0)=-4, respectively. Thus initial conditions for the error system (28) are e1(0)=-2.2, e2(0)=2, and e3(0)=-4.5. Figures 6(a)–6(c) show the synchronization between Chen and Lü system. The errors ei(t) for the drive and response system are shown in Figure 6(d).Chaos synchronization between (23) and (24). (a) Signals x1, x2. (b) Signals y1, y2. (c) Signals z1, z2. (d) Error system.
(a)
(b)
(c)
(d)


## 5. Chaos Synchronization between Fractional Order Time-Delay Lorenz and Chen System

Assuming that Chen system is synchronized with Lorenz system, define the drive system as(33)dαx1dtα=a1(y1-x1),dαy1dtα=c1x1-y1-x1z1,dαz1dtα=x1y1-b1z1(t-τ)
and the Chen system as the response system
(34)dαx2dtα=a2(y2-x2)+u1(t),dαy2dtα=(c2-a2)x2-x2z2+c2y2+u2(t),dαz2dtα=x2y2-b2z2(t-τ)+u3(t).Lete1=x2-x1, e2=y2-y1, and e3=z2-z1 be error functions. For synchronization it is essential that the errors ei→0 as t→0. Then we can get
(35)dαe1dtα=a2(y2-x2)+u1(t)-a1(y1-x1),dαe2dtα=(c2-a2)x2-x2z2+c2y2+u2(t)-c1y1+x1z1+y1,dαe3dtα=x2y2-b2z2(t-τ)+u3(t)-x1y1+b1z1(t-τ).By using Matignon’s theorem [44], so we here define active control function ui(i=1,2,3) as
(36)u1(t)=-8e1-7e1(t-τ)-6e2-5e2(t-τ)-4e3-3e3(t-τ)-a2(y2-x2)+a1(y1-x1),u2(t)=-4.8e2-2.7e1(t-τ)-2.9e3-1.3e3(t-τ)-(c2-a2)x2+x2z2-c2y2+c1y1-x1z1-y1,u3(t)=-2e3-e3(t-τ)-x2y2+b2z2(t-τ)+x1y1-b1z1(t-τ).
Substituting (36) into (35), we achieve
(37)dαe1dtα=-8e2-7e2(t-τ)-6e2-5e2(t-τ)-4e3-3e3(t-τ),dαe2dtα=-4.8e2-2.7e2(t-τ)-2.9e3-1.3e3(t-τ),dαe3dtα=-2e3-e3(t-τ).
Then we start to prove that the error system (37) is asymptotically stable; that is, the system (33) and the system (34) with the active control law (36) are synchronized.Taking the Laplace transformation in two sides of (37), letting Ei(s)=L(ei(t))(i=1,2,3), and utilizing L(dαei/dtα)=sαEi(s)-sα-1ei(0) (see [18]), we obtain
(38)sαE1(s)-sα-1e1(0)=-8E1(s)-7E1(s)e-sτ-6E2(s)-5E2(s)e-sτ-4E3(s)-3E3(s)e-sτ,sαE2(s)-sα-1e2(0)=-4.8E2(s)-2.7E2(s)e-sτ-2.9E3(s)-1.3E3(s)e-sτ,sαE3(s)-sα-1e3(0)=-2E3(s)-E3(s)e-sτ.
Equation (38) can be rewritten as follows:
(39)E1(s)=sα-1e1(0)+(-6-5e-sτ)E2(s)+(-4-3e-sτ)E3(s)sα+8+7e-sτ,E2(s)=sα-1e2(0)+(-2.9-1.3e-sτ)E3(s)sα+4.8+2.7e-sτ,E3(s)=sα-1e3(0)sα+2+e-sτ.
By use of the final-value theorem of the Laplace transformation, we have
(40)lim⁡t→+∞e3(t)=lim⁡s→0+sE3(s)=sαe3(0)sα+2+e-sτ=0.
At the same time, from (39), we can get
(41)lim⁡t→+∞e2(t)=lim⁡s→0+sE2(s)=sαe2(0)+s(-2.9-1.3e-sτ)E3(s)sα+4.8+2.7e-sτ=0,lim⁡t→+∞e1(t)=lim⁡s→0+sE1(s)=sαe1(0)+s(-6-5e-sτ)E2(s)+s(-4-3e-sτ)E3(s)sα+8+7e-sτ=0.
So the error system (37) is asymptotically stable and chaos synchronization between the two systems is achieved.


### 5.1. Simulation Research

Here, we take parameters for fractional order time-delay Lorenz system asa1=10, b1=8/3, and c1=28 and parameters of the Chen system are the same as in Section 3.1. The fractional order α is taken to be 0.95 and assume τ=0.4. The initial conditions for the two systems are x1(0)=-8, y1(0)=2, z1(0)=3, and x2(0)=4, y2(0)=-5, and z2(0)=6, respectively. Hence the initial conditions for the error system (37) are e1(0)=12, e2(0)=-7, and e3(0)=3. Figures 7(a)–7(c) show the synchronization between Chen and Lorenz system. The errors ei(t) for the drive and response system are shown in Figure 7(d).Chaos synchronization between (33) and (34). (a) Signals x1, x2. (b) Signals y1, y2. (c) Signals z1, z2. (d) Error system.
(a)
(b)
(c)
(d)


## 5.1. Simulation Research

Here, we take parameters for fractional order time-delay Lorenz system asa1=10, b1=8/3, and c1=28 and parameters of the Chen system are the same as in Section 3.1. The fractional order α is taken to be 0.95 and assume τ=0.4. The initial conditions for the two systems are x1(0)=-8, y1(0)=2, z1(0)=3, and x2(0)=4, y2(0)=-5, and z2(0)=6, respectively. Hence the initial conditions for the error system (37) are e1(0)=12, e2(0)=-7, and e3(0)=3. Figures 7(a)–7(c) show the synchronization between Chen and Lorenz system. The errors ei(t) for the drive and response system are shown in Figure 7(d).Chaos synchronization between (33) and (34). (a) Signals x1, x2. (b) Signals y1, y2. (c) Signals z1, z2. (d) Error system.
(a)
(b)
(c)
(d)


## 6. Effects of Orderα and Time Delay τ on Synchronization

Recently, fractional order chaotic systems have been focus of attention. Generally speaking, there is a lowest order in a fractional order chaotic system. From the literature [23–25], we know that the order α affects the behavior of fractional chaotic dynamical systems. As shown in [35, 45], the time delay τ also affects the behavior of chaotic dynamical systems. In the section, we pay attention to the effects of the order α and the delay τ on synchronization. In Table 1, some results are given about the error functions for different values of order in system (13) and system (14). Here experiments are done for fixed value of the delay τ=0.002. It is obvious that the error in synchronization firstly increases and then decreases as the order α is decreased, which is not the case in [46]. At the same time, we have summarized some results about the error functions for different values of the time delay τ in system (23) and system (24). In these observations, we take the order α=0.78. From Table 2, it is clear that the error functions e1(t) and e2(t) become bigger as the delay τ is increased. But the error function e3(t) firstly decreases and then increases.Table 1
Errors for different values ofα and t.
t
α
=
0.985
α
=
0.92
α
=
0.86
α
=
0.79
α
=
0.75
α
=
0.72
e
1
(
t
)
1
0
-
5
5
14.810
53.589
64.378
60.895
55.085
49.929
8
5.4515
20.460
25.487
25.197
23.386
21.610
10
3.4284
13.070
16.534
16.655
15.629
14.563
e
2
(
t
)
1
0
-
5
5
1.4810
5.2970
6.3961
6.1012
5.5483
5.0483
8
0.55838
2.0920
2.6062
2.5806
2.3980
2.2177
10
0.3352
1.3509
1.7069
1.7190
1.6131
1.5029
e
3
(
t
)
1
0
-
5
5
3.0197
10.877
12.963
12.079
10.794
9.6725
8
1.0866
4.0671
5.0380
4.9228
4.5231
4.1391
10
0.67821
2.5795
3.2468
3.2359
3.0081
2.7774Table 2
Errors for different values ofτ and t.
t
τ
=
0.1
τ
=
0.4
τ
=
0.8
τ
=
1
τ
=
1.5
τ
=
2
e
1
(
t
)
1
0
-
6
30
6.4058
8.1702
10.551
11.816
13.629
17.216
32
5.5652
7.1822
9.3213
10.4366
13.0955
16.1873
35
4.8842
6.1556
8.0039
8.2546
10.0036
13.2583
e
2
(
t
)
1
0
-
6
30
5.3471
10.9933
15.9716
71.8276
12800
18800
32
4.7002
9.4500
78.9665
194.0404
63455
7420
35
3.9123
7.8744
15.0866
88.1245
4950
28500
e
3
(
t
)
1
0
-
6
30
7.4512
5.4716
2.1277
0.5743
3.1265
7.0165
32
6.5074
4.4264
1.7500
0.44107
2.9125
6.4450
35
5.4538
3.7911
1.4449
0.3588
2.4521
5.3607


## 7. Conclusions

In the paper, chaos synchronization of different fractional order time-delay chaotic systems is considered. Based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method. Finally, effects of the fraction order and the time delay on synchronization are further discussed.



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*Source: 262151-2014-02-23.xml*