# Control and Synchronization of the Fractional-Order Lorenz Chaotic System via Fractional-Order Derivative

Authors: Ping Zhou; Rui Ding
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.


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## Body

## 1. Introduction

The theory of fractional-order derivatives can be dated back to the 17th century [1] and developed comprehensively in the last century due to its application in a wide variety of scientific and technological fields such as thermal, viscoelastic, acoustic, electrochemical, rheological, and polymeric disciplines [1, 2]. On the other hand, it has been shown that many fractional-order dynamical systems, as some well-known integer-order systems, can also display complex bifurcation and chaotic phenomena. For example, the fractional-order Lorenz system, the fractional-order Chen system, the fractional-order Lü system, and the fractional-order unified system also exhibit chaotic behavior. Due to its potential applications in secure communication and control processing, the fractional-order chaotic systems have been studied extensively in recent years in many aspects such as chaotic phenomena, chaotic control, chaotic synchronization, and other related studies [3–12].It is verified that the fractional-order controllers can have better disturbance rejection ratios and less sensitivity to plant parameter variations compared to the traditional controllers [13]. A fractional-order controller is presented to stabilize the unstable equilibrium points of integer orders chaos systems [13, 14]. But the previously presented in [13, 14] only discussed the control problem for integer orders chaos systems, not for fractional-order chaotic systems. Up to now, to the best of our knowledge, amongst all kinds of chaos control and chaos synchronization for the fractional-order chaotic systems, very few results on control and synchronization of fractional-order chaotic systems are presented via fractional-order derivative. Motivated by the above discussion, a novel control method for the fractional-order Lorenz chaotic system is investigated in this paper. A fractional-order controller is presented to stabilize the unstable equilibrium points of the fractional-order Lorenz chaotic system via fractional-order derivative, and a fractional-order controller is presented to synchronize the fractional-order Lorenz chaotic system via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.


## 2. The Fractional Derivatives and the Fractional-Order Lorenz Chaotic System

The Caputo definition of the fractional derivative, which sometimes is called smooth fractional derivative, is described as(2.1)Dqf(t)=1Γ(m-q)∫0tf(m)(τ)(t-τ)q+1-mdτ,m-1<q<m,
whereDqdenotes the Caputo definition of the fractional derivative. m is the smallest integer larger than q, and f(m)(t) is the m-order derivative in the usual sense. Γ(·) is gamma function.The Lorenz system, as the first chaotic model, revealed the complex and fundamental behaviors of the nonlinear dynamical systems. In 2003, I. Grigorenko and E. Grigorenko [4] pointed out that the fractional-order Lorenz system exhibits chaotic behavior for fractional-order q≥0.993. The fractional-order Lorenz system is described as follows:(2.2)Dqx1=10(x2-x1),Dqx2=28x1-x2-x1x3,Dqx3=x1x2-8x33,
where 0<q<1. In this paper, we choose q=0.995 for the fractional-order Lorenz chaotic system.Now, we discuss the numerical solution of fractional differential equations. All the numerical simulation of fractional-order system in this paper is based on [3]. We can set h=T/N, tn=nh(n=0,1,2…,N), and initial condition (x1(0),x2(0),x3(0)). So, the fractional-order Lorenz chaotic system (2.2) can be discretized as follows:(2.3)x1(n+1)=x1(0)+hqΓ(q+2)[10(x2p(n+1)-x1p(n+1))+∑j=0nα1,j,n+1×10(x2(j)-x1(j))],x2(n+1)=x2(0)+hqΓ(q+2)[∑j=0nα2,j,n+1(28x1(j)-x2(j)-x1(j)x3(j))28x1p(n+1)-x2p(n+1)-x1p(n+1)x3p(n+1)+∑j=0nα2,j,n+1(28x1(j)-x2(j)-x1(j)x3(j))],x3(n+1)=x3(0)+hqΓ(q+2)[∑j=0nα3,j,n+1(x1(j)x2(j)-8x3(j)3)x1p(n+1)x2p(n+1)-8x3p(n+1)3+∑j=0nα3,j,n+1(x1(j)x2(j)-8x3(j)3)],where(2.4)x1p(n+1)=x1(0)+1Γ(q)∑j=0nb1,j,n+1×10(x2(j)-x1(j)),x2p(n+1)=x2(0)+1Γ(q)∑j=0nb2,j,n+1(28x1(j)-x2(j)-x1(j)x3(j)),x3p(n+1)=x3(0)+1Γ(q)∑j=0nb3,j,n+1(x1(j)x2(j)-8x3(j)3),and fori=1,2,3,(2.5)αi,j,n+1={nq+1-(n-q)(n+1)q,j=0,(n-j+2)q+1+(n-j)q+1-2(n-j+1)q+1,1≤j≤n,1,j=n+1,bi,j,n+1=hqq[(n-j+1)q-(n-j)q],0≤j≤n.The error of this approximation is described as follows:(2.6)|xi(tn)-xi(n)|=o(hp),p=min⁡(2,1+q).Using the above numerical solution for fractional-order Lorenz chaotic system (2.2), the chaotic attractor of fractional-order Lorenz chaotic system (2.2) for q=0.995 is shown in Figure 1.Chaotic attractors of the fractional-order Lorenz chaotic system (2.2) for q=0.995.
(a)(b)


## 3. Stabilizing the Unstable Equilibrium Points of the Fractional-Order Lorenz Chaotic System via Fractional-Order Derivative

It is obvious that the fractional-order Lorenz chaotic system (2.2) has three unstable equilibrium points. The unstable equilibrium points are  p0=(0,0,0) and p±=(±72,±72,27). In this section, we will discuss how to stabilize the unstable equilibrium points of the fractional-order Lorenz chaotic system (2.2) via fractional-order derivative. First, let us present the stability theorem for linear commensurate fractional-order systems and nonlinear commensurate fractional-order systems.Lemma 3.1 (see [13, 15]).
The following linear commensurate fractional-order autonomous system(3.1)Dqx=Ax,x(0)=x0
is asymptotically stable if and only if |argλ|>0.5πq is satisfied for all eigenvalues (λ) of matrix A. Also, this system is stable if and only if |argλ|≥0.5πq is satisfied for all eigenvalues of matrix A, and those critical eigenvalues which satisfy |argλ|=0.5πqhave geometric multiplicity one, where 0<q<1, x∈Rn and A∈Rn×n.Lemma 3.2 (see [16, 17]).
The fixed points of the following nonlinear commensurate fractional-order autonomous system:(3.2)Dqx=f(x)
is asymptotically stable if all eigenvalues (λ) of the Jacobian matrix A=∂f/∂xevaluated at the fixed points satisfy |argλ|>0.5πq, where 0<q<1, x∈Rn, f:Rn→Rn are continuous nonlinear vector functions, and the fixed points of this nonlinear commensurate fractional-order system are calculated by solving equation f(x)=0.


### 3.1. Stabilizing the Unstable Equilibrium Pointp0=(0,0,0) via Fractional-Order Derivative

Now, let us design a controller for fractional-order Lorenz chaotic system (2.2) via fractional-order derivative, and we can obtain the following results.Theorem 3.3.
Let the controlled fractional-order Lorenz chaotic system be(3.3)Dqx1=10(x2-x1),Dqx2=28x1-x2-x1x3+u1(x1),Dqx3=x1x2-8x33,
where u1(x1)=-k11Dqx1-k12x1 is the fractional-order controller, and k1i(i=1,2) is the feedback coefficient. If k12=38+10k11and -10k11-11≤0, then system (3.3) will asymptotically converge to the unstable equilibrium point p0=(0,0,0).Proof.
The Jacobi matrix of the controlled fractional-order Lorenz chaotic system (3.3) at equilibrium p0=(0,0,0) is
(3.4)J=[-1010028+10k11-k12-10k11-1000-83].
Because k12=38+10k11, so
(3.5)J=[-10100-10-10k11-1000-83].
Its characteristic equation is(λ+8/3)[λ2+(10k11+11)λ+10(10k11+1)+100]=0, and the eigenvalues are
(3.6)λ1,2=-(10k11+11)±(10k11+11)2-40(10k11+11)2,λ3=-83.
Because -10k11-11≤0, so
(3.7)|argλi|>π2>0.5πq,(i=1,2,3).
According to Lemma3.2, it implies that the equilibrium point p0=(0,0,0) of system (3.3) is asymptotically stable, that is, the unstable equilibrium point p0=(0,0,0) in fractional-order Lorenz system (2.2) can be stabilized via fractional-order derivative. The proof is completed.For example, choosek11=0.1, then k12=39. The corresponding numerical result is shown in Figure 2, in which the initial conditions are (10,15,20)T in this paper.Time evolution of the equilibrium pointp0=(0,0,0).
(a)(b)(c)Theorem 3.4.
Consider the controlled fractional-order Lorenz chaotic system as follows:(3.8)Dqx1=10(x2-x1)+u2(x2),Dqx2=28x1-x2-x1x3,Dqx3=x1x2-8x33,
where u2(x2)=-k21Dqx2-k22x2 is the fractional-order controller, and k2i(i=1,2) is the feedback coefficient. If k22=38-k21 and -28k21-11≤0, then system (3.8) will gradually converge to the unstable equilibrium point p0=(0,0,0).Proof.
The Jacobi matrix of the controlled fractional-order Lorenz chaotic system (3.8) at equilibrium p0=(0,0,0) is
(3.9)J=[-10-28k2110-k21-k22028-1000-83].
If k22=38-k21, then the Jacobi matrix is
(3.10)J=[-10-28k21-28028-1000-83].
Its characteristic equation is(λ+8/3)[λ2+(28k21+11)λ+(28k21+10)+784]=0, and the eigenvalues are
(3.11)λ1,2=-(28k21+11)±(28k21+11)2-4[(28k21+10)+784]2,λ3=-83.
Because -28k21-11≤0, so
(3.12)|argλi|>π2>0.5πq,(i=1,2,3).
According to Lemma3.2, it implies that the equilibrium point p0=(0,0,0)of system (3.8) is asymptotically stable, that is, the unstable equilibrium point p0=(0,0,0) in fractional-order Lorenz system (2.2) can be stabilized via fractional-order derivative. The proof is completed.For example, choosek21=0.1, then k22=37.9. The corresponding numerical result is shown in Figure 3.Time evolution of the equilibrium pointp0=(0,0,0).
(a)(b)(c)


### 3.2. Stabilizing the Unstable Equilibrium Pointp±=(±72,±72,27) via Fractional-Order Derivative

Let us design a fractional-order controller for fractional-order Lorenz chaotic system (2.2), and we can yield the following results.Theorem 3.5.
Consider that the controlled fractional-order Lorenz chaotic system is(3.13)Dqx1=10(x2-x1),Dqx2=28x1-x2-x1x3+w1(x1),Dqx3=x1x2-8x33,
where w1(x1)=-k1Dqx1 is the fractional-order controller, and k1 is the feedback coefficient. If k1>(-155+24769)/60, then system (3.13) will gradually converge to the unstable equilibrium point p+=(72,72,27).Proof.
Letx=x1-72,y=x2-72,z=x3-27, and the controlled fractional-order Lorenz chaotic system (3.13) can be
(3.14)Dqx=10(y-x),Dqy=x-y-xz-72z-k1Dqx,Dqz=xy+72x+72y-8z3.
The Jacobi matrix of the fractional-order system (3.14) at equilibrium (0,0,0) is
(3.15)J=[-1010010k1+1-10k1-1-727272-83].
Its characteristic equation isλ3+a1λ2+a2λ+a3=0, where a1=10k1+41/3,a2=8(10k1+11)/3+72, anda3=1440. If k1>(-155+24769)/60, then a1>0,a2>0,a3>0, and a1a2-a3>0. So,
(3.16)|argλi|>π2>0.5πq,(i=1,2,3).
According to Lemma3.2, it implies that the equilibrium point p+=(72,72,27)of system (3.13) is asymptotically stable, that is, the unstable equilibrium point p+=(72,72,27) in fractional-order Lorenz system (2.2) can be stabilized via fractional-order derivative. The proof is completed.For example, choosek1=1. The corresponding numerical result is shown in Figure 4.Time evolution of the equilibrium pointp+=(72,72,27).
(a)(b)(c)Theorem 3.6.
Consider that the controlled fractional-order Lorenz chaotic system is(3.17)Dqx1=10(x2-x1)+w2(x2),Dqx2=28x1-x2-x1x3,Dqx3=x1x2-8x33,
where w2(x2)=-k2Dqx2 is the fractional-order controller, and k2 is the feedback coefficient. If (-7616-56765440)/1248<k2<(-7616+56765440)/1248, then system (3.17) will gradually converge to the unstable equilibrium point  p+=(72,72,27).Proof.
Letx=x1-72,y=x2-72,z=x3-27, and the controlled fractional-order Lorenz chaotic system (3.17) can be
(3.18)Dqx=10(y-x)-k2Dqy,Dqy=x-y-xz-72z,Dqz=xy+72x+72y-8z3.
The Jacobi matrix of the fractional-order system (3.18) at equilibrium (0,0,0) is
(3.19)J=[-10-k210+k2k2721-1-727272-83].
Its characteristic equation isλ3+a1λ2+a2λ+a3=0, where a1=k2+41/3,a2=8(k2+11)/3+72(1-k2), anda3=1440. If (-7616-56765440)/1248<k2<(-7616+56765440)/1248, then a1>0, a2>0,a3>0, and a1a2-a3>0. So,
(3.20)|argλi|>π2>0.5πq,(i=1,2,3).
According to Lemma3.2, it implies that the equilibrium point p+=(72,72,27)of system (3.17) is asymptotically stable, that is, the unstable equilibrium point p+=(72,72,27) in fractional-order Lorenz system (2.2) can be stabilized via fractional-order derivative. The proof is completed.For example, choosek2=-2. The corresponding numerical result is shown in Figure 5.Time evolution of the equilibrium pointp+=(72,72,27).
(a)(b)(c)Similarly, the fractional-order Lorenz chaotic system can be easily controlled to another unstable equilibrium pointp-=(-72,-72,27).Remark 3.7.
In general, there is no universal method to select the controller, and these particular controllers in our paper depend on the structure of the fractional-order chaotic system.Remark 3.8.
The differences of the present control strategy in our paper are compared to the result reported by Tavazoei and Haeri [13] as follows. First, we use the scalar controller in our paper, but they used the vector controller. Second, we discuss the control problem for fractional-order chaotic systems via fractional-order derivative, but they discussed the control problem for integer orders chaos systems via fractional-order derivative.Remark 3.9.
The differences of the present control strategy in our paper are compared to the result reported by Razminia et al. [18] as follows. We discuss the control problem for fractional-order chaotic systems via fractional-order derivative, but they discussed the control problem for fractional-order chaotic systems via state feedback, and they did not use fractional-order derivative.


## 3.1. Stabilizing the Unstable Equilibrium Pointp0=(0,0,0) via Fractional-Order Derivative

Now, let us design a controller for fractional-order Lorenz chaotic system (2.2) via fractional-order derivative, and we can obtain the following results.Theorem 3.3.
Let the controlled fractional-order Lorenz chaotic system be(3.3)Dqx1=10(x2-x1),Dqx2=28x1-x2-x1x3+u1(x1),Dqx3=x1x2-8x33,
where u1(x1)=-k11Dqx1-k12x1 is the fractional-order controller, and k1i(i=1,2) is the feedback coefficient. If k12=38+10k11and -10k11-11≤0, then system (3.3) will asymptotically converge to the unstable equilibrium point p0=(0,0,0).Proof.
The Jacobi matrix of the controlled fractional-order Lorenz chaotic system (3.3) at equilibrium p0=(0,0,0) is
(3.4)J=[-1010028+10k11-k12-10k11-1000-83].
Because k12=38+10k11, so
(3.5)J=[-10100-10-10k11-1000-83].
Its characteristic equation is(λ+8/3)[λ2+(10k11+11)λ+10(10k11+1)+100]=0, and the eigenvalues are
(3.6)λ1,2=-(10k11+11)±(10k11+11)2-40(10k11+11)2,λ3=-83.
Because -10k11-11≤0, so
(3.7)|argλi|>π2>0.5πq,(i=1,2,3).
According to Lemma3.2, it implies that the equilibrium point p0=(0,0,0) of system (3.3) is asymptotically stable, that is, the unstable equilibrium point p0=(0,0,0) in fractional-order Lorenz system (2.2) can be stabilized via fractional-order derivative. The proof is completed.For example, choosek11=0.1, then k12=39. The corresponding numerical result is shown in Figure 2, in which the initial conditions are (10,15,20)T in this paper.Time evolution of the equilibrium pointp0=(0,0,0).
(a)(b)(c)Theorem 3.4.
Consider the controlled fractional-order Lorenz chaotic system as follows:(3.8)Dqx1=10(x2-x1)+u2(x2),Dqx2=28x1-x2-x1x3,Dqx3=x1x2-8x33,
where u2(x2)=-k21Dqx2-k22x2 is the fractional-order controller, and k2i(i=1,2) is the feedback coefficient. If k22=38-k21 and -28k21-11≤0, then system (3.8) will gradually converge to the unstable equilibrium point p0=(0,0,0).Proof.
The Jacobi matrix of the controlled fractional-order Lorenz chaotic system (3.8) at equilibrium p0=(0,0,0) is
(3.9)J=[-10-28k2110-k21-k22028-1000-83].
If k22=38-k21, then the Jacobi matrix is
(3.10)J=[-10-28k21-28028-1000-83].
Its characteristic equation is(λ+8/3)[λ2+(28k21+11)λ+(28k21+10)+784]=0, and the eigenvalues are
(3.11)λ1,2=-(28k21+11)±(28k21+11)2-4[(28k21+10)+784]2,λ3=-83.
Because -28k21-11≤0, so
(3.12)|argλi|>π2>0.5πq,(i=1,2,3).
According to Lemma3.2, it implies that the equilibrium point p0=(0,0,0)of system (3.8) is asymptotically stable, that is, the unstable equilibrium point p0=(0,0,0) in fractional-order Lorenz system (2.2) can be stabilized via fractional-order derivative. The proof is completed.For example, choosek21=0.1, then k22=37.9. The corresponding numerical result is shown in Figure 3.Time evolution of the equilibrium pointp0=(0,0,0).
(a)(b)(c)


## 3.2. Stabilizing the Unstable Equilibrium Pointp±=(±72,±72,27) via Fractional-Order Derivative

Let us design a fractional-order controller for fractional-order Lorenz chaotic system (2.2), and we can yield the following results.Theorem 3.5.
Consider that the controlled fractional-order Lorenz chaotic system is(3.13)Dqx1=10(x2-x1),Dqx2=28x1-x2-x1x3+w1(x1),Dqx3=x1x2-8x33,
where w1(x1)=-k1Dqx1 is the fractional-order controller, and k1 is the feedback coefficient. If k1>(-155+24769)/60, then system (3.13) will gradually converge to the unstable equilibrium point p+=(72,72,27).Proof.
Letx=x1-72,y=x2-72,z=x3-27, and the controlled fractional-order Lorenz chaotic system (3.13) can be
(3.14)Dqx=10(y-x),Dqy=x-y-xz-72z-k1Dqx,Dqz=xy+72x+72y-8z3.
The Jacobi matrix of the fractional-order system (3.14) at equilibrium (0,0,0) is
(3.15)J=[-1010010k1+1-10k1-1-727272-83].
Its characteristic equation isλ3+a1λ2+a2λ+a3=0, where a1=10k1+41/3,a2=8(10k1+11)/3+72, anda3=1440. If k1>(-155+24769)/60, then a1>0,a2>0,a3>0, and a1a2-a3>0. So,
(3.16)|argλi|>π2>0.5πq,(i=1,2,3).
According to Lemma3.2, it implies that the equilibrium point p+=(72,72,27)of system (3.13) is asymptotically stable, that is, the unstable equilibrium point p+=(72,72,27) in fractional-order Lorenz system (2.2) can be stabilized via fractional-order derivative. The proof is completed.For example, choosek1=1. The corresponding numerical result is shown in Figure 4.Time evolution of the equilibrium pointp+=(72,72,27).
(a)(b)(c)Theorem 3.6.
Consider that the controlled fractional-order Lorenz chaotic system is(3.17)Dqx1=10(x2-x1)+w2(x2),Dqx2=28x1-x2-x1x3,Dqx3=x1x2-8x33,
where w2(x2)=-k2Dqx2 is the fractional-order controller, and k2 is the feedback coefficient. If (-7616-56765440)/1248<k2<(-7616+56765440)/1248, then system (3.17) will gradually converge to the unstable equilibrium point  p+=(72,72,27).Proof.
Letx=x1-72,y=x2-72,z=x3-27, and the controlled fractional-order Lorenz chaotic system (3.17) can be
(3.18)Dqx=10(y-x)-k2Dqy,Dqy=x-y-xz-72z,Dqz=xy+72x+72y-8z3.
The Jacobi matrix of the fractional-order system (3.18) at equilibrium (0,0,0) is
(3.19)J=[-10-k210+k2k2721-1-727272-83].
Its characteristic equation isλ3+a1λ2+a2λ+a3=0, where a1=k2+41/3,a2=8(k2+11)/3+72(1-k2), anda3=1440. If (-7616-56765440)/1248<k2<(-7616+56765440)/1248, then a1>0, a2>0,a3>0, and a1a2-a3>0. So,
(3.20)|argλi|>π2>0.5πq,(i=1,2,3).
According to Lemma3.2, it implies that the equilibrium point p+=(72,72,27)of system (3.17) is asymptotically stable, that is, the unstable equilibrium point p+=(72,72,27) in fractional-order Lorenz system (2.2) can be stabilized via fractional-order derivative. The proof is completed.For example, choosek2=-2. The corresponding numerical result is shown in Figure 5.Time evolution of the equilibrium pointp+=(72,72,27).
(a)(b)(c)Similarly, the fractional-order Lorenz chaotic system can be easily controlled to another unstable equilibrium pointp-=(-72,-72,27).Remark 3.7.
In general, there is no universal method to select the controller, and these particular controllers in our paper depend on the structure of the fractional-order chaotic system.Remark 3.8.
The differences of the present control strategy in our paper are compared to the result reported by Tavazoei and Haeri [13] as follows. First, we use the scalar controller in our paper, but they used the vector controller. Second, we discuss the control problem for fractional-order chaotic systems via fractional-order derivative, but they discussed the control problem for integer orders chaos systems via fractional-order derivative.Remark 3.9.
The differences of the present control strategy in our paper are compared to the result reported by Razminia et al. [18] as follows. We discuss the control problem for fractional-order chaotic systems via fractional-order derivative, but they discussed the control problem for fractional-order chaotic systems via state feedback, and they did not use fractional-order derivative.


## 4. Synchronizing the Fractional-Order Lorenz Chaotic System via Fractional-Order Derivative

Now, we design a feedback controller for the fractional-order Lorenz chaotic system (2.2) via fractional-order derivative and obtain the controlled response system (4.1) (4.1)Dqy1=10(y2-y1),Dqy2=28y1-y2-y1y3+V,Dqy3=y1y2-8y33,
where V=k1(Dqy1-Dqx1)+k2(y1-x1)+y1y3-x1x3 is the fractional-order controller, and ki(i=1,2) is the feedback coefficient. Now, we can yield the following theorem.Theorem 4.1.
If the feedback coefficientski(i=1,2) satisfy
(4.2)28-10k1+k2=-10,-1+10(k1-1)<0,
then the fractional-order Lorenz chaotic system (2.2) and the controlled fractional-order Lorenz chaotic system (4.1) achieved synchronization via fractional-order derivative.Proof.
Define the synchronization error variables as follows:(4.3)ei=yi-xi,(i=1,2,3).
By subtracting (2.2) from (4.1), we obtain
(4.4)(Dqe1Dqe2Dqe3)=A(e1e2e3),
where
(4.5)A=[-1010028-10k1+k2-1+10k10x2y1-83].
Because28-10k1+k2=-10, so
(4.6)A=[-10100-10-1+10k10x2y1-83].
Therefore, the eigenvalues are(4.7)λ1,2=β±β2+40β2,λ3=-83<0,
where β=-1+10(k1-1).
Becauseβ=-1+10(k1-1)<0, then all the eigenvalues of matrix A have negative real part. Therefore,
(4.8)|argλi(A)|>π2>0.5πq,(i=1,2,3).
According to Lemma3.2, it implies that the equilibrium point (0,0,0) of error system (4.4) is asymptotically stable, that is, limt→+∞ei=0(i=1,2,3). So, the fractional-order Lorenz chaotic system (2.2) and the controlled fractional-order Lorenz chaotic system (4.1) achieved synchronization via fractional-order derivative. The proof is completed.Next, in order to verify the effectiveness and feasibility of the proposed synchronization scheme, the corresponding numerical simulations are given. For example, choosek1=-0.1 and k2=-39; the time variation of synchronization error is shown in Figure 6. The initial conditions are x(0)=(10,20,30)T, and y(0)=(20,35,50)T respectively.The time variation of synchronization error.
(a)(b)(c)


## 5. Conclusion

Using fractional-order derivative, we can stabilize the unstable equilibrium points of the fractional-order Lorenz chaotic system and realize chaos synchronization for the fractional-order Lorenz chaotic system. The control technique in our paper is simple and theoretically rigorous. Some examples are also given to illustrate the effectiveness of the theoretical result. This proposed control method is different from the previous works and can be applied to other fractional-order chaotic systems.



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