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

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.

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 as𝐷^𝑞𝑓(𝑡)=1Γ(𝑚−𝑞)^𝑡0𝑓^(𝑚)(𝜏)(𝑡−𝜏)^{𝑞+1−𝑚}𝑑𝜏,𝑚−1<𝑞<𝑚,(2.1) where𝐷^𝑞denotes 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:𝐷_𝑞𝐷^𝑞𝑥₁=10𝑥₂−𝑥₁,𝐷𝑥_2𝑞=28𝑥₁−𝑥₂−𝑥₁𝑥₃,𝑥₃=𝑥₁𝑥₂−8𝑥³3,(2.2) 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, 𝑡_𝑛=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:𝑥₁(𝑛+1)=𝑥₁(0)+ℎ^𝑞Γ(𝑞+2)10𝑥^𝑝2(𝑛+1)−𝑥^𝑝1(𝑛+1)+^𝑛_𝑗=0𝑥𝛼_{1,𝑗,𝑛+1}×10𝑥₂(𝑗)−𝑥₁(𝑗)₂,(𝑛+1)=𝑥₂(0)+ℎ^𝑞Γ(𝑞+2)28𝑥^𝑝1+(𝑛+1)−𝑥^𝑝2(𝑛+1)−𝑥^𝑝1(𝑛+1)𝑥^𝑝3(𝑛+1)_𝑛_𝑗=0𝑥𝛼_{2,𝑗,𝑛+1}28𝑥₁(𝑗)−𝑥₂(𝑗)−𝑥₁(𝑗)𝑥₃(𝑗)₃,(𝑛+1)=𝑥₃(0)+ℎ^𝑞Γ(𝑞+2)𝑥^𝑝1(𝑛+1)𝑥^𝑝2(𝑛+1)−8𝑥^𝑝3(𝑛+1)+_𝑛3_𝑗=0𝛼_{3,𝑗,𝑛+1}𝑥¹(𝑗)𝑥₂(𝑗)−8𝑥³(𝑗)3,(2.3)

where𝑥^𝑝1(𝑛+1)=𝑥₁(0)+1Γ(𝑞)^𝑛_𝑗=0𝑥𝑏_{1,𝑗,𝑛+1}×10𝑥₂(𝑗)−𝑥₁(𝑗)_𝑝2,(𝑛+1)=𝑥₂(0)+1Γ(𝑞)^𝑛_𝑗=0𝑥𝑏_{2,𝑗,𝑛+1}28𝑥₁(𝑗)−𝑥₂(𝑗)−𝑥₁(𝑗)𝑥₃(𝑗)_𝑝3,(𝑛+1)=𝑥₃(0)+1Γ(𝑞)^𝑛_𝑗=0𝑏_{3,𝑗,𝑛+1}𝑥¹(𝑗)𝑥₂(𝑗)−8𝑥³(𝑗)3,(2.4)

and for 𝑖=1,2,3,𝛼_{𝑖,𝑗,𝑛+1}=⎧⎪⎨⎪⎩𝑛^𝑞+1(𝑛−−𝑗+(𝑛2−)𝑞_𝑞+1)(𝑛+1)^𝑞,𝑗=0,𝑏1,+(𝑛−𝑗)^𝑞+1−2(𝑛−𝑗+1)^𝑞+1,1𝑗=≤𝑛𝑗+≤1𝑛,_{𝑖,𝑗,𝑛+1},=ℎ^𝑞𝑞(𝑛−𝑗+1)^𝑞−(𝑛−𝑗)^𝑞,0≤𝑗≤𝑛.(2.5)

The error of this approximation is described as follows:||𝑥_𝑖𝑡_𝑛−𝑥_𝑖(𝑛)||=𝑜ℎ^𝑝,𝑝=min(2,1+𝑞).(2.6)

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.

(a)

(b)

Figure 1 

Chaotic attractors of the fractional-order Lorenz chaotic system (2.2) for 𝑞=0.995.

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 𝐷^𝑞𝑥=𝐴𝑥,𝑥(0)=𝑥₀(3.1) 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∈𝑅^𝑛 and 𝐴∈𝑅^𝑛×𝑛.

Lemma 3.2 (see [16, 17]). The fixed points of the following nonlinear commensurate fractional-order autonomous system: 𝐷^𝑞𝑥=𝑓(𝑥)(3.2) 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∈𝑅^𝑛, 𝑓∶𝑅^𝑛→𝑅^𝑛 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 Point 𝑝₀=(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 𝐷_𝑞𝐷^𝑞𝑥₁=10𝑥₂−𝑥₁,𝑥₂=𝐷28_𝑞𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑢₁𝑥₁,𝑥₃=𝑥₁𝑥₂−8𝑥³3,(3.3) where 𝑢₁(𝑥₁)=−𝑘₁₁𝐷^𝑞𝑥₁−𝑘₁₂𝑥₁ is the fractional-order controller, and 𝑘_1𝑖(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 𝐽=⎡⎢⎢⎢⎢⎣28+1−0𝑘010₁₁−𝑘₁₂−10𝑘100₁₁−1−0083⎤⎥⎥⎥⎥⎦.(3.4) Because 𝑘₁₂=38+10𝑘₁₁, so 𝐽=⎡⎢⎢⎢⎢⎣−−01100−10𝑘100₁₁−1−0083⎤⎥⎥⎥⎥⎦.(3.5) Its characteristic equation is(𝜆+8/3)[𝜆²+(10𝑘₁₁+11)𝜆+10(10𝑘₁₁+1)+100]=0, and the eigenvalues are 𝜆_1,2=−10𝑘¹¹+11±10𝑘₁₁+11²−4010𝑘₁₁+112,𝜆³=−83.(3.6) Because −10𝑘₁₁−11≤0, so ||arg𝜆_𝑖||>𝜋2>0.5𝜋𝑞,(𝑖=1,2,3).(3.7)
According to Lemma 3.2, 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, choose 𝑘₁₁=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.

(a)

(b)

(c)

Figure 2 

Time evolution of the equilibrium point 𝑝0=(0,0,0).

Theorem 3.4. Consider the controlled fractional-order Lorenz chaotic system as follows: 𝐷^𝑞𝐷𝑥_1𝑞=10𝑥₂−𝑥₁+𝑢₂𝑥₂,𝐷𝑥_2𝑞=28𝑥₁−𝑥₂−𝑥₁𝑥₃,𝑥₃=𝑥₁𝑥₂−8𝑥³3,(3.8) where 𝑢₂(𝑥₂)=−𝑘₂₁𝐷^𝑞𝑥₂−𝑘₂₂𝑥₂ is the fractional-order controller, and 𝑘_2𝑖(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 𝐽=⎡⎢⎢⎢⎢⎣−10−28𝑘²¹20810−𝑘−01₂₁−𝑘₂₂−0083⎤⎥⎥⎥⎥⎦.(3.9) If 𝑘₂₂=38−𝑘₂₁, then the Jacobi matrix is 𝐽=⎡⎢⎢⎢⎢⎣−10−28𝑘²¹208−−0218−0083⎤⎥⎥⎥⎥⎦.(3.10) Its characteristic equation is(𝜆+8/3)[𝜆²+(28𝑘₂₁+11)𝜆+(28𝑘₂₁+10)+784]=0, and the eigenvalues are 𝜆_1,2=−28𝑘²¹+11±28𝑘₂₁+11²−428𝑘₂₁+10+7842,𝜆³=−83.(3.11) Because −28𝑘₂₁−11≤0, so ||arg𝜆_𝑖||>𝜋2>0.5𝜋𝑞,(𝑖=1,2,3).(3.12)
According to Lemma 3.2, 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, choose 𝑘₂₁=0.1, then 𝑘₂₂=37.9. The corresponding numerical result is shown in Figure 3.

(a)

(b)

(c)

Figure 3 

Time evolution of the equilibrium point 𝑝0=(0,0,0).

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 𝐷_𝑞𝐷^𝑞𝑥₁=10𝑥₂−𝑥₁,𝑥₂=2𝐷8_𝑞𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑤₁𝑥₁,𝑥₃=𝑥₁𝑥₂−8𝑥³3,(3.13) where 𝑤₁(𝑥₁)=−𝑘₁𝐷^𝑞𝑥₁ 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. Let 𝑥=𝑥₁−√72,𝑦=𝑥₂−√72,𝑧=𝑥₃−27, and the controlled fractional-order Lorenz chaotic system (3.13) can be 𝐷𝐷^𝑞𝑥=10(𝑦−𝑥),_𝑞𝐷𝑦=𝑥−𝑦−𝑥𝑧−√72𝑧−𝑘₁𝐷^𝑞𝑥,_𝑞𝑧=𝑥𝑦+√72𝑥+√72𝑦−8𝑧3.(3.14)
The Jacobi matrix of the fractional-order system (3.14) at equilibrium (0,0,0) is 𝐽=⎡⎢⎢⎢⎢⎣10√−𝑘₁10+1−101𝑘0₁−1−√07272√72−83⎤⎥⎥⎥⎥⎦.(3.15)
Its characteristic equation is 𝜆³+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, ||arg𝜆_𝑖||>𝜋2>0.5𝜋𝑞,(𝑖=1,2,3).(3.16)
According to Lemma 3.2, 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, choose 𝑘₁=1. The corresponding numerical result is shown in Figure 4.

(a)

(b)

(c)

Figure 4 

Time evolution of the equilibrium point 𝑝+=(√72,√72,27).

Theorem 3.6. Consider that the controlled fractional-order Lorenz chaotic system is 𝐷^𝑞𝑥𝐷_1𝑞=10𝑥₂−𝑥₁+𝑤₂𝑥₂,𝐷𝑥_2𝑞=28𝑥₁−𝑥₂−𝑥₁𝑥₃,𝑥₃=𝑥₁𝑥₂−8𝑥³3,(3.17) where 𝑤₂(𝑥₂)=−𝑘₂𝐷^𝑞𝑥₂ 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. Let 𝑥=𝑥₁−√72,𝑦=𝑥₂−√72,𝑧=𝑥₃−27, and the controlled fractional-order Lorenz chaotic system (3.17) can be 𝐷𝐷^𝑞𝑥=10(𝑦−𝑥)−𝑘₂𝐷^𝑞𝑦,_𝑞𝐷𝑦=𝑥−𝑦−𝑥𝑧−√72𝑧,_𝑞𝑧=𝑥𝑦+√72𝑥+√72𝑦−8𝑧3.(3.18)
The Jacobi matrix of the fractional-order system (3.18) at equilibrium (0,0,0) is 𝐽=⎡⎢⎢⎢⎢⎣−10−𝑘²110−+1𝑘₂𝑘−₂√√72√7272√72−83⎤⎥⎥⎥⎥⎦.(3.19)
Its characteristic equation is 𝜆³+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, ||arg𝜆_𝑖||>𝜋2>0.5𝜋𝑞,(𝑖=1,2,3).(3.20)
According to Lemma 3.2, 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, choose 𝑘₂=−2. The corresponding numerical result is shown in Figure 5.

(a)

(b)

(c)

Figure 5 

Time evolution of the equilibrium point 𝑝+=(√72,√72,27).

Similarly, the fractional-order Lorenz chaotic system can be easily controlled to another unstable equilibrium point 𝑝₋=(−√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) 𝐷_𝑞𝐷^𝑞𝑦₁=10𝑦₂−𝑦₁,𝑦₂𝐷=28𝑦₁−𝑦₂−𝑦₁𝑦₃+𝑉,_𝑞𝑦₃=𝑦₁𝑦₂−8𝑦³3,(4.1) where 𝑉=𝑘₁(𝐷^𝑞𝑦₁−𝐷^𝑞𝑥₁)+𝑘₂(𝑦₁−𝑥₁)+𝑦₁𝑦₃−𝑥₁𝑥₃ is the fractional-order controller, and 𝑘_𝑖(𝑖=1,2) is the feedback coefficient. Now, we can yield the following theorem.

Theorem 4.1. If the feedback coefficients 𝑘_𝑖(𝑖=1,2) satisfy 28−10𝑘₁+𝑘₂=−10,−1+10𝑘₁−1<0,(4.2) 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: 𝑒_𝑖=𝑦_𝑖−𝑥_𝑖,(𝑖=1,2,3).(4.3)
By subtracting (2.2) from (4.1), we obtain ⎛⎜⎜⎜⎜⎝𝐷^𝑞𝑒₁𝐷^𝑞𝑒₂𝐷^𝑞𝑒₃⎞⎟⎟⎟⎟⎠=𝐴⎛⎜⎜⎜⎜⎝𝑒₁𝑒₂𝑒₃⎞⎟⎟⎟⎟⎠,(4.4) where 𝐴=⎡⎢⎢⎢⎢⎣28−1−𝑥01𝑘₂0₁+𝑘₂−1+1010𝑘₁00𝑦₁−83⎤⎥⎥⎥⎥⎦.(4.5)
Because 28−10𝑘₁+𝑘₂=−10, so 𝐴=⎡⎢⎢⎢⎢⎣−−𝑥11₂00−1+1010𝑘₁00𝑦₁−83⎤⎥⎥⎥⎥⎦.(4.6)
Therefore, the eigenvalues are 𝜆_1,2=𝛽±√𝛽²+40𝛽2,𝜆³=−83<0,(4.7) where 𝛽=−1+10(𝑘₁−1).
Because 𝛽=−1+10(𝑘₁−1)<0, then all the eigenvalues of matrix 𝐴 have negative real part. Therefore, ||arg𝜆_𝑖(𝐴)||>𝜋2>0.5𝜋𝑞,(𝑖=1,2,3).(4.8)
According to Lemma 3.2, it implies that the equilibrium point (0,0,0) of error system (4.4) is asymptotically stable, that is, lim_𝑡→+∞𝑒_𝑖=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, choose 𝑘₁=−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.

(a)

(b)

(c)

Figure 6 

The time variation of synchronization error.

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.

Acknowledgments

The authors are very grateful to the reviewers for their valuable comments and suggestions. This work was supported by the Foundation of Science and Technology project of Chongqing Education Commission (KJ110525) and by the National Natural Science Foundation of China (61104150).