Discrete Dynamics in Nature and Society

PDF
Discrete Dynamics in Nature and Society / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 191063 | 10 pages | https://doi.org/10.1155/2012/191063

Projective Synchronization of N-Dimensional Chaotic Fractional-Order Systems via Linear State Error Feedback Control

Baogui Xin 1,2 and Tong Chen1

1Nonlinear Dynamics and Chaos Group, School of Management, Tianjin University, Tianjin 300072, China

2Center for Applied Mathematics, School of Economics and Management, Shandong University of Science and Technology, Qingdao 266510, China

Academic Editor: Her-Terng Yau
Received04 Apr 2012
Accepted16 Jun 2012
Published24 Jul 2012

Abstract

Based on linear feedback control technique, a projective synchronization scheme of N-dimensional chaotic fractional-order systems is proposed, which consists of master and slave fractional-order financial systems coupled by linear state error variables. It is shown that the slave system can be projectively synchronized with the master system constructed by state transformation. Based on the stability theory of linear fractional order systems, a suitable controller for achieving synchronization is designed. The given scheme is applied to achieve projective synchronization of chaotic fractional-order financial systems. Numerical simulations are given to verify the effectiveness of the proposed projective synchronization scheme.

1. Introduction

The fractional calculus, as a very old mathematical topic, has been in existence for more than 300 years [1], but it has not been widely used in the science and engineering for many years, because its geometrical or physical interpretation has been not widely accepted [2, 3]. However, due to the long memory advantage, in the recent past, the fractional calculus has been widely applied to diffusion processes [4], Sprott chaotic systems [5], happiness and love [6], economics and finances [7, 8], and so on.

Chaos synchronization has been widely investigated in science and engineering such as humanistic community [9], physical science [10], and secure communications [11]. The chaos projective synchronization was first reported by Mainieri and Rehacek [12]. This type of projective synchronization is interesting due to its property of proportionally diminished or enlarged synchronizing responses, but the early work was limited to a certain kind of nonlinear systems with partly linear properties. Chaos projective synchronization has been an active research topic in nonlinear science until Wen and Xu [13, 14] proposed an observer-based control method and showed “no special limitation” to nonlinear systems themselves to achieve this type of chaos synchronization. Wen and coauthors tried to explore the potential applications of projective synchronization to noise reduction in mechanical engineering [15, 16] or design bifurcation solutions based on the property of projective synchronization [17]. Synchronization of fractional-order chaotic systems was first presented by Deng and Li [18]. There has been an increasing interest in fractional-order chaos synchronization during the last few years because of its potentials in both theory and applications [19]. Peng et al. [20] proposed the generalized projective synchronization scheme of fractional order chaotic systems via a transmitted signal. Shao [21] proposed a method to achieve general projective synchronization of two fractional order Rossler systems. Odibat et al. [22] studied synchronization of 3-dimensional chaotic fractional-order systems via linear control. The advantage of the linear feedback controller is that it is robust and linear, and moreover, it is easier to be designed and implemented for chaos synchronization than standard PID feedback controller, sliding mode controller, nonlinear feedback controller, and so on [23–25].

Huang and Li [26] reported an integer order financial model as follows: ̇𝑥=𝑧+(𝑦−𝑎)𝑥,̇𝑦=1−𝑏𝑦−𝑥2,̇𝑧=−𝑥−𝑐𝑧,(1.1) where 𝑥 is the interest rate, 𝑦 is the investment demand, 𝑧 is the price index, 𝑎 is the saving amount, 𝑏 is the cost per investment, 𝑐 is the demand elasticity of commercial markets, and all three constants 𝑎,𝑏,𝑐≥0.

Chen [7] considered the generalization of system (1.1) for the fractional-order model which takes the following form: 𝑑𝑞1𝑥𝑑𝑡𝑞1𝑑=𝑧+(𝑦−𝑎)𝑥,𝑞2𝑦𝑑𝑡𝑞2=1−𝑏𝑦−𝑥2,𝑑𝑞3𝑧𝑑𝑡𝑞3=−𝑥−𝑐𝑧,(1.2) where the 𝑞𝑖th-order fractional derivative is given by the following Caputo definition, 𝑖=1,2,3.

Definition 1.1 (see [27]). The 𝑞th-order fractional derivative of function  𝑓(𝑡) with respect to 𝑡 and the terminal value 0 is written as 𝑑𝑞𝑓(𝑡)𝑑𝑡𝑞=1Γ(𝑚−𝑞)𝑡0𝑓𝑚(𝜏)(𝑡−𝜏)𝑞−𝑚+1𝑑𝜏,(1.3) where 𝑚 is an integer and satisfies 𝑚−1≤𝑞<𝑚.

Remark 1.2. If 𝑞1=𝑞2=𝑞3=1, the system (1.2) degenerates into the system (1.1).
The remainder of this paper is organized as follows. In Section 2, a projective synchronization scheme of n-dimensional chaotic fractional-order systems is proposed. In Section 3, a projective synchronization scheme of chaotic fractional-order financial systems is studied. In Section 4, the Adams-Bashforth-Moulton predictor-corrector scheme of a fractional-order system is described. Numerical simulations are given in Section 5 to show the effectiveness of the proposed synchronization scheme. Finally, the paper is concluded in Section 6.

2. A Projective Synchronization Scheme of 𝑛-Dimensional Chaotic Fractional-Order Systems

Definition 2.1. The projective synchronization discussed in this paper is defined as two relative chaotic dynamical systems can be synchronous with a desired scaling factor.
Consider a fractional-order chaotic system as follows: 𝑑𝑞𝑥(𝑡)𝑑𝑡𝑞=𝐴𝑥(𝑡)+𝐶+𝑓(𝑥(𝑡)),0<𝑞<1,(2.1) where 𝑥=(𝑥1,𝑥2,…,𝑥𝑛)𝑇∈ℝ𝑛 is an n-dimensional state vector of the system, 𝐴 is an 𝑛×𝑛 linear constant matrix, 𝐶 is an 𝑛×1 linear constant matrix, 𝑓∶ℝ𝑛→ℝ𝑛 is a continuous nonlinear vector function.
For the given system (2.1), one can construct the following new system 𝑑𝑞𝑦(𝑡)𝑑𝑡𝑞=𝐴𝑦(𝑡)+𝛼(𝐶+𝑓(𝑥(𝑡)))+𝑢(𝑡),0<𝑞<1,(2.2) where 𝑦=(𝑦1,𝑦2,…,𝑦𝑛)𝑇∈ℝ𝑛 is an n-dimensional state vector of the system, 𝑓∶ℝ𝑛→ℝ𝑛 is a continuous nonlinear vector function, 𝐴,𝐶 are linear constant matrix, 𝛼 is a desired scaling factor, 𝑢(𝑡) is a linear state error feedback controller.
The synchronization error between the master system (2.1) and the slave system (2.2) is defined as 𝑒(𝑡)=𝑦(𝑡)−𝛼𝑥(𝑡),𝑖=1,2,…,𝑛.(2.3) The linear state error feedback controller is defined as 𝑢(𝑡)=𝐴𝑒(𝑡),(2.4) where 𝐴 is an 𝑛×𝑛 linear constant matrix.
Then the error system can be written as 𝑑𝑞𝑒(𝑡)𝑑𝑡𝑞=𝑑𝑞𝑦(𝑡)𝑑𝑡𝑞𝑑−𝛼𝑞𝑥(𝑡)𝑑𝑡𝑞=𝐴𝑒(𝑡)+𝑢(𝑡)=𝐵𝑒(𝑡),(2.5) where 𝐴𝐵=𝐴+ is an 𝑛×𝑛 linear constant matrix. Obviously the orginal point is the equilibrium point of system (2.5).
According to the stability criterion of linear fractional-order dynamical system, one can directly obtain the following theorem.

Theorem 2.2. If 𝐵 is an upper or lower triangular matrix and all eigenvalues 𝜆𝐼,𝜆2,…,𝜆𝑛 satisfy 𝜆𝑖,𝜆2,…,𝜆𝑛<0, then the equilibrium point of synchronization error 𝑒(𝑡) is asymptotically stable and lim𝑡→∞𝑒(𝑡)=0, that is, the master system (2.1) and the slave system (2.2) achieve projective synchronization.

Remark 2.3. If 𝛼=1 and 𝑛=3, the above synchronization scheme is similar to the synchronization scheme in [28].

Remark 2.4. If 𝛼=1 and 𝑛=3, the above synchronization scheme degenerates into the synchronization scheme proposed by Odibat et al. [22].

Remark 2.5. If 𝑛=3, the above synchronization scheme degenerates into the synchronization scheme proposed by Xin et al. [29].

3. A Projective Synchronization Scheme of Chaotic Fractional-Order Financial Systems

In order to investigate the synchronization behaviors of two chaotic fractional-order financial systems, one can set a master-slave configuration with a master system given by the fractional-order financial systems (1.2) and with a slave system (denoted by the subscript 𝑠) as follows: 𝑑𝑞1𝑥𝑠𝑑𝑡𝑞1=−𝑎𝑥𝑠+𝑧𝑠+𝛼𝑥𝑦+𝑢1,𝑑𝑞2𝑦𝑠𝑑𝑡𝑞2=−𝑏𝑦𝑠+𝛼1−𝑥2+𝑢2,𝑑𝑞3𝑧𝑠𝑑𝑡𝑞3=−𝑥𝑠−𝑐𝑧𝑠+𝑢3,(3.1) where 𝑥𝑠,𝑦𝑠,𝑧𝑠∈ℝ𝑛 have the same meanings as 𝑥,𝑦,𝑧  of system (1.2), 𝛼 is a desired scaling factor, 𝑢1,𝑢2,𝑢3 are linear state error feedback controllers.

Proposition 3.1. The drive system (1.2) and the slave system (3.1) will approach global synchronization for any initial condition if anyone of the following control laws holds: (1)𝑢1=𝑎𝑢𝑥𝑠−𝛼𝑥−𝑧𝑠+𝛼𝑧,𝑢2=𝑏𝑢𝑦𝑠−𝛼𝑦,𝑢3=𝑐𝑢𝑧𝑠(−𝛼𝑧,(3.2)2)𝑢1=𝑎𝑢𝑧𝑠−𝛼𝑧,𝑢2=𝑏𝑢𝑦𝑠−𝛼𝑦,𝑢3=𝑥𝑠−𝛼𝑥+𝑐𝑢𝑧𝑠−𝛼𝑧,(3.3) where 𝑎𝑢<𝑎, 𝑏𝑢<𝑏 and 𝑐𝑢<𝑐.

Proof. The synchronization errors between the master system (1.2) and the slave system (3.1) are defined as  𝑒𝑥=𝑥𝑠−𝛼𝑥,  𝑒𝑦=𝑦𝑠−𝛼𝑦,  𝑒𝑧=𝑧𝑠−𝛼𝑧. Subtracting (1.2) from (3.1) yields the error system as follows: 𝑑𝑞1𝑒𝑥𝑑𝑡𝑞1=𝑒𝑧−𝑎𝑒𝑥+𝑢1,𝑑𝑞2𝑒𝑦𝑑𝑡𝑞2=−𝑏𝑒𝑦+𝑢2,𝑑𝑞3𝑒𝑧𝑑𝑡𝑞3=−𝑒𝑥−𝑐𝑒𝑧+𝑢3.(3.4) For the first control law in Proposition 3.1, substituting (3.2) into the error system (3.4), the system (3.4) can be rewritten as follows: 𝑑𝑞1𝑒𝑥𝑑𝑡𝑞1=𝑎𝑢𝑒−𝑎𝑥,𝑑𝑞2𝑒𝑦𝑑𝑡𝑞2=𝑏𝑢𝑒−𝑏𝑦,𝑑𝑞3𝑒𝑧𝑑𝑡𝑞3=−𝑒𝑥+𝑐𝑢𝑒−𝑐𝑧,(3.5) which has one equilibrium point at 𝐸∗=(0,0,0). Its Jacobian matrix evaluated at equilibrium point 𝐸∗ is given by 𝐽𝐸∗=⎛⎜⎜⎜⎜⎝𝑎𝑢−𝑎000𝑏𝑢−𝑏0−10𝑐𝑢⎞⎟⎟⎟⎟⎠−𝑐,(3.6) which is a lower triangular matrix and its eigenvalues satisfy 𝜆1,𝜆2,𝜆3<0.
For the second control law in Proposition 3.1, substituting (3.3) into the error system (3.4), the system (3.4) can be rewritten as follows: 𝑑𝑞1𝑒𝑥𝑑𝑡𝑞1=𝑎𝑢𝑒−𝑎𝑥+𝑒𝑧,𝑑𝑞2𝑒𝑦𝑑𝑡𝑞2=𝑏𝑢𝑒−𝑏𝑦,𝑑𝑞3𝑒𝑧𝑑𝑡𝑞3=𝑐𝑢𝑒−𝑐𝑧,(3.7) which has one equilibrium point at 𝐸∗=(0,0,0). Its Jacobian matrix evaluated at equilibrium point 𝐸∗ is given by 𝐽𝐸∗=⎛⎜⎜⎜⎜⎝𝑎𝑢−𝑎010𝑏𝑢−𝑏000𝑐𝑢⎞⎟⎟⎟⎟⎠−𝑐,(3.8) which is an upper triangular matrix and its eigenvalues satisfy  𝜆1,𝜆2,𝜆3<0.
It follows from Theorem 2.2 that system (3.4) is asymptotically stable, that is, the master system (1.2) and the slave system (3.1) are synchronized finally.
The Proposition 3.1 is proved.

4. Numerical Method for Solving System (1.2)

An improved Adams-Bashforth-Moulton predictor-corrector scheme [30] can be employed to solve fractional-order ordinary differential equations. The improved predictor-corrector scheme of system (1.2) can be described as follows.

With the initial value (𝑥(𝑘)0,𝑦(𝑘)0,𝑧(𝑘)0), 𝑘=0,1,…,[𝑚]−1, system (1.2) is equivalent to the Volterra integral equations as follows: 𝑥(𝑡)=[𝑚]−1𝑘=0𝑥(𝑘)0𝑡𝑘+1𝑘!Γ𝑞1𝑡0(𝑡−𝜏)𝑞1−1(𝑧(𝜏)+(𝑦(𝜏)−𝑎)𝑥(𝜏))𝑑𝜏,𝑦(𝑡)=[𝑚]−1𝑘=0𝑦(𝑘)0𝑡𝑘+1𝑘!Γ𝑞2𝑡0(𝑡−𝜏)𝑞2−11−𝑏𝑦(𝜏)−𝑥2(𝜏)𝑑𝜏,𝑧(𝑡)=[𝑚]−1𝑘=0𝑧(𝑘)0𝑡𝑘+1𝑘!Γ𝑞3𝑡0(𝑡−𝜏)𝑞3−1(−𝑥(𝜏)−𝑐𝑧(𝜏))𝑑𝜏.(4.1)

Consider the uniform grid {𝑡𝑛=𝑛ℎ,𝑛=0,1,…,𝑁} for some integers 𝑁∈𝑍+ and ℎ=𝑇/𝑁, system (4.1) can be approximated to the following difference equations: 𝑥𝑛+1=𝑥0+ℎ𝑞1Γ𝑞1𝑧+2𝑝𝑛+1+𝑦𝑝𝑛+1𝑥−𝑎𝑝𝑛+1+ℎ𝑞1Γ𝑞1+2𝑛𝑗=0𝛼1,𝑗,𝑛+1𝑧𝑗+𝑦𝑗𝑥−𝑎𝑗,𝑦𝑛+1=𝑦0+ℎ𝑞2Γ𝑞2+21−𝑏𝑦𝑝𝑛+1−𝑥2𝑝𝑛+1+ℎ𝑞2Γ𝑞2+2𝑛𝑗=0𝛼2,𝑗,𝑛+11−𝑏𝑦𝑗−𝑥2𝑗,𝑧𝑛+1=𝑧0+ℎ𝑞3Γ𝑞3+2−𝑥𝑝𝑛+1−𝑐𝑧𝑝𝑛+1+ℎ𝑞3Γ𝑞3+2𝑛𝑗=0𝛼3,𝑗,𝑛+1−𝑥𝑗−𝑐𝑧𝑗,(4.2) where 𝑥𝑝𝑛+1=𝑥0+1Γ𝑞1𝑛𝑗=0𝛽1,𝑗,𝑛+1𝑧𝑗+𝑦𝑗𝑥−𝑎𝑗,𝑦𝑝𝑛+1=𝑦0+1Γ𝑞2𝑛𝑗=0𝛽2,𝑗,𝑛+11−𝑏𝑦𝑗−𝑥2𝑗,𝑧𝑝𝑛+1=𝑧0+1Γ𝑞3𝑛𝑗=0𝛽3,𝑗,𝑛+1−𝑥𝑗−𝑐𝑧𝑗,𝛼𝑖,𝑗,𝑛+1=⎧⎪⎨⎪⎩𝑛𝑞𝑖+1−𝑛−𝑞𝑖(𝑛+1)𝑞𝑖,𝑗=0,(𝑛−𝑗+2)𝑞𝑖+1+(𝑛−𝑗)𝑞𝑖+1−2(𝑛−𝑗+1)𝑞𝑖+1𝛽,1≤𝑗≤𝑛,1,𝑗=𝑛+1,𝑖,𝑗,𝑛+1=ℎ𝑞𝑖𝑞𝑖(𝑛−𝑗+1)𝑞𝑖−ℎ𝑞𝑖𝑞𝑖(𝑛−𝑗)𝑞𝑖,0≤𝑗≤𝑛,𝑖=1,2,3.(4.3) Errors of the above method are Δ𝑥=max𝑗=0,1,…,𝑁||𝑥𝑡𝑗−𝑥ℎ𝑡𝑗||ℎ=𝑂𝑝1,Δ𝑦=max𝑗=0,1,…,𝑁||𝑦𝑡𝑗−𝑦ℎ𝑡𝑗||ℎ=𝑂𝑝2,Δ𝑧=max𝑗=0,1,…,𝑁||𝑧𝑡𝑗−𝑧ℎ𝑡𝑗||ℎ=𝑂𝑝3,(4.4) where 𝑝𝑖=min(2,1+𝑞𝑖).

5. Numerical Simulations

Based on the Adams-Bashforth-Moulton predictor-corrector scheme, one can let the master system (1.2) and the slave system (3.1) with parameters 𝑎=1, 𝑏=0.1, 𝑐=1.2, 𝑞1=0.88, 𝑞2=0.98, 𝑞3=0.96, 𝑎=0.5, initial values 𝑥(0)=3, 𝑦(0)=4, 𝑧(0)=1, 𝑥𝑠(0)=0.5, 𝑦𝑠(0)=0, 𝑧𝑠(0)=2.5. The following numerical simulations are carried out to illustrate the main results.

From the first control law of Proposition 3.1, the linear controllers have the following form: 𝑢1=𝑧−𝛼𝑧𝑠, 𝑢2=0, 𝑢3=0. The chaotic attractors of the master system (1.2) and the slave system (3.1) are shown in Figure 1(a). Synchronization errors between systems (1.2) and (3.1) are shown in Figure 1(b). Time evolutions of 𝑥, 𝑥𝑠, 𝑦, 𝑦𝑠, 𝑧 and 𝑧𝑠 are shown in Figures 1(c)–1(e), respectively. From Figures 1(a)–1(e), it is clear that the projective synchronization is achieved for all these values.

(a) Chaotic attractors of systems (1.2) and(3.1)
(a) Chaotic attractors of systems (1.2) and(3.1)
(b) Errors between systems (1.2) and(3.1)
(b) Errors between systems (1.2) and(3.1)
(c) Time evolutions of 𝑥 and 𝑥 𝑠
(c) Time evolutions of 𝑥 and 𝑥 𝑠
(d) Time evolutions of 𝑦 and 𝑦 𝑠
(d) Time evolutions of 𝑦 and 𝑦 𝑠
(e) Time evolutions of 𝑧 and 𝑧 𝑠
(e) Time evolutions of 𝑧 and 𝑧 𝑠
Figure 1 
Synchronization errors between the master system (1.2) and the slave system (3.1) with 𝑎 = 1 , 𝑏 = 0 . 1 , 𝑐 = 1 . 2 , 𝑞 1 = 0 . 8 8 , 𝑞 2 = 0 . 9 8 , 𝑞 3 = 0 . 9 6 , 𝑎 = 0 . 5 , 𝑥 ( 0 ) = 3 , 𝑦 ( 0 ) = 4 , 𝑧 ( 0 ) = 1 , 𝑥 𝑠 ( 0 ) = 0 . 5 , 𝑦 𝑠 ( 0 ) = 0 , 𝑧 𝑠 ( 0 ) = 2 . 5 .

From the second control law of Proposition 3.1, the linear controllers have the following form: 𝑢1=0, 𝑢2=0, 𝑢3=𝑥𝑠−𝛼𝑥. The chaotic attractors of the master system (1.2) and the slave system (3.1) are shown in Figure 2(a). Synchronization errors between systems (1.2) and (3.1) are shown in Figure 2(b). Time evolutions of 𝑥, 𝑥𝑠, 𝑦, 𝑦𝑠, 𝑧, and 𝑧𝑠 are shown in Figures 2(c)–2(e), respectively. From Figures 2(a)–2(e), it is clear that the projective synchronization is achieved for all these values.

(a)   Chaotic attractors of systems (1.2) and(3.1)
(a)   Chaotic attractors of systems (1.2) and(3.1)
(b) Errors between systems (1.2) and(3.1)
(b) Errors between systems (1.2) and(3.1)
(c) Time evolutions of 𝑥 and 𝑥 𝑠
(c) Time evolutions of 𝑥 and 𝑥 𝑠
(d) Time evolutions of 𝑦 and 𝑦 𝑠
(d) Time evolutions of 𝑦 and 𝑦 𝑠
(e) Time evolutions of 𝑧 and 𝑧 𝑠
(e) Time evolutions of 𝑧 and 𝑧 𝑠
Figure 2 
Synchronization errors between the master system (1.2) and the slave system (3.1) with 𝑎 = 1 , 𝑏 = 0 . 1 , 𝑐 = 1 . 2 , 𝑞 1 = 0 . 8 8 , 𝑞 2 = 0 . 9 8 , 𝑞 3 = 0 . 9 6 , 𝑎 = 0 . 5 , 𝑥 ( 0 ) = 3 , 𝑦 ( 0 ) = 4 , 𝑧 ( 0 ) = 1 , 𝑥 𝑠 ( 0 ) = 0 . 5 , 𝑦 𝑠 ( 0 ) = 0 , 𝑧 𝑠 ( 0 ) = 2 . 5 .

6. Conclusions

In this paper, we propose a projective synchronization scheme of n-dimensional chaotic fractional-order systems via line error feedback control, and apply the scheme to achieve synchronization of the chaotic fractional-order financial systems. Numerical simulations validate the main results of this work.

Acknowledgment

This work was supported in part by Excellent Young Scientist Foundation of Shandong Province (Grant no. BS2011SF018), National Social Science Foundation of China (Grant no. 12BJY103), Humanities and Social Sciences Foundation of the Ministry of Education of China (Grant no. 11YJCZH200), and Research Project of “SUST Spring Bud” (Grant no. 2010AZZ067).

References

  1. B. Ross, A Brief History and Exposition of the Fundamental Theory of Fractional Calculus, vol. 457 of Lecture Notes in Mathematics, Springer, New York, NY, USA, 1975. View at: Zentralblatt MATH
  2. R. Hilfer, Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at: Publisher Site | Zentralblatt MATH
  3. R. E. Gutiérrez, J. M. Rosário, and J. MacHado, “Fractional order calculus: basic concepts and engineering applications,” Mathematical Problems in Engineering, vol. 2010, Article ID 375858, 19 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. X. Jiang, M. Xu, and H. Qi, “The fractional diffusion model with an absorption term and modified Fick's law for non-local transport processes,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 262–269, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. G. H. Erjaee and M. Alnasr, “Phase synchronization in coupled Sprott chaotic systems presented by fractional differential equations,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 753746, 10 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. L. Song, S. Xu, and J. Yang, “Dynamical models of happiness with fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 616–628, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. W. C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons & Fractals, vol. 36, no. 5, pp. 1305–1314, 2008. View at: Publisher Site | Google Scholar
  8. B. G. Xin, T. Chen, and Y. Q. Liu, “Complexity evolvement of a chaotic fractional-order financial system,” Acta Physica Sinica, vol. 60, no. 4, Article ID 048901, 2011. View at: Google Scholar
  9. L. Song and J. Yang, “Chaos control and synchronization of dynamical model of happiness with fractional order,” in Proceedings of the 4th IEEE Conference on Industrial Electronics and Applications (ICIEA '09), pp. 919–924, May 2009. View at: Publisher Site | Google Scholar
  10. B. Ratajska-Gadomska and W. Gadomski, “On control of chaos and synchronization in the vibronic laser,” Optics Express, vol. 17, no. 16, pp. 14166–14171, 2009. View at: Publisher Site | Google Scholar
  11. J. Cai, M. Lin, and Z. Yuan, “Secure communication using practical synchronization between two different chaotic systems with uncertainties,” Mathematical & Computational Applications, vol. 15, no. 2, pp. 166–175, 2010. View at: Google Scholar | Zentralblatt MATH
  12. R. Mainieri and J. Rehacek, “Projective synchronization in three-dimensional chaotic systems,” Physical Review Letters, vol. 82, no. 15, pp. 3042–3045, 1999. View at: Publisher Site | Google Scholar
  13. G. L. Wen and D. Xu, “Observer-based control for full-state projective synchronization of a general class of chaotic maps in any dimension,” Physics Letters A, vol. 333, no. 5-6, pp. 420–425, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. G. Wen and D. Xu, “Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems,” Chaos, Solitons & Fractals, vol. 26, no. 1, pp. 71–77, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. G. Wen, Y. Lu, Z. Zhang, C. Ma, H. Yin, and Z. Cui, “Line spectra reduction and vibration isolation via modified projective synchronization for acoustic stealth of submarines,” Journal of Sound and Vibration, vol. 324, no. 3-5, pp. 954–961, 2009. View at: Publisher Site | Google Scholar
  16. G. Wen, S. Yao, Z. Zhang et al., “Vibration control for active seat suspension system based on projective chaos synchronisation,” International Journal of Vehicle Design, vol. 58, no. 1, pp. 1–14, 2012. View at: Publisher Site | Google Scholar
  17. G. Wen, “Designing Hopf limit circle to dynamical systems via modified projective synchronization,” Nonlinear Dynamics, vol. 63, no. 3, pp. 387–393, 2011. View at: Publisher Site | Google Scholar
  18. W. H. Deng and C. P. Li, “Chaos synchronization of the fractional Lü system,” Physica A, vol. 353, no. 1–4, pp. 61–72, 2005. View at: Publisher Site | Google Scholar
  19. W. Deng, “Generalized synchronization in fractional order systems,” Physical Review E, vol. 75, no. 5, Article ID 056201, 2007. View at: Publisher Site | Google Scholar
  20. G. Peng, Y. Jiang, and F. Chen, “Generalized projective synchronization of fractional order chaotic systems,” Physica A, vol. 387, no. 14, pp. 3738–3746, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  21. S. Shao, “Controlling general projective synchronization of fractional order Rossler systems,” Chaos, Solitons & Fractals, vol. 39, no. 4, pp. 1572–1577, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  22. Z. M. Odibat, N. Corson, M. A. Aziz-Alaoui, and C. Bertelle, “Synchronization of chaotic fractional-order systems via linear control,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 20, no. 1, pp. 81–97, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. B. Xin, T. Chen, and Y. Liu, “Synchronization of chaotic fractional-order WINDMI systems via linear state error feedback control,” Mathematical Problems in Engineering, vol. 2010, Article ID 859685, 10 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  24. Z. Wei, “Synchronization of coupled nonidentical fractional-order hyperchaotic systems,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 430724, 9 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  25. M. M. El-Dessoky and E. Saleh, “Generalized projective synchronization for different hyperchaotic dynamical systems,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 437156, 19 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  26. D. Huang and H. Li, Theory and Method of the Nonlinear Economics, Sichuan University Press, Chengdu, China, 1993.
  27. I. Pdlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  28. L. Chen, Y. Chai, and R. Wu, “Control and synchronization of fractional-order financial system based on linear control,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 958393, 21 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  29. B. Xin, T. Chen, and Y. Liu, “Projective synchronization of chaotic fractional-order energy resources demand-supply systems via linear control,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4479–4486, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  30. W. Deng, “Numerical algorithm for the time fractional Fokker-Planck equation,” Journal of Computational Physics, vol. 227, no. 2, pp. 1510–1522, 2007. View at: Publisher Site | Google Scholar

Copyright © 2012 Baogui Xin and Tong Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

768 Views | 1132 Downloads | 6 Citations
PDF Download Citation Citation
Download other formatsMore
ePub
XML
Order printed copiesOrder