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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 191063, 10 pages
http://dx.doi.org/10.1155/2012/191063
Research Article

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

Baogui Xin1,2 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

Received 4 April 2012; Accepted 16 June 2012

Academic Editor: Her-Terng Yau

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.

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.

fig1
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, 𝑞1=0.88, 𝑞2=0.98, 𝑞3=0.96, 𝑎=0.5, 𝑥(0)=3, 𝑦(0)=4, 𝑧(0)=1, 𝑥𝑠(0)=0.5, 𝑦𝑠(0)=0, 𝑧𝑠(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.

fig2
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, 𝑞1=0.88, 𝑞2=0.98, 𝑞3=0.96, 𝑎=0.5, 𝑥(0)=3, 𝑦(0)=4, 𝑧(0)=1, 𝑥𝑠(0)=0.5, 𝑦𝑠(0)=0, 𝑧𝑠(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).

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