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

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: 𝑥𝑦̇̇==𝑧1+−(𝑏𝑦𝑦−−𝑎𝑥)₂𝑥,𝑧̇=−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: 𝑑^𝑞1𝑥𝑑𝑡^𝑞1𝑑=𝑧+(𝑦−𝑎)𝑥,_𝑞2𝑦𝑑𝑡^𝑞2𝑑_𝑞3=1−𝑏𝑦−𝑥²,𝑧𝑑𝑡^𝑞3=−𝑥−𝑐𝑧,(1.2) where the 𝑞_𝑖th-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 𝑑^𝑞𝑓(𝑡)𝑑𝑡^𝑞=1Γ(𝑚−𝑞)^𝑡0𝑓^𝑚(𝜏)(𝑡−𝜏)^{𝑞−𝑚+1}𝑑𝜏,(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: 𝑑^𝑞𝑥(𝑡)𝑑𝑡^𝑞=𝐴𝑥(𝑡)+𝐶+𝑓(𝑥(𝑡)),0<𝑞<1,(2.1) where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈ℝ^𝑛 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∶ℝ^𝑛→ℝ^𝑛 is a continuous nonlinear vector function.
For the given system (2.1), one can construct the following new system 𝑑^𝑞𝑦(𝑡)𝑑𝑡^𝑞=𝐴𝑦(𝑡)+𝛼(𝐶+𝑓(𝑥(𝑡)))+𝑢(𝑡),0<𝑞<1,(2.2) where 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇∈ℝ^𝑛 is an n-dimensional state vector of the system, 𝑓∶ℝ^𝑛→ℝ^𝑛 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)=𝐴𝑒(𝑡),(2.4) where 𝐴 is an 𝑛×𝑛 linear constant matrix.
Then the error system can be written as 𝑑^𝑞𝑒(𝑡)𝑑𝑡^𝑞=𝑑^𝑞𝑦(𝑡)𝑑𝑡^𝑞−𝛼𝑑^𝑞𝑥(𝑡)𝑑𝑡^𝑞=𝐴𝑒(𝑡)+𝑢(𝑡)=𝐵𝑒(𝑡),(2.5) where 𝐵=𝐴+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 𝜆_𝐼,𝜆₂,…,𝜆_𝑛 satisfy 𝜆_𝑖,𝜆₂,…,𝜆_𝑛<0, then the equilibrium point of synchronization error 𝑒(𝑡) is asymptotically stable and lim_𝑡→∞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: 𝑑^𝑞1𝑥_𝑠𝑑𝑡^𝑞1𝑑_𝑞2=−𝑎𝑥_𝑠+𝑧_𝑠+𝛼𝑥𝑦+𝑢₁,𝑦_𝑠𝑑𝑡^𝑞2𝑑=_𝑞3−𝑏𝑦_𝑠+𝛼1−𝑥²+𝑢₂,𝑧_𝑠𝑑𝑡^𝑞3=−𝑥_𝑠−𝑐𝑧_𝑠+𝑢₃,(3.1) where 𝑥_𝑠,𝑦_𝑠,𝑧_𝑠∈ℝ^𝑛 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: ((12))𝑢𝑢₁₁=𝑎_𝑢𝑥_𝑠−𝛼𝑥−𝑧_𝑠+𝛼𝑧,𝑢₂=𝑏_𝑢𝑦_𝑠−𝛼𝑦,𝑢₃=𝑐_𝑢𝑧_𝑠−𝛼𝑧,(3.2)=𝑎_𝑢𝑧_𝑠−𝛼𝑧,𝑢₂=𝑏_𝑢𝑦_𝑠−𝛼𝑦,𝑢₃=𝑥_𝑠−𝛼𝑥+𝑐_𝑢𝑧_𝑠−𝛼𝑧,(3.3) where 𝑎_𝑢<𝑎, 𝑏_𝑢<𝑏 and 𝑐_𝑢<𝑐.

Proof. The synchronization errors between the master system (1.2) and the slave system (3.1) are defined as  𝑒_𝑥=𝑥_𝑠−𝛼𝑥,  𝑒_𝑦=𝑦_𝑠−𝛼𝑦,  𝑒_𝑧=𝑧_𝑠−𝛼𝑧. Subtracting (1.2) from (3.1) yields the error system as follows: 𝑑^𝑞1𝑒_𝑥𝑑𝑡^𝑞1𝑑_𝑞2=𝑒_𝑧−𝑎𝑒_𝑥+𝑢₁,𝑒_𝑦𝑑𝑡^𝑞2𝑑_𝑞3=−𝑏𝑒_𝑦+𝑢₂,𝑒_𝑧𝑑𝑡^𝑞3=−𝑒_𝑥−𝑐𝑒_𝑧+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: 𝑑^𝑞1𝑒_𝑥𝑑𝑡^𝑞1𝑑_𝑞2=𝑎_𝑢−𝑎𝑒_𝑥,𝑒_𝑦𝑑𝑡^𝑞2𝑑_𝑞3=𝑏_𝑢−𝑏𝑒_𝑦,𝑒_𝑧𝑑𝑡^𝑞3=−𝑒_𝑥+𝑐_𝑢−𝑐𝑒_𝑧,(3.5) which has one equilibrium point at 𝐸^∗=(0,0,0). Its Jacobian matrix evaluated at equilibrium point 𝐸^∗ is given by 𝐽𝐸^∗=⎛⎜⎜⎜⎜⎝𝑎^𝑢0−𝑎𝑏_𝑢00−10−𝑏𝑐_𝑢0−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: 𝑑^𝑞1𝑒_𝑥𝑑𝑡^𝑞1𝑑_𝑞2=𝑎_𝑢−𝑎𝑒_𝑥+𝑒_𝑧,𝑒_𝑦𝑑𝑡^𝑞2𝑑_𝑞3=𝑏_𝑢−𝑏𝑒_𝑦,𝑒_𝑧𝑑𝑡^𝑞3=𝑐_𝑢−𝑐𝑒_𝑧,(3.7) which has one equilibrium point at 𝐸^∗=(0,0,0). Its Jacobian matrix evaluated at equilibrium point 𝐸^∗ is given by 𝐽𝐸^∗=⎛⎜⎜⎜⎜⎝𝑎^𝑢0−𝑎𝑏_𝑢0100−𝑏𝑐_𝑢0−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]−1_𝑘=0𝑥^(𝑘)₀𝑡^𝑘𝑘!+Γ1𝑞₁^𝑡0𝑦(𝑡)=_[𝑚]−1(𝑡−𝜏)^𝑞1−1(𝑧(𝜏)+(𝑦(𝜏)−𝑎)𝑥(𝜏))𝑑𝜏,_𝑘=0𝑦^(𝑘)₀𝑡^𝑘𝑘!+Γ1𝑞₂^𝑡0𝑧(𝑡)=_[𝑚]−1(𝑡−𝜏)^𝑞2−11−𝑏𝑦(𝜏)−𝑥²(𝜏)𝑑𝜏,_𝑘=0𝑧^(𝑘)₀𝑡^𝑘𝑘!+Γ1𝑞₃^𝑡0(𝑡−𝜏)^𝑞3−1(−𝑥(𝜏)−𝑐𝑧(𝜏))𝑑𝜏.(4.1)

Consider the uniform grid {𝑡_𝑛=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: 𝑥_𝑛+1=𝑥₀+ℎ^𝑞1Γ𝑞₁+2𝑧^𝑝𝑛+1+𝑦^𝑝𝑛+1−𝑎𝑥^𝑝𝑛+1+ℎ^𝑞1Γ𝑞₁+2^𝑛_𝑗=0𝑦_𝑛+1𝛼_{1,𝑗,𝑛+1}𝑧_𝑗+𝑦_𝑗−𝑎𝑥_𝑗,=𝑦₀+ℎ^𝑞2Γ𝑞₂+21−𝑏𝑦^𝑝𝑛+1−𝑥^2𝑝𝑛+1+ℎ^𝑞2Γ𝑞₂+2^𝑛_𝑗=0𝑧_𝑛+1𝛼_{2,𝑗,𝑛+1}1−𝑏𝑦_𝑗−𝑥^2𝑗,=𝑧₀+ℎ^𝑞3Γ𝑞₃+2−𝑥^𝑝𝑛+1−𝑐𝑧^𝑝𝑛+1+ℎ^𝑞3Γ𝑞₃+2^𝑛_𝑗=0𝛼_{3,𝑗,𝑛+1}−𝑥_𝑗−𝑐𝑧_𝑗,(4.2) where 𝑥^𝑝𝑛+1=𝑥₀+1Γ𝑞₁^𝑛_𝑗=0𝑦_𝑝𝑛+1𝛽_{1,𝑗,𝑛+1}𝑧_𝑗+𝑦_𝑗−𝑎𝑥_𝑗,=𝑦₀+1Γ𝑞₂^𝑛_𝑗=0𝑧_𝑝𝑛+1𝛽_{2,𝑗,𝑛+1}1−𝑏𝑦_𝑗−𝑥^2𝑗,=𝑧₀+1Γ𝑞₃^𝑛_𝑗=0𝛼_{𝑖,𝑗,𝑛+1}𝛽_{3,𝑗,𝑛+1}−𝑥_𝑗−𝑐𝑧_𝑗,=⎧⎪⎨⎪⎩𝑛^𝑞𝑖+1(𝑛−𝑗−+2𝑛)−𝑞_{𝑖𝑞𝑖+1}(𝑛+1)^𝑞𝑖,𝑗=0,𝛽1,+(𝑛−𝑗)^𝑞𝑖+1−2(𝑛−𝑗+1)^𝑞𝑖+1,1𝑗=≤𝑛𝑗+≤1𝑛,_{𝑖,𝑗,𝑛+1},=ℎ^𝑞𝑖𝑞_𝑖(𝑛−𝑗+1)^𝑞𝑖−ℎ^𝑞𝑖𝑞_𝑖(𝑛−𝑗)^𝑞𝑖,0≤𝑗≤𝑛,𝑖=1,2,3.(4.3) Errors of the above method are ΔΔ𝑥𝑦==mmaaxx_{𝑗=0,1,…,𝑁𝑗=0,1,…,𝑁}||𝑥𝑡_𝑗−𝑥_ℎ𝑡_𝑗||=𝑂ℎ^𝑝1,Δ𝑧=max_{𝑗=0,1,…,𝑁}||𝑦𝑡_𝑗−𝑦_ℎ𝑡_𝑗||=𝑂ℎ^𝑝2,||𝑧𝑡_𝑗−𝑧_ℎ𝑡_𝑗||=𝑂ℎ^𝑝3,(4.4) where 𝑝_𝑖=min(2,1+𝑞_𝑖).

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, 𝑥_𝑠(0)=0.5, 𝑦_𝑠(0)=0, 𝑧_𝑠(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−𝛼𝑧_𝑠, 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, 𝑥_𝑠, 𝑦, 𝑦_𝑠, 𝑧 and 𝑧_𝑠 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)

(b) Errors between systems (1.2) and(3.1)

(c) Time evolutions of 𝑥 and 𝑥𝑠

(d) Time evolutions of 𝑦 and 𝑦𝑠

(e) Time evolutions of 𝑧 and 𝑧𝑠

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=𝑥_𝑠−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, 𝑥_𝑠, 𝑦, 𝑦_𝑠, 𝑧, and 𝑧_𝑠 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)

(b) Errors between systems (1.2) and(3.1)

(c) Time evolutions of 𝑥 and 𝑥𝑠

(d) Time evolutions of 𝑦 and 𝑦𝑠

(e) Time evolutions of 𝑧 and 𝑧𝑠

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).