# Phase and Antiphase Synchronization between 3-Cell CNN and Volta Fractional-Order Chaotic Systems via Active Control

Authors: Zahra Yaghoubi; Hassan Zarabadipour
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

Synchronization of fractional-order chaotic dynamical systems is receiving increasing attention owing to its interesting applications in secure communications of analog and digital signals and cryptographic systems. In this paper, a drive-response synchronization method is studied for “phase and antiphase synchronization” of a class of fractional-order chaotic systems via active control method, using the 3-cell and Volta systems as an example. These examples are used to illustrate the effectiveness of the synchronization method.


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## Body

## 1. Introduction

The theory of fractional calculus is a 300-year-old topic which can trace back to Leibniz, Riemann, Liouville, Grünwald, and Letnikov [1, 2]. However, the fractional calculus did not attract much attention for a long time. Nowadays, the past three decades have witnessed significant progress on fractional calculus, because the applications of fractional calculus were found in more and more scientific fields, covering mechanics, physics, engineering, informatics, and materials. Nowadays, it has been found that some fractional-order differential systems such as the fractional-order jerk model [3], the fractional-order Rössler system [4], and the fractional-order Arneodo system [5] can demonstrate chaotic behavior.Recently, synchronization of fractional-order chaotic systems has started to attract increasing attention due to its potential applications in secure communication and control processing [6, 7]. The concept of synchronization can be extended to generalized synchronization [8], complete synchronization [9], lag synchronization [10], phase synchronization, antiphase synchronization [11], and so on.Synchronization of fractional-order chaotic systems was first studied by Deng and Li [12] who carried out synchronization in case of the fractional Lü system. Further, they have investigated synchronization of fractional Chen system [13].In this paper, phase and anti-phase synchronization using is introduced, which is used to “phase and anti-phase synchronization” for a class of fractional-order chaotic systems using active control method [14].The outline of the rest of the paper is organized as follows. First, Section2 provides a brief review of the fractional derivative and the numerical algorithm of fractional-order differential equation. Section 3 is devoted to 3-Cell and Volta systems description. Next, in Section 4, the definition of phase and anti-phase synchronization is introduced. In Section 5, the proposed method is applied to synchronize two examples of fractional-order chaotic systems. Finally, Section 6 is the brief conclusion.


## 2. Fractional Derivative and Numerical Algorithm of Fractional Differential Equation

There are many definitions of fractional derivatives [15, 16]. Many authors formally use the Riemann-Liouville fractional derivatives, defined by(2.1)Dαx(t)=dmdtmJm-αx(t),α>0,
where m=[α], that is, m is the first integer which is not less than α·Jβ is the β-order Riemann-Liouville integral operator, and Γ(·) is the gamma function which is described as follows:(2.2)Jαx(t)=1Γ(α)∫0t(t-τ)α-1f(τ)dτ,Γ(z)=∫0∞tz-1e-tdt.In this paper, the following definition is used:(2.3)D*αx(t)=Jm-αx(m)(t),α>0.It is common practice to call operatorD*αthe Caputo differential operator of order α [17].The numerical calculation of a fractional differential equation is not so simple as that of an ordinary differential equation. Here, we choose the Caputo version and use a predictor-corrector algorithm for fractional differential equations [18], which is the generalization of Adams-Bashforth-Moulton one. When α>0, the algorithm is universal. The following is a brief introduction of the algorithm. The differential equation(2.4)D*αx(t)=f(t,x(t)),0≤t≤T,x(k)(0)=x0(k),k=0,1,…,m-1.
is equivalent to the Volterra integral equation(2.5)x(t)=∑k-0[a]-1tkk!x0(k)+1Γ(α)∫0t(t-τ)α-1f(τ,x(τ))dτ.


## 3. Systems Description

Chua and Yang introduced the cellular neural network (CNN) in 1988 as a nonlinear dynamical system composed by an array of elementary and locally interacting nonlinear subsystems, so called cells [19].Arena et al. introduced a new class of the CNN with fractional- (noninteger-) order cells [20].Hartley et al. introduced a fractional-order Chua’s system [21]. From this consideration, the idea of developing a fractional-order CNN arose. This system is described as follows:(3.1)Dαx(t)=-x(t)+p1f(x(t))-sf(y(t))-sf(z(t)),Dαy(t)=-y(t)-sf(x(t))+p2f(y(t))-rf(z(t)),Dαz(t)=-z(t)-sf(x(t))+rf(y(t))+p3f(z(t)),
wheref(x)=0.5(|x+1|-|x-1|). In Figure 1 is shown the chaotic behavior for fractional-order chaotic system (3.1), where system parameters are p1=1.24, s=3.21, p2=1.1, r=4.4, and p3=1, commensurate order of the derivatives is α=0.99, and the initial conditions are x(0)=0.1, y(0)=0.1, and z(0)=0.1 for the simulation time Tsim=100 s and time step h=0.005.Figure 1
Chaotic attractor of 3-cell CNN’s system (3.1).Petráš [22, 23] has pointed out that system (3.2) shows chaotic behavior for suitable a, b, and c. Fractional-order Volta system can be written in the form of (3.2) as(3.2)Dαx(t)=-x(t)-ay(t)-z(t)y(t),Dαy(t)=-y(t)-bx(t)-x(t)z(t),Dαz(t)=cz(t)+x(t)y(t)+1.In Figure2 is shown the chaotic behavior for fractional-order chaotic system (3.2), where system parameters are a=19, b=11, and c=0.73, commensurate order of the derivatives is α=0.99, and the initial conditions are x(0)=8, y(0)=2, and z(0)=1 for the simulation time Tsim=20 s and time step h=0.0005.Figure 2
Chaotic attractor of Volta’s system (3.2).


## 4. Phase Synchronization

In this section, we study the phase synchronization between the two fractional-order 3-cell CNN and Volta systems by means of active control.Consider 3-cell CNN system as the drive system(4.1)Dαx1(t)=-x1(t)+p1f(x1(t))-sf(y1(t))-sf(z1(t)),Dαy1(t)=-y1(t)-sf(x1(t))+p2f(y1(t))-rf(z1(t)),Dαz1(t)=-z1(t)-sf(x1(t))+rf(y1(t))+p3f(z1(t)).
and Volta system as the response system(4.2)Dαx2(t)=-x2(t)-ay2(t)-z2(t)y2(t)+u1(t),Dαy2(t)=-y2(t)-bx2(t)-x2(t)z2(t)+u2(t),Dαz2(t)=cz2(t)+x2(t)y2(t)+1+u3(t).Define the error functions ase1=x2-x1, e2=y2-y1, and e3=z2-z1. For phase synchronization, it is essential that the errors ei tend to a zero ast→∞. In order to determine the control functions ui, we subtract (4.1) from (4.2) and obtain(4.3)Dαe1(t)=-e1-ae2-ay1-e3e2-z1e2-e3y1-z1y1-p1f(x1)+sf(y1)+sf(z1)+u1(t),Dαe2(t)=-e2-be1-bx1-e1z1-x1z1-e1e3-x1e3+sf(x1)-p2f(y1)+rf(z1)+u2(t),Dαe3(t)=1+ce3+cz1+e1e2+x1e2+e1y1+x1y1+z1+sf(x1)-rf(y1)-p3f(z1)+u3(t).Choosing the control functions(4.4)u1(t)=ay1+e3e2+z1e2+e3y1+z1y1+p1f(x1)-sf(y1)-sf(z1)+V1(t),u2(t)=bx1+e1z1+x1z1+e1e3+x1e3-sf(x1)+p2f(y1)-rf(z1)+V2(t),u3(t)=-cz1-e1e2-x1e2+e1y1-x1y1-z1-sf(x1)+rf(y1)+p3f(z1)+V3(t).
Equation (4.3) leads to(4.5)Dαe1(t)=-e1-ae2+V1(t),Dαe2(t)=-e2-be1+V2(t),Dαe3(t)=1-ce3+V3(t).The linear functionsV1, V2, and V3 are given by(4.6)V1(t)=e1+ae2+λ1e1,V2(t)=e2+be1+λ2e2,V3(t)=-1+ce3+λ3e3,
where λ1, λ2, and λ3 are the eigenvalues of the linear system (4.5).


### 4.1. Simulation Results

Parameters of 3-cell CNN and Volta systems arep1=1.24, s=3.21, p2=1.1, r=4.4, and p3=1 and a=19, b=11, and c=0.73, respectively. The initial conditions for drive and response systems are x1(0)=0.1, y1(0)=0.1, and z1(0)=0.1 and x2(0)=8, y2(0)=2, and z2(0)=1, respectively. By choosing (λ1, λ2, λ3) = (−1, −1, −1), the control functions can be determined, and phase synchronization between signals (x1, x2), (y1, y2), and (z1, z2) will be achieved, respectively. Numerical results are illustrated in Figures 3(a)–3(c) for fractional-order α=0.99. The curves of synchronization errors are shown in Figure 4, and the phase diagrams of (4.1) and (4.2) are plotted together in Figure 5.Phase synchronization with fractional-orderα=0.99 for signals (x1,x2) in (a), (y1,y2) in (b), and (z1,z2) in (c).
(a)(b)(c)Figure 4
Synchronization errors of drive system (4.1) and response system (4.2).Figure 5
The attractors of drive system (4.1) and response system (4.2).


## 4.1. Simulation Results

Parameters of 3-cell CNN and Volta systems arep1=1.24, s=3.21, p2=1.1, r=4.4, and p3=1 and a=19, b=11, and c=0.73, respectively. The initial conditions for drive and response systems are x1(0)=0.1, y1(0)=0.1, and z1(0)=0.1 and x2(0)=8, y2(0)=2, and z2(0)=1, respectively. By choosing (λ1, λ2, λ3) = (−1, −1, −1), the control functions can be determined, and phase synchronization between signals (x1, x2), (y1, y2), and (z1, z2) will be achieved, respectively. Numerical results are illustrated in Figures 3(a)–3(c) for fractional-order α=0.99. The curves of synchronization errors are shown in Figure 4, and the phase diagrams of (4.1) and (4.2) are plotted together in Figure 5.Phase synchronization with fractional-orderα=0.99 for signals (x1,x2) in (a), (y1,y2) in (b), and (z1,z2) in (c).
(a)(b)(c)Figure 4
Synchronization errors of drive system (4.1) and response system (4.2).Figure 5
The attractors of drive system (4.1) and response system (4.2).


## 5. Antiphase Synchronization

In this section, we study the anti-phase synchronization between the two fractional-order 3-cell CNN and Volta systems by means of active control.Consider 3-cell CNN system as the drive system(5.1)Dαx1(t)=-x1(t)+p1f(x1(t))-sf(y1(t))-sf(z1(t)),Dαy1(t)=-y1(t)-sf(x1(t))+p2f(y1(t))-rf(z1(t)),Dαz1(t)=-z1(t)-sf(x1(t))+rf(y1(t))+p3f(z1(t)).
and Volta system as the response system(5.2)Dαx2(t)=-x2(t)-ay2(t)-z2(t)y2(t)+u1(t),Dαy2(t)=-y2(t)-bx2(t)-x2(t)z2(t)+u2(t),Dαz2(t)=cz2(t)+x2(t)y2(t)+1+u3(t).Define the error functions ase1=x2+x1, e2=y2+y1, and e3=z2+z1. For phase synchronization, it is essential that the errors ei tend to a zero ast→∞. In order to determine the control functions ui, we subtract (5.1) from (5.2) and obtain(5.3)Dαe1(t)=-e1-ae2-ay1-e3e2-z1e2-e3y1-z1y1-2x1+p1f(x1)-sf(y1)-sf(z1)+u1(t),Dαe2(t)=-e2-be1-bx1-e1z1-x1z1-e1e3-x1e3-2y1-sf(x1)+p2f(y1)-rf(z1)+u2(t),Dαe3(t)=1+ce3+cz1+e1e2+x1e2+e1y1+x1y1-z1-sf(x1)+rf(y1)+p3f(z1)+u3(t).Choosing the control functions(5.4)u1(t)=ay1+e3e2+z1e2+e3y1+z1y1+2x1-p1f(x1)+sf(y1)+sf(z1)+V1(t),u2(t)=bx1+e1z1+x1z1+e1e3+x1e3+2y1+sf(x1)-p2f(y1)+rf(z1)+V2(t),u3(t)=-e1e2-x1e2-e1y1-x1y1-cz1+z1+sf(x1)-rf(y1)-p3f(z1)+V3(t).
Equation(5.3) leads to(5.5)Dαe1(t)=-e1-ae2+V1(t),Dαe2(t)=-e2-be1+V2(t),Dαe3(t)=1+ce3+V3(t).The linear functionsV1, V2, and V3 are given by(5.6)V1(t)=e1+ae2+λ1e1,V2(t)=e2+be1+λ2e2,V3(t)=-1-ce3+λ3e3,
where λ1, λ2, and λ3 are the eigenvalues of the linear system (5.5).


### 5.1. Simulation Results

Parameters of 3-cell CNN and Volta systems arep1=1.24, s=3.21, p2=1.1, r=4.4, and p3=1 and a=19, b=11, and c=0.73, respectively. The initial conditions for drive and response systems are x1(0)=0.1, y1(0)=0.1, and z1(0)=0.1, and x2(0) = 8, y2(0)=2, and z2(0)=1, respectively. By choosing (y1,y2,y3) = (−1, −1, −1), the control functions can be determined and phase synchronization between signals (x1,x2), (y1,y2), and (z1,z2) will be achieved, respectively. Numerical results are illustrated in Figures 6(a)–6(c) for fractional-order α=0.99. The curves of synchronization errors are shown in Figure 7, and the phase diagrams of (5.1) and (5.2) are plotted together in Figure 8.Antiphase synchronization with fractional-orderα=0.99 for signals (x1,x2) in (a), (y1,y2) in (b), and (z1,z2) in (c).
(a)(b)(c)Figure 7
Synchronization errors of drive system (5.1) and response system (5.2).Figure 8
The attractors of drive system (5.1) and response system (5.2).


## 5.1. Simulation Results

Parameters of 3-cell CNN and Volta systems arep1=1.24, s=3.21, p2=1.1, r=4.4, and p3=1 and a=19, b=11, and c=0.73, respectively. The initial conditions for drive and response systems are x1(0)=0.1, y1(0)=0.1, and z1(0)=0.1, and x2(0) = 8, y2(0)=2, and z2(0)=1, respectively. By choosing (y1,y2,y3) = (−1, −1, −1), the control functions can be determined and phase synchronization between signals (x1,x2), (y1,y2), and (z1,z2) will be achieved, respectively. Numerical results are illustrated in Figures 6(a)–6(c) for fractional-order α=0.99. The curves of synchronization errors are shown in Figure 7, and the phase diagrams of (5.1) and (5.2) are plotted together in Figure 8.Antiphase synchronization with fractional-orderα=0.99 for signals (x1,x2) in (a), (y1,y2) in (b), and (z1,z2) in (c).
(a)(b)(c)Figure 7
Synchronization errors of drive system (5.1) and response system (5.2).Figure 8
The attractors of drive system (5.1) and response system (5.2).


## 6. Conclusion

This paper investigated the phase and anti-phase synchronization for the fractional-order chaotic systems. Based on the stability criterion of the fractional-order system and tracking control, a synchronization approach is proposed. Finally, the phase and anti-phase synchronization between the fractional-order 3-cell CNN system and fractional-order Volta system are used to demonstrate the effectiveness of phase and anti-phase synchronization schemes.



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