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

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.


Introduction
The theory of fractional calculus is a 300-year-old topic which can trace back to Leibniz, Riemann, Liouville, Gr ünwald, 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 fractionalorder differential systems such as the fractional-order jerk model 3 , the fractional-order R össler 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.

Mathematical Problems in Engineering
Synchronization of fractional-order chaotic systems was first studied by Deng and Li 12 who carried out synchronization in case of the fractional L ü 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, Section 2 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.

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 where m α , that is, m is the first integer which is not less than α•J β is the β-order Riemann-Liouville integral operator, and Γ • is the gamma function which is described as follows:

2.2
In this paper, the following definition is used: It is common practice to call operator D α * the 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 predictorcorrector 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 is equivalent to the Volterra integral equation 2.5

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: In Figure 1 is shown the chaotic behavior for fractionalorder chaotic system 3.1 , where system parameters are p 1 1.24, s 3.21, p 2 1.1, r 4.4, and p 3 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 T sim 100 s and time step h 0.005.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 In Figure 2 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 T sim 20 s and time step h 0.0005.

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 and Volta system as the response system

4.2
Define the error functions as e 1 x 2 − x 1 , e 2 y 2 − y 1 , and e 3 z 2 − z 1 .For phase synchronization, it is essential that the errors e i tend to a zero as t → ∞.In order to determine the control functions u i , we subtract 4.1 from 4.2 and obtain Choosing the control functions

4.5
The linear functions V 1 , V 2 , and V 3 are given by

Simulation Results
Parameters 1, respectively.By choosing λ 1 , λ 2 , λ 3 −1, −1, −1 , the control functions can be determined, and phase synchronization between signals x 1 , x 2 , y 1 , y 2 , and z 1 , z 2 will be achieved, respectively.Numerical results are illustrated in Figures 3 a -3 c for fractionalorder 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.

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 and Volta system as the response system

5.2
Mathematical Problems in Engineering Define the error functions as e 1 x 2 x 1 , e 2 y 2 y 1 , and e 3 z 2 z 1 .For phase synchronization, it is essential that the errors e i tend to a zero as t → ∞.In order to determine the control functions u i , we subtract 5.1 from 5.2 and obtain Choosing the control functions

5.4
Mathematical Problems in Engineering

5.5
The linear functions V 1 , V 2 , and V 3 are given by
Mathematical Problems in Engineering

Conclusion
This paper investigated the phase and anti-phase synchronization for the fractionalorder 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

Figure 3 :
Figure 3: Phase synchronization with fractional-order α 0.99 for signals x 1 , x 2 in a , y 1 , y 2 in b , and z 1 , z 2 in c .

Figure 6 :
Figure 6: Antiphase synchronization with fractional-order α 0.99 for signals x 1 , x 2 in a , y 1 , y 2 in b , and z 1 , z 2 in c .
of 3-cell CNN and Volta systems are p 1 1.24, s 3.21, p 2 1.1, r 4.4, and p 3 1 and a 19, b 11, and c 0.73, respectively.The initial conditions for drive and response systems are x 1 .By choosing y 1 , y 2 , y 3−1, −1, −1 , the control functions can be determined and phase synchronization between signals x 1 , x 2 , y 1 , y 2 , and z 1 , z 2 will be achieved, respectively.Numerical results are illustrated in Figures6 a -6 cfor fractionalorder α 0.99.The curves of synchronization errors are shown in Figure7, and the phase diagrams of 5.1 and 5.2 are plotted together in Figure8.