Antisynchronization of Nonidentical Fractional-Order Chaotic Systems Using Active Control

Antisynchronization phenomena are studied in nonidentical fractional-order differential systems. The characteristic feature of antisynchronization is that the sum of relevant state-variables vanishes for sufficiently large value of time variable. Active control method is used first time in the literature to achieve antisynchronization between fractional-order Lorenz and Financial systems, Financial and Chen systems, and Lü and Financial systems. The stability analysis is carried out using classical results. We also provide numerical results to verify the effectiveness of the proposed theory.


Introduction
In their pioneering work 1, 2 , Pecora and Carroll have shown that chaotic systems can be synchronized by introducing appropriate coupling.The notion of synchronization of chaos has further been explored in secure communications of analog and digital signals 3 and for developing safe and reliable cryptographic systems 4 .For the synchronization of chaotic systems, a variety of approaches have been proposed which include nonlinear feedback 5 , adaptive 6, 7 , and active controls 8, 9 .
Antisynchronization AS is a phenomenon in which the state vectors of the synchronized systems have the same amplitude but opposite signs to those of the driving system.Hence the sum of two signals converges to zero when AS appears.Antisynchronization has applications in lasers 10 , in periodic oscillators, and in communication systems.Using AS to lasers, one may generate not only drop-outs of the intensity but also short pulses of high intensity, which results in the pulses of special shapes.

Fractional Calculus
Basic definitions and properties of fractional derivative/integrals are given below 16,17,46 .

2.4
The IVP 2.4 is equivalent to the Volterra integral equation Consider the uniform grid {t n nh/n 0, 1, . . ., N} for some integer N and h : T/N.Let y h t n be approximation to y t n .Assume that we have already calculated approximations y h t j , j 1, 2, . . ., n, and we want to obtain y h t n 1 by means of the equation where

2.7
International Journal of Differential Equations The preliminary approximation y P h t n 1 is called predictor and is given by where Error in this method is max j 0,1,...,N y t j − y h t j O h p , 2.10 where p min 2, 1 α .

System Description
The fractional-order Lorenz system 50, 51 is described by where σ 10 is the Prandtl number, r 28 is the Rayleigh number over the critical Rayleigh number, and μ 8/3 gives the size of the region approximated by the system.The minimum effective dimension for this system is 2.97 51 .
In 52 Chen proposed the financial system to fractional-order where a 3, b 0.1, and c 1.The minimum effective dimension for which the system exhibits chaos is given by 2.32 52 .
Li and Peng 53 studied chaos in fractional-order Chen system where a 1 35, b 1 3, and c 1 27.The minimum effective dimension reported is 2.92 53 .
Fractional-order L ü system is the lowest-order chaotic system among all the chaotic systems reported in the literature 54 .The minimum effective dimension reported is 0.30.The system is given by

Antisynchronization between Fractional-Order Lorenz and Financial System
In this section, we study the antisynchronization between Lorenz and Financial systems.Assuming that the Lorenz system drives the Financial system, we define the drive master and response slave systems as follows:

4.2
The unknown terms u 1 , u

International Journal of Differential Equations
We define active control functions u i t as

4.5
The terms V i t are linear functions of the error terms e i t .With the choice of u i t given by 4.5 , the error system 4.5 becomes

4.6
The control terms V i t are chosen so that the system 4.6 becomes stable.There is not a unique choice for such functions.We choose where A is a 3 × 3 real matrix, chosen so that for all eigenvalues λ i of the system 4.
then the eigenvalues of the linear system 4.6 are −1, −1, and −1.Hence the condition 4.8 is satisfied for α < 2. Since we consider only the values α ≤ 1, we get the required antisynchronization.

Simulation and Results
Parameters of the Lorenz system are taken as σ 10, r 28, μ 8/3 and Financial system as a 3, b 0.1, c 1.The fractional-order α is taken to be 0.99 for which both the systems are chaotic.The initial conditions for drive and response system are x 1 0 10, y 1 0 5,

Antisynchronization between Financial and Chen Systems of Fractional Order
Assuming that Chen system is antisynchronized with Financial system; define the drive system as 5.1 and the response system as

5.2
Let e 1 x 1 x 2 , e 2 y 1 y 2 , and e 3 z 1 z 2 be error functions.For antisynchronization, it is essential that the errors e i → 0 as t → ∞.Note that x 1 e 2 e 1 e 2 − b 1 e 3 u 6 t .

5.3
The control functions are chosen as

5.4
The linear functions V 4 , V 5 , V 6 are given by

5.5
With the values given in 5.4 and 5.5 , the error system 5.3 becomes It can be observed that the coefficient matrix of the error system 5.6 has eigenvalues −1, −1, −1.So the system is stable and antisynchronization is achieved.

Simulations and Results
We take parameters for fractional-order Chen system as a 1 35, b 1 3, c 1 27.Parameters for the Financial system are same as given in Section 4.1.Experiments are done for fixed value of fractional-order α 0.95, which is same for drive and response system 5.1 and 5.2 .The initial conditions for the systems 5.1 and 5.2 are x 1 0 2, y 1 0 3, z 1 0 2 and x 2 0 10, y 2 0 25, z 2 0 36, respectively.For the error system 5.6 , the initial conditions turns out to be e 1 0 12, e 2 0 28, e 3 0 38.The simulation results are summarized in Figure 2. Antisynchronization between fractional Financial and Chen system is shown in Figure 2

Antisynchronization between Fractional L ü and Financial System
In this case, consider L ü system as the drive system 6.1 and the response system as the Financial system

International Journal of Differential Equations
Let e 1 x 1 x 2 , e 2 y 1 y 2 , and e 3 z 1 z 2 be error functions.For antisynchronization, it is essential that the errors e i → 0 as t → ∞.To achieve this one should choose the control terms u 7 , u 8 , u 9 properly.The error system thus becomes

6.3
The control functions are chosen as

6.4
The linear functions V 7 , V 8 , V 9 are given by V 9 e 1 c − 1 e 3 .

6.5
With the values given in 6.4 and 6.5 , the error system 6.3 becomes It can be observed that the coefficient matrix of the error system 6.6 has eigenvalues −1, −1, −1.So the system is stable and antisynchronization is achieved.

Simulations and Results
Parameters for the L ü system are a 2 35, b 2 3, c 2 28, whereas parameters for Financial system are unaltered.The initial conditions for drive system are x 1 0 0.2, y 1 0 0, z 1 0 0.5, whereas the initial conditions for response system are x 2 0 2, y 2 0 3, z 2 0 2. Hence the initial conditions for the error system 6.6 are e 1 0 2.2, e 2 0 3, e 3 0 2.5.We perform the numerical simulations for fractional order α, namely, 0.91 of the drive system 6.1 and response system 6.2 .Figures 3 a , 3 b , and 3 c show antisynchronization between fractional L ü and Financial system for α 0.91.

Conclusions
Antisynchronization of nonidentical fractional-order chaotic systems has been done first time in the literature using active control.The fractional Financial system is controlled by fractional Lorenz system, the fractional Chen system is controlled by fractional Financial system, and the fractional Financial system is controlled by fractional L ü system.
Figure 3 d shows the errors e 1 t solid line , e 2 t dashed line , and e 3 t dot-dashed line in the antisynchronization for α 0.91.Mathematica 7 has been used for computations in the present paper.