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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.

In their pioneering work [

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 [

Active control method is used to AS for two identical integer order systems by Ho et al. [

Fractional calculus deals with derivatives and integration of arbitrary order [

Synchronization of fractional-order chaotic systems was first studied by Deng and Li [

In the present paper, we study the antisynchronization of the following fractional systems using active control method: (i) Lorenz with Financial, (ii) Financial with Chen, and (iii) Lü with Financial.

Basic definitions and properties of fractional derivative/integrals are given below [

A real function

A real function

Let

The (left-sided) Caputo fractional derivative of

Numerical methods used for solving ODEs have to be modified for solving fractional differential equations (FDEs). A modification of Adams-Bashforth-Moulton algorithm is proposed by Diethelm et al. in [

Consider for

The fractional-order Lorenz system [

In [

Li and Peng [

Fractional-order Lü system is the lowest-order chaotic system among all the chaotic systems reported in the literature [

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:

Parameters of the Lorenz system are taken as

(a) Signals

Assuming that Chen system is antisynchronized with Financial system; define the drive system as

We take parameters for fractional-order Chen system as

(a) Signals

In this case, consider Lü system as the drive system

Let

Parameters for the Lü system are

(a)

Mathematica 7 has been used for computations in the present paper.

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 Lü system.

V. Daftardar-Gejji acknowledges the Department of Science and Technology, N. Delhi, India for the Research Grants (project no. SR/S2/HEP-024/2009).