SYNCHRONIZATION BETWEEN TWO NON IDENTICAL FRACTIONAL ORDER HYPERCHAOTIC SYSTEMS

Abstract. In this paper we have investigated chaos synchronization between the two nonidentical fractional order hyperchaotic systems using feedback control technique.The hyperchaotic system introduced by Xin and Ling has been synchronized with the Lü like hyperchaotic system.The analytical conditions for the synchronization of this pair of different fractional order hyperchaotic systems are derived by using Laplace transform. Numerical simulations are carried out using Matlab to show the effectiveness of the method.


Introduction
Chaotic systems are characterized by their sensitive dependence on initial conditions.Chaos synchronization has attracted a great deal of attention since the seminal work by Pecora and Carroll [1] in which they established a chaos synchronization scheme for two identical chaotic systems with different initial conditions.
Recently, the study of dynamics of fractional-order chaotic systems has received interest of many researchers.Yu and Li in [20] used Laplace transformation theory and variational iteration method to study Rössler system, Wu, Lu and Shen discussed the synchronization of a new fractional-order hyperchaotic system via active control [21], Wang, Yu and Diao in [22] studied the hybrid projective synchronization between fractional-order chaotic systems of different dimensions, Sahab and Ziabari [23] analyzed the chaos between two different hyperchaotic systems by generalized backstepping method, S.T. Mohammad and H. Mohammad in [24] proposed a controller based on active sliding mode theory to synchronize the chaotic fractional order systems, Zhang and Lu introduced a new type of hybrid synchronization called full state hybrid lag projective synchronization and applied it to the Rössler system and the hyperchaotic Lorenz system to verify their results numerically [25].A. Ouannas in [26] studied the Q-S synchronization of chaotic dynamical systems in continuous-time, Boutefnouchet, Taghvafard and Erjaee in their paper [27] discussed the phase synchronization in coupled chaotic systems.This paper is organized as follows: in section 2, the fractional order derivative and its approximation is given.In section 3, the synchronization between the two non-identical fractional-order hyperchaotic systems using feedback control method is discussed.Section 4 presentS the numerical results to verify the effectiveness of the method.Finally, the conclusion is given in section 5.

Fractional order Derivative and its Approximation
Fractional calculus is a generalization of integration and differentiation to a non-integer-order integro-differential operator a D q t defined by where q is the fractional order which can be a complex number,R(q) denotes the real part of q and a < t, where a is the fixed lower terminal and t is the moving upper terminal.
There are two commonly used definitions for fractional derivatives [28], they are Grunward -Letnikov definition and Riemann-Liouville definition.The Riemann-Liouville definition is given by where η is the first integer that is not less than q , J β is the β -order Riemann-Liouville integral operator defined as follows: The Laplace transform of the Riemann-Liouville fractional derivative is given by where L means the Laplace transform and s is a complex variable.Assuming the initial conditions to be zero, the above equation reduces to Thus the fractional integral operator of order "q" can be represented by the transfer function The standard definitions of fractional order calculus do not allow direct implemetation of the fractional operators in time-domain simulations.An efficient method to circumvent this problem is to approximate fractional operators by using standard integer-order operators.In [29], an effective algorithm is developed to approximate fractional order transfer functions, which has been adopted in [30][31][32] and has sufficient accuracy for time domain implementations.In table 1 of [30], approximations for 1 s q with q from 0.1 to 0.9 in steps 0.1 were given with errors of approximately 2dB.We will use these approximations in the simulations.
3. Synchronization between the new fractional order hyperchaotic system and L ü like fractional order hyperchaotic system.
In this section, our goal is to achieve synchronization between the new fractional order hyperchaotic system and Lü like fractional order hyperchaotic system.The drive and response systems are given as follows: As a drive system, consider a fractional order hyperchaotic system proposed by Xin and Ling in [33] where where (y 1 , y 2 , y 3 , y 4 ) ∈ R 4 , u 1 , u 2 , u 3 and u 4 are the linear or nonlinear control functions to be Define the error functions as Subtracting (1) from (2) and using (3), we get Choosing the control functions u i , i = 1, 2, 3, 4 as the error system (4) reduces to where k 1 , k 2 , k 4 ≥ 0 are real parameters.

Conclusion
Chaos synchronization between two fractional order hyperchaotic systems has been studied using feedback control technique for a short interval of time.The hyperchaotic system introduced by Xin and Ling has been used to drive the Lü like hyperchaotic system.Numerical simulations are carried out to show the effectiveness of the method.

FIGURE 3 .
figures (3) and (4) respectively.It can be seen from figure4that the trajectories of the drive and response systems aymptotically synchronize and that from figure 5 that the error system converges to zero which shows that the systems (1) and (2) are synchronized.

FIGURE 4 .
FIGURE 4. The chaotic tracjectories of the drive and response system after the controllers are applied.

FIGURE 5 .
FIGURE 5. Error functions comparison of four state variables versus the time t.