Synchronization for a Class of Fractional-Order Hyperchaotic System and Its Application

A new controller design method is proposed to synchronize the fractional-order hyperchaotic system through the stability theory of fractional calculus; the synchronization between two identical fractional-order Chen hyperchaotic systems is realized by designing only two suitable controllers in the response system. Furthermore, this control scheme can be used in secure communication via the technology of chaotic masking using the complex nonperiodic information as trial message, and the useful information can be recovered at the receiver. Numerical simulations coincide with the theoretical analysis.


Introduction
It is well known that the fractional calculus has a long mathematical history, but its applications to physics and engineering are just recent subject of interest 1 .In recent years, more and more researchers focused on the control of fractional-order chaotic systems and its dynamic behavior 2-4 .In 5 , chaos and hyperchaos in the fractional-order R ȏssler equations were studied, in which chaos can exist in the fractional-order R ȏssler equation with order as low as 2.4, and hyperchaos exists in the fractional-order R ȏssler hyperchaos equation with order as low as 3.8.In 6 , the chaotic behavior with the lowest order 3.72 in the fractional-order hyperchaotic Chen system is presented.
For decades, the complex dynamics and synchronization of chaotic systems have attracted much attention 7-21 since the seminal paper by Pecora and Carroll in 1990 22 .The chaotic control and synchronization of fractional-order systems are concerned extremely 23-27 .The fractional-order system, compared to the integer-order system,

Fractional Derivative and Its Approximation
There are several definitions of fractional derivatives 1 .In our work, we use the best known Caputo derivative 30 defined by where n is the first integer which is not less than α, and J β is the β-order Riemann-Liouville integer operator which is described as follows: where Γ • is the gamma function and 0 < β 1.Given a fractional-order chaotic system, the drive master system is d q X 1 dt q F X 1 .

2.3
Here , and q ∈ 0, 1 .The fractional-order q ∈ n×1 may be unequal.The equilibrium points of system 2.3 can be derived by solving following equation: The stability of fractional-order system has been thoroughly investigated, and necessary and sufficient conditions have been presented in 26 : where X 2 ∈ n×1 , and u t is the control functions.Suppose that the error between the system 2.3 and the system 2.5 is e t X 1 t − X 2 t ; then the fractional error system can be obtained as 2.6 The master-slave synchronization of two chaotic systems is tightly associated with stability of the error dynamics; in this paper, the control function method is presented.Before discussing the method, we first give some useful preliminaries which are of great help to the proof of the forthcoming theorem.Lemma 2.1 see 36 .System 2.3 is locally asymptotically stable if all the eigenvalues λ 1 , . . ., λ n of the Jacobian matrix of all equilibrium point satisfy In nature, if q > 2/π | arctan Im λ / Re λ |, the system 2.3 is locally asymptotically stable and behave chaotic for all the variations where the eigenvalues of the matrix satisfy | arg λ | > qπ/2.The stable and unstable regions for q ∈ 0, 1 are depicted in Figure 1.Obviously, the stable region of a fractional-order system is normally larger than its corresponding integer-order system: whose stable region is the left half plane.Based on this, we obtain the following corollary immediately.
Corollary 2.2.The fractional-order system 2.3 with order q ∈ 0, 1 is asymptotically stable if the corresponding integer system 2.8 is stable.

The Fractional-Order Chen System
Now, consider the fractional-order hyperchaotic Chen system as follows:

3.2
The corresponding Jacobian matrix is as follows: Then, the eigenvalues of the Jacobian matrix are obtained: S 0 : λ 1 −39.7356,λ 2 16.7356, λ 3 −3.0000,λ 4 0.5000; It is easy to show that eigenvalues from S 0 to S 4 hold if they satisfy q > 2/π | arctan 82.25/289.93 | based on Lemma 2.1, and the fractional-order hyperchaotic Chen is chaotic.From the phase portrait of chaotic attractor at α 0.96 as shown in Figure 2 we can find that the system 3.1 exists with chaotic behavior indeed.

The Synchronization of the Two Identical Incommensurate
Fractional-Order Chen Systems Now, we will study synchronization between two identical fractional-order hyperchaotic Chen systems.The fractional-order hyperchaotic Chen system as the drive system is expressed by

4.1
And the corresponding response system is written by

4.2
Here, u u 1 , u 2 T is the control function.Our aim is to design the controller u u 1 , u 2 T that will make the system 4.2 achieve synchronization with the system 4.1 .In order to facilitate the following analysis, we set the errors between the system 4.2 and system 4.1 :

4.4
Then, consider the following control function:

4.7
We can obtain the Jacobian matrix of the error system with linear system 4.7 as follows: Obviously, the system 4.7 , in which all eigenvalues −a, −k 1 , −d, and −k 2 are less than zero with k 1 and k 2 to positive constant, must be stable without a doubt.And the corresponding fractional-order system 4.6 is asymptotically stable according to Corollary 2.2 given in the second section.So, the errors lime 1 t , lime 2 t , lime 3 t , and lime 4 t will converge to zero when t → ∞.Therefore, synchronization of the two identical fractional-order hyperchaotic systems is achieved.

Simulation Results
In the numerical simulations, we set the parameters of the system 4.1 and 4.2 as a 35, b 3, c 28, and r 0.5 for drive system and response system with α 0.96 and the coefficient of control function k 1 15 and k 2 20.The initial conditions of the drive and response systems are taken arbitrarily as x 1 0 10, y 1 0 10, z 1 0 10, and w 1 0 10; and x 2 0 −10, y 2 0 −10, z 2 0 3, and w 2 0 −10.Numerical results show that the synchronization of two identical fractional-order hyperchaotic system is achieved as shown in Figure 3.The experiments coincided with the theory analysis.Drive chaos subsystem Response chaos subsystem

Application to Secure Communication
In this section, to verify and demonstrate the effectiveness of the proposed method, we will display the numerical results for fractional-order hyperchaotic Chen systems in secure communication.Based on the theory of the communication, the block schematic of secure communication scheme with the synchronization scheme is depicted in Figure 4, where x 1 t is the chaotic state variable of drive system for the transmitter, x 2 t is the chaotic state variable of response system, s t is the transmitted message signal with complex nonperiodic mode, which is added to the variable x 1 t , mixed signal m t s t x 1 t , and s 0 t is the recovered signal after synchronization between the chaotic state variable x 2 t and x 1 t at the receiver terminal end.
The complex nonperiodic information, which has the typical representative to accurately simulate the real transmission signal with generally complex and disorder, is chosen as the transmitted useful message during the numerical experiment in order to reinforce the feasibility of the scheme.The mixed signals Figure 6 have a good masking effectiveness, are completely different from the original signals Figure 5 , which reached the purpose of safety, and are not to be cracked on the processing of signal transmit.Simulation results show that the system effectively restored the useful signals after about 1.2 s as depicted in Figure 7. Therefore, we can find that, even adopting other irregular signals as transmitted information by the synchronization and cover-up technology, the useful signals can be recovered with no distortion at the receiver end; namely, the decoded information s 0 t coincides with the transmitted signal s t .

Conclusions
In this paper, a new method of designing controller to synchronize a class of fractionalorder hyperchaotic system is presented, and the synchronization between two identical fractional-order systems has been realized via designing only two controllers.The simulation results show that the control method is reliable.Moreover, the complex nonperiodic information signals can be recovered with no distortion when the scheme is applied to secure communication.Numerical experiments for the secure communication system indicate that the synchronization works quite well, which may has potential applications in many interdisciplinary fields.Future work on this topic should include transmission of highfrequency digital signal as well as in-depth studies on application to secure communication.

Figure 2 :
Figure 2: The phase portrait of system 3.1 chaotic attractor α 0.96 with a x, y, z and b y, z, w.

Figure 3 :
Figure 3: The synchronization of system 4.2 and system 4.1 with α 0.96: a the synchronization of x 1 − x 2 ; b the synchronization of y 1 − y 2 ; c the synchronization of z 1 − z 2 ; d the synchronization of w 1 − w 2 .

Figure 4 :
Figure 4: Chaotic masking technology for communication system.

Figure 7 :
Figure 7: The original message and the recovered message.