Modified Function Projective Synchronization between Different Dimension Fractional-Order Chaotic Systems

and Applied Analysis 3 where ‖ · ‖ is the Euclidean norm,M x is am×n real matrix, and matrix elementMij x i 1, 2, . . . m, j 1, 2, . . . , n are continuous bounded functions. ei yi − ∑n j 1Mijxj i 1, 2, . . . m are called MFPS error. Remark 2.2. According to the view of tracking control, M x x can be chosen as a reference signal. The MFPS in our paper is transformed into the problem of tracking control, that is the output signal y in system 2.2 follows the reference signalM x x. In order to achieve the output signal y follows the reference signal M x x. Now, we define a compensation controller C1 x ∈ R for response system 2.2 via fractional-order derivative dr M x x /dtr . The compensation controller is shown as follows: C1 x dr M x x dtqr − Fr M x x , 2.4 and let controller C x, y as follows: C ( x, y ) C1 x C2 ( x, y ) , 2.5 where C2 x, y ∈ R is a vector function which will be designed later. By controller 2.5 and compensation controller 2.4 , the response system 2.2 can be changed as follows: dr e dtqr D1 ( x, y ) e C2 ( x, y ) , 2.6 where D1 x, y e Fr y − Fr M x x , and D1 x, y ∈ Rm×m. So, the MFPS between drive system 2.1 and response system 2.2 is transformed into the following problem: choose a suitable vector function C2 x, y such that system 2.6 is asymptotically converged to zero. In what followswe present the stability theorem for nonlinear fractional-order systems of commensurate order 22–25 . Consider the following nonlinear commensurate fractionalorder autonomous system


Introduction
Fractional-order calculus, which can be dated back to the 17th century 1, 2 . However, only in the last few decades, its application to physics and engineering has been addressed. So, the fractional-order calculus has attracted increasing attention only recently. On the other hand, complex bifurcation and chaotic phenomena have been found in many fractionalorder dynamical systems. For example, the fractional-order Lorenz chaotic system 3 , the fractional-order unified chaotic system 4 , the fractional-order Chua chaotic circuit 5 , the fractional-order modified Duffing chaotic system 6 , and the fractional-order Rössler chaotic system 7, 8 , and so on.
Nowadays, synchronization of chaotic systems and fractional-order chaotic systems has attracted much attention because of its applications in secure communication and control processing 9-21 . Many approaches have been reported for the synchronization of chaotic systems and fractional-order chaotic systems [12][13][14][15][16][17][18][19] . In 1999, Mainieri and Rehacek proposed projective synchronization PS 12 for chaotic systems, which has

The MFPS Scheme for Different Dimension Fractional-Order Chaotic Systems
The fractional-order chaotic drive and response systems with different dimension are defined as follows, respectively: where q d 0 < q d < 1 and q r 0 < q r < 1 are fractional order, and q d may be different with q r .
x ∈ R n , y ∈ R m n / m are state vectors of the drive system 2.1 and response system 2.2 , respectively. F d : R n → R n , F r : R m → R m are two continuous nonlinear vector functions, and C x, y ∈ R m is a controller which will be designed later. According to the view of tracking control, M x x can be chosen as a reference signal. The MFPS in our paper is transformed into the problem of tracking control, that is the output signal y in system 2.2 follows the reference signal M x x.
In order to achieve the output signal y follows the reference signal M x x. Now, we define a compensation controller C 1 x ∈ R m for response system 2.2 via fractional-order derivative d q r M x x /dt q r . The compensation controller is shown as follows: and let controller C x, y as follows: where C 2 x, y ∈ R m is a vector function which will be designed later. By controller 2.5 and compensation controller 2.4 , the response system 2.2 can be changed as follows: where D 1 x, y e F r y − F r M x x , and D 1 x, y ∈ R m×m . So, the MFPS between drive system 2.1 and response system 2.2 is transformed into the following problem: choose a suitable vector function C 2 x, y such that system 2.6 is asymptotically converged to zero.
In what follows we present the stability theorem for nonlinear fractional-order systems of commensurate order 22-25 . Consider the following nonlinear commensurate fractionalorder autonomous system the fixed points of system 2.7 is asymptotically stable if all eigenvalues λ of the Jacobian matrix A ∂f/∂x evaluated at the fixed points satisfy | arg λ| > 0.5πq. Where 0 < q < 1, x ∈ R n , f : R n → R n are continuous nonlinear functions, and the fixed points of this nonlinear commensurate fractional-order system are calculated by solving equation f x 0. Now, the following theorem is given based on the above discussion in order to achieve the MFPS between the drive system 2.1 and the response system 2.2 . Theorem 2.3. Choose the control vector C 2 x, y D 2 x, y e, and if D 1 x, y D 2 x, y satisfy the following conditions: Abstract and Applied Analysis then the modified function projective synchronization (MFPS) between 2.1 and 2.2 can be achieved. Where D 2 x, y ∈ R m×m , and d ij i, j 1, 2, . . . m, for all d ij ∈ R are the matrix element of matrix D 1 x, y D 2 x, y .
Proof. Using C 2 x, y D 2 x, y e, so fractional-order system 2.6 can be rewritten as follows: Suppose λ is one of the eigenvalues of matrix D 1 x, y D 2 x, y and the corresponding non-zero eigenvector is ψ, that is, Take conjugate transpose H on both sides of 2.9 , we yield This indicates that the modified function projective synchronization between drive system 2.1 and response system 2.2 will be obtained. The proof is completed.
Remark 2.4. Theorem 2.3 indicates that the condition of the MFPS between drive system 2.1 and response system 2.2 are | arg λ D 1 x, y D 2 x, y | > 0.5q r π. So, in practical applications, we can easily choose the matrix D 2 x, y according to the matrix D 1 x, y . Moreover, in order to reserve all the nonlinear terms in response system or error system, the controller in our work may be complex than the controller reported by 16,17 . But, all the nonlinear terms in response system or error system are absorbed in 16, 17 .
Remark 2.5. Perhaps our result can be extended to the modified function projective synchronization of complex networks of fractional order chaotic systems 26-28 and the complex fractional-order multi scroll chaotic systems 29-31 . But, the modified function projective synchronization for complex networks and complex fractional-order multi-scroll chaotic systems would be much more complex. Further work on this issue is an ongoing research topic in our group.

Applications
In this section, to illustrate the effectiveness of the proposed MFPS scheme for different dimension fractional-order chaotic systems. Two groups of examples are considered and their numerical simulations are performed.

The MFPS between 3-Dimensional Fractional-Order Lorenz System and 4-Dimensional Fractional-Order Hyperchaotic System
The fractional-order Lorenz 3 system is described as follows: D q r y 1 10 y 2 − y 1 D q r y 2 28y 1 − y 2 − y 1 y 3 D q r y 3 y 1 y 2 − 8y 3 3 .

3.1
The fractional-order Lorenz system exhibits chaotic behavior 3 for q r ≥ 0.993. The chaotic attractor for q r 0.995 is shown in Figure 1. Recently, Pan et al. constructed a hyperchaotic system 17 . Its corresponded fractional-order system is described as follows:

3.2
The hyperchaotic attractor of system 3.2 for q d 0.95 is shown in Figure 2.
Consider the fractional-order hyperchaotic system 3.2 with fractional-order q d 0.95 as drive system, and the fractional-order Loren system with fractional-order q r 0.995 as response system. According to the above mentioned, we can obtain

3.3
Abstract and Applied Analysis 7 Now, we can choose So,

3.5
According to the above theorem, the MFPS between the 3-dimensional fractional-order Lorenz system 3.1 and the 4-dimensional fractional-order hyperchaotic system 3.2 can be achieved. For example, choose M x . The corresponding numerical result is shown in Figure 3, in which the initial conditions are x 0 2, 1, 2, 1 T , and y 0 18, 13, 13.5 T , respectively.

The MFPS between 4-Dimensional Fractional-Order Hyperchaotic Lǔ System and 3-Dimensional Fractional-Order Arneodo Chaotic System
In 2002, Lü and Chen reported a new chaotic system 32 , which be called Lü chaotic system. The Lü chaotic system is different from the Lorenz and Chen system. Based on Lü chaotic system, the hyperchaotic Lü chaotic system and the fractional-order hyperchaotic Lü system have been constructed recently. The fractional-order hyperchaotic Lü system 16 is described by the following D q r y 1 36 y 2 − y 1 y 4 D q r y 2 20y 2 − y 1 y 3 D q r y 3 y 1 y 2 − 3y 3 D q r y 4 y 1 y 3 − y 4 .

3.6
The hyperchaotic attractor of system 3.6 for q r 0.96 is shown in Figure 4.
The fractional order Arneodo chaotic system 16 is defined as follows:

3.7
The chaotic attractor of system 3.7 for q d 0.998 is shown in Figure 5. Consider the fractional-order Arneodo chaotic system 3.7 with fractional-order q d 0.998 as drive system, and the fractional-order hyperchaotic Lǔ system 3.6 with fractionalorder q r 0.96 as response system. According to the above mentioned, we can yield

3.10
According to above theorem, the MFPS between the 4-dimensional fractional-order hyperchaotic Lǔ system 3.6 and the 3-dimensional fractional-order Arneodo chaotic system 3.7 can be achieved. For example, choose M x . The corresponding numerical result is shown in Figure 6, in which the initial conditions are x 0 2, 2, 2 T , and y 0 11, 10, 11, 2 T , respectively.

Conclusions
In this paper, based on the stability theory of the fractional-order system and the tracking control, a modified function projective synchronization scheme for different dimension fractional-order chaotic systems is addressed. The derived method in the present paper shows that the modified function projective synchronization between drive system and response system with different dimensions can be achieved. The modified function projective synchronization between 3-dimensional fractional-order Lorenz system and 4-dimensional  Figure 6: The modified function projective synchronization errors between the fractional-order system 3.6 and the following fractional-order system: fractional-order hyperchaotic system, and the modified function projective synchronization between the 4-dimensional fractional-order hyperchaotic Lǔ system, and the 3-dimensional fractional-order Arneodo chaotic system, are chosen to illustrate the proposed method. Numerical experiments shows that the present method works very well, which can be used for other chaotic systems.