Chaos Synchronization between Two Different Fractional Systems of Lorenz Family

This work investigates chaos synchronization between two different fractional order chaotic systems of Lorenz family. The fractional order Lü system is controlled to be the fractional order Chen system, and the fractional order Chen system is controlled to be the fractional order Lorenzlike system. The analytical conditions for the synchronization of these pairs of different fractional order chaotic systems are derived by utilizing Laplace transform. Numerical simulations are used to verify the theoretical analysis using different values of the fractional order parameter.


Introduction
Fractional calculus has been known since the early 17th century 1, 2 .It has useful applications in many fields of science like physics 3 , engineering 4 , mathematical biology 5, 6 , and finance 7, 8 .
The fractional order derivatives have many definitions; one of them is the Riemann-Liouville definition 9 which is given by where J θ is the θ-order Riemann-Liouville integral operator which is given as Mathematical Problems in Engineering However, the most common definition is the Caputo definition 10 , since it is widely used in real applications: where f l represents the l-order derivative of f t and l α ; this means that l is the first integer which is not less than α.The operator D α * is called "the Caputo differential operator of order α."Hence, I choose the Caputo type throughout this paper.
On the other hand, chaos has been studied and developed with much interest by scientists since the birth of Lorenz chaotic attractor in 1963 11 .Chen attractor is similar to Lorenz attractor but not topologically equivalent 12 .Recently, L ü et al. found a new chaotic system which connects the Lorenz and Chen attractors, according to the conditions formulated by Vaněček and Čelikovsk ý, and it is called L ü system 13 .Afterwards, chaos in fractional order dynamical systems has become an interesting topic.In 14 chaotic behaviors of the fractional order Lorenz system are studied.Moreover, chaotic behaviors have also been found in the fractional order Chen system 15 and the fractional order L ü system 16 .Furthermore, Chaos synchronization in fractional order chaotic systems starts to attract increasing attention 16-20 .However, it has been studied very well in the case of integer order chaotic systems, due to its potential applications in physical, chemical, and biological systems 21-24 and secure communications 25 .
The generalized synchronization between two different fractional order systems is investigated in 26 .However, in this paper, I investigate the conditions of chaos synchronization between two different fractional order chaotic systems of Lorenz family by designing suitable linear controllers.I give examples to achieve chaos synchronization of two pairs of different fractional order chaotic systems fractional Chen & fractional L ü, fractional Lorenz-like, and fractional Chen in drive-response structure.Conditions for achieving chaos synchronization using linear control method are further discussed using Laplace transform theory.

Systems Description
The fractional order Chen system isgiven as follows: Here and throughout, a, b, c 35, 3, 28 where α is the fractional order.In the following I choose α 0.9 at which system 2.1 exhibits chaotic attractor see Figure 1 .
The fractional order L ü system is given as follows Here and throughout, r, p, q 35, 28, 3 .By choosing α 0.9, system 2.2 has chaotic attractor see Figure 2 .
The fractional order Lorenz-like system 27 is described by which has a chaotic attractor as shown in Figure 3 when β 2.8, γ 10.6, ρ 14, σ 20, and α 0.9.It should be also noted that, the systems 2.1 , 2.2 , and 2.3 are still chaotic at the fractional order values α 0.95 and α 0.99.

Synchronization between Two Different Fractional Order Systems
Consider the master-slave or drive-response synchronization scheme of two autonomous different fractional order chaotic systems: where α is the fractional order, X ∈ R n , Y ∈ R n represent the states of the drive and response systems, respectively, f : R n → R n , g : R n → R n are the vector fields of the drive and response systems, respectively.The aim is to choose a suitable linear control function U t u 1 , . . ., u n T such that the states of the drive and response systems are synchronized i.e., lim t → ∞ X − Y 0, where • is the Euclidean norm .

Synchronization between Chen and L ü Fractional Order Systems
In this subsection, the goal is to achieve chaos synchronization between the fractional order Chen system and the fractional order L ü system by using the fractional order Chen system to drive the fractional order L ü system.The drive and response systems are given as follows: where u 1 , u 2 , and u 3 are the linear control functions.Define the error variables as follows:

3.5
Now, by letting where k 1 , k 2 ≥ 0, then the error system 3.5 is reduced to d α e 3 dt α − q k 2 e 3 y m e 1 x m e 2 e 1 e 2 .

3.7
By taking the Laplace transform in both sides of 3.7 , letting E i s L{e i t } where i 1, 2, 3 , and applying L{d α e i /dt α } s α E i s − s α−1 e i 0 , we obtain

3.8
Proposition 3.1.If E 1 s , E 2 s are bounded and p − k 1 / 0, then the drive and response systems 3.2 and 3.3 will be synchronized under a suitable choice of k 1 and k 2 .

Mathematical Problems in Engineering
Using the final value theorem of the Laplace transform, it follows that lim

3.10
Since Consequently, the synchronization between the drive and response systems 3.2 and 3.3 is achieved.

Numerical Results
An efficient method for solving fractional order differential equations is the predictorcorrectors scheme or more precisely, PECE Predict, Evaluate, Correct, Evaluate technique which has been investigated in 28, 29 , and represents a generalization of the Adams-Bashforth-Moulton algorithm.It is used throughout this paper.
Based on the above mentioned discretization scheme, the drive and response systems 3.2 and 3.3 are integrated numerically with the fractional orders α 0.9, 0.95, 0.99 and using the initial values x m 0 15, y m 0 20, z m 0 29 and x s 0 10, y s 0 15, z s 0 25.From Figure 4, it is clear that the synchronization is achieved for all these values of fractional order when k 1 20 and k 2 10.

Synchronization between Lorenz-Like and Chen Fractional Order Systems
In this case it is assumed that, the fractional order Lorenz-like system drives the fractional order Chen system.The drive and response systems are defined as follows:

Mathematical Problems in Engineering
Now, by choosing 3.16 where k 1 , k 2 ≥ 0, then the error system 3.15 is rewritten as 3.17 Take Laplace transform in both sides of 3.17 , let E i s L{e i t }, where i 1, 2, 3 , and apply L{d α e i /dt α } s α E i s − s α−1 e i 0 .After that, by doing similar analysis like the previous subsection, we obtain Thus, the states of the drive system 3.12 are synchronized with the states of the response system 3.13 , as the controllers 3.16 are activated.

Numerical Results
Numerical simulations are carried out to integrate the drive and response systems 3.12 and 3.13 using the predictor-correctors scheme, with the fractional orders α 0.9, 0.95, 0.99 and the initial values x m 0 10, y m 0 16, z m 0 25 and x s 0 15, y s 0 20, z s 0 29.Thus, the drive and response systems 3.12 and 3.13 are synchronized in such a successful way for all at the above-mentioned fractional orders values, using the linear controllers 3.16 with k 1 20 and k 2 10 see Figure 5 .

Conclusion
Chaos synchronization between two different fractional order chaotic systems has been studied using linear control technique.Fractional order Chen system has been used to drive fractional order L ü system, and fractional order Lorenz-like system has been used to drive fractional order Chen system.Conditions for chaos synchronization have been investigated theoretically by using Laplace transform.Numerical simulations have been carried out using different fractional order values to show the effectiveness of the proposed synchronization techniques.

e 1 1 , d α e 2 dt α pe 2 −
x s − x m , e 2 y s − y m , e 3 z s − z m .3.4 By subtracting 3.2 from 3.3 and using 3.4 , we obtain d α e 1 dt α r e 2 − e 1 r − a y m − x m u z m e 1 − x m e 3 − e 1 e 3 − c − a x m p − c y m u 2 , d α e 3 dt α −qe 3 y m e 1 x m e 2 e 1 e 2 − q − b z m u 3 .
E 1 s , E 2 s are bounded and p − k 1 / 0 then lim t → ∞ e 1 t lim t → ∞ e 2 t 0. Now, owing to the attractiveness of the attractors of systems 2.1 and 2.2 , there exists η > 0 such that |x i t | ≤ η < ∞, |y i t | ≤ η < ∞, and |z i t | ≤ η < ∞ where i refers to the index of the drive or response variables.Therefore, lim t → ∞ e 3 t If we assume that c − k 1 / 0 and E 1 s , E 2 s are bounded, then it follows that lim t → ∞ e 1 t lim t → ∞ e 2 t 0. Now, owing to the attractiveness of the attractors of systems 2.1 and 2.3 , there exists ξ > 0 such that |x i t | ≤ ξ < ∞, |y i t | ≤ ξ < ∞, and |z i t | ≤ ξ < ∞ where i refers to the index of the drive or response variables.Therefore, lim t → ∞ e 3 t