On the Q – S Chaos Synchronization of Fractional-Order Discrete-Time Systems : General Method and Examples

Copyright © 2018 Adel Ouannas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we propose two control strategies for the Q–S synchronization of fractional-order discrete-time chaotic systems. Assuming that the dimension of the response system m is higher than that of the drive system n, the first control scheme achieves n-dimensional synchronization whereas the second deals with the m-dimensional case. The stability of the proposed schemes is established by means of the linearization method. Numerical results are presented to confirm the findings of the study.


Introduction
Since the concept of chaos synchronization was first proposed nearly three decades ago, dynamical systems exhibiting chaotic behavior have attracted considerable attention in a number of fields.In addition to the original continuoustime chaotic systems such as Lorenz and Chen systems, discrete-time chaotic systems or maps have been proposed and studied.Among the most widely used 2-component discrete systems of these are the Hénon map [1], the Lozi map [2], and the flow model [3].Higher-dimensional systems have also been studied such as the generalized Hénon map [4] and the Stefanski map [5].All of these maps are classified as integer-order signifying that the difference order is integer.
Although fractional calculus was studied by mathematicians as early as the nineteenth century, it wasn't until recently that it gained popularity in applied science and engineering.For the longest time, the study and application of fractional calculus was limited to continuous time.However, most recently, researchers have diverted their attention to the discrete-time case and attempted to put together a complete theoretical framework for the subject.Perhaps one of the earliest works is that of [6].Among the most interesting and relevant works on fractional discrete calculus in the last decade is that of [7] where the authors introduce a backward fractional difference operator.In [8], the author discusses the discrete-time difference counterparts of conventional Riemann and Caputo derivatives.More on the general notation of fractional discrete-time calculus can be found in [9,10].Furthermore, the numerical formulas corresponding to a fractional difference systems can be found in [11].Recent studies have examined the stability conditions for fractional discrete-time systems including [12].Most recently, some advances have been made in relation to chaotic fractional discrete-time systems and their applications.A number of fractional difference maps including the fractional logistic map [13], the fractional sine map [14], fractional Hénon map [15], and fractional cubic logistic map [16].
Chaos synchronization in its simplest form refers to the control of a response chaotic system to force its trajectory towards that of a drive [17].In this case, the error defined as the difference between the states of the drive and those of the response towards zero as  → ∞.Numerous other types of synchronization have been proposed in the literature, whereby a general function of the two state vectors is forced to zero instead.The synchronization of integer-order discretetime systems subject to different definitions of the error has been studied in the literature such as hybrid synchronization [18], generalized synchronization [19], and inverse full state hybrid projective synchronization [20].As for fractionalorder discrete systems, the literature related to their synchronization is still very limited including synchronization of the logistic map [21], synchronization based on the stability condition [22], synchronization of linearly coupled fractional Hénon maps [23], and impulsive synchronization of fractional-order discrete-time hyperchaotic systems [24].
In this paper, we are concerned with the - synchronization of fractional-order discrete-time chaotic systems with different dimensions and non-identical orders.In this scheme, two functions  and  are used to condition the response and drive states, respectively, such that different values for  and  lead to different types of synchronization.- synchronization was first proposed by Yan in 2005 [25] for continuous-time dynamical systems.The author developed a backstepping approach with a strict feedback form to synchronize two identical systems.The author, then, extended the - scheme to discrete-time systems [26].In the years that followed, several algorithms were proposed for the - synchronization of integer-order continuous-time systems including [27][28][29].Several studies also looked at the fractional continuous-time case such as [30].Finally, in [31,32], the authors investigated - synchronization between integer-order and fractional-order continuous-time systems with different dimensions.
In the next section of this paper, we will present the notation adopted in our study and formulate the - synchronization problem for a fractional-order discrete-time chaotic drive-response pair.We denote the number of states for the drive and response by  and , respectively.Sections 3 and 4 present the control laws for the -dimensional and -dimensional cases, respectively.Section 5 presents some numerical results related to two particular examples.Finally, Section 6 provides a general summary of the main findings of this study.

Problem Formulation
Consider the following drive and response maps described as where () = ( 1 (), . . .,   ())  and () = ( 1 (), . . .,   ())  denote the drive and response state vectors, respectively,  ∈ R × ,  ∈ R × ,  : R  → R  ,  : R  → R  are nonlinear functions and  = (  ) 1≤≤ is a vector controller to be determined by the control laws of the synchronization scheme.Note that N  denotes the set of natural numbers starting from  and 0 <  ≤ 1.
The notation  Δ   () denotes the Caputo type delta difference of () defined over N  [8].This operator can be defined as for  ∉ N,  ∈ N +− , and  = [] + 1.The operator Δ   () denotes the -th fractional sum of   : N  → R defined in [7] as with  > 0, N  = {,  + 1,  + 2, . ..}, and  () is given in terms of the Gamma function Γ as For the purpose of this study, we will assume  = 0 and thus the subscript may be ignored.With these notations in mind, we can go ahead and define the − synchronization of our drive-response pair.The following definition is consistent with the literature reviewed earlier in this paper.( It is easy to observe from Definition 1 that, depending on our choice of the pair ((()), (())), we may end up with one of many synchronization types, namely,  For reasons that will become clear later on, suppose that  can be divided into two parts: where Q is an invertible  ×  matrix and q : R  → R  is a nonlinear function.In the following two sections, we will present two distinct synchronization schemes of dimensions  =  and  = , respectively.

𝑄−𝑆 Synchronization in Dimension 𝑚
The - synchronization error defined in ( 5) can be written in the following form: where Q is an invertible  ×  matrix, q : R  → R  , and  : R  → R  .The  order Caputo fractional difference of ( 8) is of the following form: This can be further simplified to where  ∈ R × is an appropriately chosen control matrix and R = ( − )  () + Q () + Q ( ()) The following remark is important.
Remark 2. Note that our aim in synchronization is to find a suitable controller  that forces the synchronization error to zero asymptotically.Since  is part of the response system, the function (()) hinders our process.To overcome this problem, we assumed that (()) can be decomposed into a linear part (matrix Q) and some other nonlinear function q(()).The reasoning behind this is that matrix Q is constant and thus may be factored out of the fractional difference operator.
Before stating the proposed control law and establishing its stability, it is important to state the following theorem, which is essential for our proof.Interested readers are referred to [33] for the proof of this result.
With this stability result, we are now ready to present our synchronization scheme.

Theorem 4. The drive-response pair (1) is globally 𝑄-𝑆 synchronized in dimension 𝑚 subject to
where Q −1 is the inverse of Q and the control matrix  is selected such that all the eigenvalues  of  −  satisfy Proof.By substituting ( 14) into (10), we obtain a new formulation for the error system: Δ   () = ( − )  ( +  − 1) .
Now, it is easy to see that subject to (15), all eigenvalues  of matrix  −  satisfy and Therefore, by means of Theorem 3, we establish the global asymptotic stability of the zero solution.Consequently, the drive-response maps (1) are globally  −  synchronized in dimension .

𝑄−𝑆 Synchronization in Dimension 𝑛
Let us now move to establish  −  synchronization in dimension .In this section, we assume  < .We define the following notations: and  : R  → R  .
We set the last  −  controller components to zero, i.e., Then, the error system reduces to where Q is an invertible  ×  matrix and Û = ( 1 , . . .,   )  .This can be further simplified to where  ∈ R × is an appropriate control matrix and T = ( − )  () + Q () + Q ( ()) To achieve - synchronization between the driveresponse maps (1) in dimension , we propose choosing Substituting ( 24) into ( 22) yields the error dynamics The following theorem can be proven in the exact same way as Theorem 4.
Theorem 5.By selecting control matrix  such that all the eigenvalues  of matrix  −  satisfy −2  <  < 0, the drive and response maps (1) are globally - synchronized in  dimensions subject to (20) and (24).

Numerical Examples
In order to show the validity of the proposed control strategies, let us consider some numerical examples.The 2D fractional Hénon map given by is chosen as the driving map.The fractional Hénon map (26) was proposed in [34].The authors showed that this map has a chaotic attractor, for instance, when ( 1 ,  1 ) = (1.4,0.3) and  = 0.984.In order to unify the notation, (26) can be rewritten as where The chaotic behavior of the drive map is depicted in Figure 1.As for the response map, we choose the 3D fractionalorder generalized Hénon map [35], which is of the following form: This system exhibits a chaotic behavior subject to certain conditions.We choose as an example the parameters ( 2 ,  2 ) = (0.99, 0.2) and  = 0.984, which has chaotic trajectories as shown in Figure 2. Map (29) can be rewritten as where and  = ( We will take 3and 2-dimensional - synchronization cases separately following the control laws of Theorems 4 and 5, respectively.Case 1.Let us, first, consider the 3-dimensional - synchronization case.Based on the approach stated in Section 3, the error system is given by where and We may rewrite  in the following form: where and It is easy to see that According to our approach described in Section 3, there exists a control matrix  such that all the eigenvalues  −  satisfy the condition of Theorem 4. We may simply choose the following: We start by calculating R by means of ( 11) and then  by means of (14).The resulting error system is given by The time evolution of the errors is depicted in Figure 3 for the initial values: The errors clearly converge to zero in sufficient time indicating that the drive map (26) and the response map (29) become - synchronized in 3 dimensions.
Case 2. As for the two-dimensional case, we follow the approach described in Section 4. The error system is given by where and We can write where and This yields According to Theorem 5, the control matrix  can be selected as ) , which clearly satisfies the condition of Theorem 5. Next, we calculate T as specified in (23) and Û as in (24).The complete control vector  is formed by appending a zero to the end of Û.The resulting error system is given by  Δ   1 () = − 1 ( +  − 1) ,  Δ   2 () =  1 ( +  − 1) − 1 2  2 ( +  − 1) . (50) The convergence of the errors to zero is depicted in Figure 4 for the initial conditions: 2 (0) ) = ( 0.75 1.14 ) . (51)

Concluding Remarks
This paper has studied - synchronization of fractionalorder discrete-time chaotic systems of different dimensions.The - scheme aims for the general definition of the error towards zero in finite time, thereby covering a range of different synchronization types.We assume an -dimensional drive map and an -dimensional response map with the condition  > .Two different control laws have been derived and the asymptotic stability of their zero solution investigated through the linearization method.The two strategies correspond to the synchronization dimensions  =  and  = .Computer simulation results have been presented, whereby a 3-dimensional fractional-order generalized Hénon map response synchronized the 2-dimensional fractional-order Hénon drive.The two proposed schemes were utilized to achieve 2and 3-dimensional synchronization.The errors have been shown to converge towards zero in sufficient time.

Figure 3 :
Figure 3: The evolution of errors over time for Case 1.

Figure 4 :
Figure 4: The evolution of errors over time for Case 2.