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

Authors: Adel Ouannas; Amina-Aicha Khennaoui; Giuseppe Grassi; Samir Bendoukha
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

In this paper, we propose two control strategies for theQ–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.


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## Body

## 1. 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 continuous-time 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 t→∞. 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 discrete-time 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 fractional-order 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 theQ–S synchronization of fractional-order discrete-time chaotic systems with different dimensions and non–identical orders. In this scheme, two functions Q and S are used to condition the response and drive states, respectively, such that different values for Q and S lead to different types of synchronization. Q–S 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 Q–S scheme to discrete-time systems [26]. In the years that followed, several algorithms were proposed for the Q–S synchronization of integer-order continuous-time systems including [27–29]. Several studies also looked at the fractional continuous-time case such as [30]. Finally, in [31, 32], the authors investigated Q–S 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 theQ–S synchronization problem for a fractional-order discrete-time chaotic drive-response pair. We denote the number of states for the drive and response by n and m, respectively. Sections 3 and 4 present the control laws for the m–dimensional and n–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.


## 2. Problem Formulation

Consider the following drive and response maps described as(1)ΔaαCXt=AXt+υ-1+fXt+υ-1,∀t∈Na+1-α,ΔaβCYt=BYt+υ-1+gYt+υ-1+U,∀t∈Na+1-βwhere Xt=x1t,…,xntT and Yt=y1t,…,ymtT denote the drive and response state vectors, respectively, A∈Rn×n, B∈Rm×m, f:Rn→Rn, g:Rm→Rm are nonlinear functions and U=ui1≤i≤m is a vector controller to be determined by the control laws of the synchronization scheme. Note that Na denotes the set of natural numbers starting from a and 0<υ≤1.The notationΔaυCXt denotes the Caputo type delta difference of Xt defined over Na [8]. This operator can be defined as(2)ΔaυCXt=Δa-n-υΔnXt=1Γn-υ∑s=at-n-υt-s-1n-υ-1ΔsnXs,for υ∉N, t∈Na+n-υ, and n=υ+1. The operator ΔsnXt denotes the υ–th fractional sum of xi:Na→R defined in [7] as(3)Δa-υXt=1Γυ∑s=at-υt-s-1υ-1Xs,with υ>0, Na=a,a+1,a+2,…, and tυ is given in terms of the Gamma function Γ as(4)tυ=Γt+1Γt+1-υ.For the purpose of this study, we will assumea=0 and thus the subscript may be ignored. With these notations in mind, we can go ahead and define the Q-S synchronization of our drive-response pair. The following definition is consistent with the literature reviewed earlier in this paper.Definition 1.
The drive-response pair (1) is Q-S synchronized in the dth dimension if there exist a controller U=ui1≤i≤m and two functions Q:Rm→Rd and S:Rn→Rd such that(5)limt→+∞⁡et≔QYt-SXt=0.It is easy to observe from Definition1 that, depending on our choice of the pair QYt,SXt, we may end up with one of many synchronization types, namely, (6)Yt,Xt⟶Complete  synchronization,Yt,-Xt⟶Anti  synchronization,Yt,θXt,θ∈R∗⟶Projective  synchronization,Yt,MXt,M∈Rm×n⟶Matrix  projective  synchronization,MYt,Xt,M∈Rn×m⟶Inverse  matrix  projective  synchronization,Yt,SXt⟶Generalized  synchronization,QYt,Xt⟶Inverse  generalized  synchronization.For reasons that will become clear later on, suppose thatQ can be divided into two parts:(7)QYt=QYt+qYt,where Q is an invertible d×d matrix and q:Rd→Rd is a nonlinear function. In the following two sections, we will present two distinct synchronization schemes of dimensions d=m and d=n, respectively.


## 3.Q-S Synchronization in Dimension m

TheQ–S synchronization error defined in (5) can be written in the following form:(8)et=QYt+qYt-SXt,where Q is an invertible m×m matrix, q:Rm→Rm, and S:Rn→Rm. The β order Caputo fractional difference of (8) is of the following form:(9)ΔβCet=ΔβCQYt+qYt-SXt=ΔβCQYt+ΔβCqYt-ΔβCSXt=QΔβCYt+ΔβCqYt-ΔβCSXt=QBYt+β-1+gYt+β-1+U+ΔβCqYt-ΔβCSXt=QBYt+β-1+QgYt+β-1+Q×U+ΔβCqYt-ΔβCSXt.This can be further simplified to(10)ΔβCet=B-Cet+β-1+R+Q×U,where C∈Rm×m is an appropriately chosen control matrix and(11)R=C-Bet+QBYt+QgYt+ΔβCqYt-ΔβCSXt.The following remark is important.Remark 2.
Note that our aim in synchronization is to find a suitable controllerU that forces the synchronization error to zero asymptotically. Since U is part of the response system, the function QYt hinders our process. To overcome this problem, we assumed that QYt can be decomposed into a linear part (matrix Q) and some other nonlinear function qYt. 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.Theorem 3.
The zero equilibrium of the linear fractional-order discrete-time system:(12)ΔaυCet=Met+υ-1,where e(t)=e1(t),…,en(t)T, 0<υ≤1, M∈Rn×n and ∀t∈Na+1-υ, is asymptotically stable, if(13)λ∈z∈C:z<2cos⁡arg⁡z-π2-υυandarg⁡z>υπ2,for all the eigenvalues λ of M.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(14)U=-Q-1×R,where Q-1 is the inverse of Q and the control matrix C is selected such that all the eigenvalues λ of B-C satisfy(15)-2β<λ<0.Proof.
By substituting (14) into (10), we obtain a new formulation for the error system:(16)ΔβCet=B-Cet+β-1.Now, it is easy to see that subject to (15), all eigenvalues λ of matrix B-C satisfy (17)arg⁡λ=π>βπ2,and(18)λ<2cos⁡arg⁡λ-π2-ββ.Therefore, by means of Theorem 3, we establish the global asymptotic stability of the zero solution. Consequently, the drive-response maps (1) are globally Q-S synchronized in dimension m.


## 4.Q-S Synchronization in Dimension n

Let us now move to establishQ-S synchronization in dimension n. In this section, we assume n<m. We define the following notations: (19)Q∈Rn×m,q:Rm→Rn,andS:Rn→Rn.We set the last m-n controller components to zero, i.e., (20)ui=0forn<i≤m.Then, the error system reduces to(21)ΔβCet=QBYt+β-1+QgYt+β-1+Q^×U^+ΔβCqYt-ΔβCSXt,where Q^ is an invertible n×n matrix and U^=u1,…,unT. This can be further simplified to(22)ΔCβet=A-Let+T+Q^×U^,where L∈Rn×n is an appropriate control matrix and(23)T=L-Aet+QBYt+QgYt+ΔβCqYt-ΔβCSXt.To achieveQ–S synchronization between the drive-response maps (1) in dimension n, we propose choosing(24)U^=-Q^-1T.Substituting (24) into (22) yields the error dynamics(25)ΔβCet=A-Let+β-1.The following theorem can be proven in the exact same way as Theorem 4.Theorem 5.
By selecting control matrixL such that all the eigenvalues λ of matrix A-L satisfy -2β<λ<0, the drive and response maps (1) are globally Q–S synchronized in n dimensions subject to (20) and (24).


## 5. 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(26)ΔαCx1t=x2t+α-1+1-a1x12t+α-1-x1t+α-1,ΔαCx2t=b1x1t+α-1-x2t+α-1,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 a1,b1=1.4,0.3 and α=0.984. In order to unify the notation, (26) can be rewritten as(27)ΔαCXt=AXt+fXt,where(28)A=-11b1-1andf=1-a1x12t0.The chaotic behavior of the drive map is depicted in Figure 1.Figure 1
Phase space plot for the fractional Hénon map witha1,b1=1.4,0.3, α=0.984, and x10=x20=0.As for the response map, we choose the 3D fractional-order generalized Hénon map [35], which is of the following form:(29)ΔβCy1t=-y1t+β-1-b2y3t+β-1+u1t+β-1,ΔβCy2t=b2y3t+β-1+y1t+β-1-y2t+β-1+u2t+β-1,ΔβCy3t=1+y2t+β-1-a2y32t+β-1-y3t+β-1+u3t+β-1.This system exhibits a chaotic behavior subject to certain conditions. We choose as an example the parameters a2,b2=0.99,0.2 and β=0.984, which has chaotic trajectories as shown in Figure 2. Map (29) can be rewritten as(30)ΔβCYt=BYt+gYt,where(31)B=-10-b21-1b201-1andg=001-a2y32t.Figure 2
Phase portraits for the fractional generalized Hénon map witha2,b2=0.99,0.2, β=0.984 and y10,y20,y30=0.1,0.2,0.5.We will take3- and 2-dimensional Q–S synchronization cases separately following the control laws of Theorems 4 and 5, respectively.Case 1.
Let us, first, consider the3-dimensional Q–S synchronization case. Based on the approach stated in Section 3, the error system is given by(32)e1t,e2t,e3tT=Qy1t,y2t,y3t-Sx1t,x2t,where (33)Qy1t,y2t,y3t=y1t,y1t+y2t,y3t3+y2tT,and (34)Sx1t,x2t=x1t,x2t,x1t+x2t.We may rewrite Q in the following form:(35)Qy1t,y2t,y3t=Q×y1t,y2t,y3tT+qy1t,y2t,y3twhere(36)Q=100120003and(37)qy1t,y2t,y3t=0,0,y2ty3tT.It is easy to see that(38)Q-1=100-121200013.According to our approach described in Section 3, there exists a control matrix C such that all the eigenvalues B-C satisfy the condition of Theorem 4. We may simply choose the following:(39)C=000100010.We start by calculating R by means of (11) and then U by means of (14). The resulting error system is given by(40)ΔβCe1t=-e1t+β-1-b2e3t+β-1,ΔβCe2t=-e2t+β-1+b2e3t+β-1,ΔβCe3t=-e3t+β-1.The time evolution of the errors is depicted in Figure 3 for the initial values: (41)e10e20e30=0.10.31.6.The errors clearly converge to zero in sufficient time indicating that the drive map (26) and the response map (29) become Q–S synchronized in 3 dimensions.Figure 3
The evolution of errors over time for Case1.Case 2.
As for the two-dimensional case, we follow the approach described in Section4. The error system is given by(42)e1t,e2t,e3tT=Qy1t,y2t,y3t-Sx1t,x2t,where (43)Qy1t,y2t,y3t=y1t+2y2t+y32t,-y1t+3y3tT,and(44)Sx1t,x2t=x1t,x1tx2t.We can write(45)Qy1t,y2t,y3t=Q×y1t,y2t,y3tT+qy1t,y2t,y3t,where(46)Q=120-103,and(47)qy1t,y2t,y3t=y32t,0T.This yields(48)Q^=12-10andQ^-1=12-10.According to Theorem 5, the control matrix L can be selected as(49)L=01-1+b1-12,which clearly satisfies the condition of Theorem 5. Next, we calculate T as specified in (23) and U^ as in (24). The complete control vector U is formed by appending a zero to the end of U^. The resulting error system is given by(50)ΔβCe1t=-e1t+β-1,ΔβCe2t=e1t+β-1-12e2t+β-1.The convergence of the errors to zero is depicted in Figure 4 for the initial conditions: (51)e10e20=0.751.14.Figure 4
The evolution of errors over time for Case2.


## 6. Concluding Remarks

This paper has studiedQ–S synchronization of fractional-order discrete-time chaotic systems of different dimensions. The Q–S 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 n-dimensional drive map and an m-dimensional response map with the condition m>n. 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 d=n and d=m.Computer simulation results have been presented, whereby a3-dimensional fractional-order generalized Hénon map response synchronized the 2-dimensional fractional-order Hénon drive. The two proposed schemes were utilized to achieve 2- and 3-dimensional synchronization. The errors have been shown to converge towards zero in sufficient time.



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