# Compound Generalized Function Projective Synchronization for Fractional-Order Chaotic Systems

Authors: Chunde Yang; Hao Cai; Ping Zhou
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

A modified function projective synchronization for fractional-order chaotic system, called compound generalized function projective synchronization (CGFPS), is proposed theoretically in this paper. There are one scaling-drive system, more than one base-drive system, and one response system in the scheme of CGFPS, and the scaling function matrices come from multidrive systems. The proposed CGFPS technique is based on the stability theory of fractional-order system. Moreover, we achieve the CGFPS between three-driver chaotic systems, that is, the fractional-order Arneodo chaotic system, the fractional-order Chen chaotic system, and the fractional-order Lu chaotic system, and one response chaotic system, that is, the fractional-order Lorenz chaotic system. Numerical experiments are demonstrated to verify the effectiveness of the CGFPS scheme.


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## Body

## 1. Introduction

Chaotic phenomenon in nonlinear system has attracted more and more attentions in recent years [1, 2]. In 1990,chaos synchronization was introduced by Pecora and Carroll for the first time [3], and after that, chaos synchronization has attracted a lot of attention over the last two decades. This is due to its potential applications in many areas of physics and engineering science [3], such as complex networks control [4], signal processing [5], information transmission [6], authenticated encryption [7], and secure communication [8, 9]. On the other hand, as physical interpretation of the fractional derivative becomes clear, many real-world physical systems can be more accurately described by fractional-order differential equations [10, 11]. Chaotic phenomenon has been also observed in many physical fractional-order systems, for example, microelectromechanical systems [12], gyroscopes [13], and electronic circuits [14, 15]. Meanwhile, more and more attentions [4, 7, 16, 17] have been paid on synchronization and control of fractional-order chaotic system in both theoretical and applied perspectives, since it is usually a prerequisite of many practical applications in chaos engineering, especially in authenticated encryption [7] and chaos communications [18].Up to now, various types of chaos synchronization have been proposed, such as complete synchronization (CS) [3, 8], generalized synchronization (GS) [9, 19], projective synchronization (PS) [20], function projective synchronization (FPS) [21–23], combination synchronization [24], and compound synchronization [25]. Among all these schemes, PS and FPS have been extensively studied in recent years because the two schemes can not only obtain faster communication but also enhance the security of communications with its proportional feature. The PS scheme indicates that the drive system and response system could be arrived to synchronize with a scaling constant factor. The FPS scheme, generalized from the PS scheme, means that the driver system and response system could be synchronized with a scaling function matrix instead of the scaling factor. Since the scaling function matrix can be also used as additional secret codes, the FPS scheme can enhance the security in secure communication.However, the response system could be synchronized to the drive system with a function matrix in FPS scheme, and there are only one drive system and one response system. Therefore, the FPS between multidrive systems and one response system, where the scaling function matrix comes from multidrive systems, is interesting and general topic. It is obvious that multidrive systems and one response system synchronization in FPS scheme can additionally enhance the security of communication; this is due to the fact that the transmitted signals can be split into several parts, and each part may be loaded in different drive systems; or the transmitted signals can be divided in time into different intervals, and the signals in different intervals may be loaded in different drive systems [24, 25]. Motivated by the abovementioned, a new function projective synchronization scheme, called compound generalized function projective synchronization (briefly denoted by CGFPS), is proposed for fractional-order chaotic systems in this paper. The CGFPS scheme means that there are multidrive systems and one response system, and the scaling function matrix comes from multidrive systems. This CGFPS technique is based on the stability theory of fractional-order systems. Some examples are demonstrated to verify the effectiveness of the CGFPS scheme.The layout of this paper is organized as follows. In Section2, the CGFPS scheme for fractional-order chaotic systems is presented. In Section 3, numerical experiments are used to verify the effectiveness of the CGFPS scheme. Finally, the conclusion ends the paper in Section 4.


## 2. The Compound Generalized Function Projective Synchronization (CGFPS) for Fractional-Order Chaotic Systems

In this section, we will demonstrate the CGFPS scheme for fractional-order chaotic systems. As is well known, there are several definitions about fractional derivative. This paper will use the Caputo definition of the fractional derivative, described as follows [26]:(1)Dqft=1Γm-q∫0tfmτt-τq+1-mdτ,m-1<q<m,where Dq is called the Caputo operator, m is the first integer which is not less than q, and f(m)(t) is the m-order derivative for f(t); that is, f(m)(t)=dmf(t)/dtm.Consider the fractional-order multidrive chaotic systems and one fractional-order response chaotic system described as follows:(2)Dqd0z0=d0z0,(3)Dqdizi=dizii=1,2,…,m,(4)Dqry=ry+σz0,z1,…,zm,y,where system (2) is an n-dimensional scaling-drive system [25], each system in (3) is an n-dimensional base-drive system [25], and system (4) is an n-dimensional response system. 0<qdi<1 (i=0,1,2,…,m) and 0<qr<1 are fractional-order. zi=(zi1,zi2,…,zin)T (i=0,1,…,m) and y=(y1,y2,…,yn)T are state vectors of fractional-order chaotic systems (2)–(4). di (i=0,1,2,…,m) :Rn→Rn and r:Rn→Rn are both differential nonlinear vector functions. σ(z0,z1,…,zm,y)∈Rn×1 is a controller which will be designed later. In the base-drive systems described in (3), each system is an individual chaotic system, which is not related to any other chaotic systems in (3), and the state variables in ith chaotic system in (3) are (zi1,zi2,…,zin).Before we give a detailed definition of the CGFPS scheme, let us first recall the definitions of the FPS scheme and the compound synchronization scheme.Definition 1 (see [21–23]).
For the scaling-drive system (2) and the response system (4), in which the vector controller σ(z0,z1,…,zm,y) is changed as σ(z0,y) in response system (4), it is said to be function projective synchronization (FPS) if there exists scaling function matrix K(z0)∈Rn×n such that(5)limt→+∞⁡e=limt→+∞⁡Kz0z0-y=0,where · represents the Euclidean norm.Definition 2 (see [25]).
For the scaling-drive system (2), the base-drive systems (3), and the response system (4), it is said to be compound synchronization if there exists constant real matrix Ki∈Rn×n (i=1,2,…,m) such that(6)limt→+∞⁡e=limt→+∞⁡∑i=1mKiziz0-y=0,where · represents the Euclidean norm.Now, we establish the CGFPS scheme for fractional-order chaotic systems.Definition 3.
For the scaling-drive system (2), the base-drive systems (3), and the response system (4), it is said to be compound generalized function projective synchronization if there exist constant real matrix Li∈Rn×n (i=0,1,2,…,m) and scaling function matrix Ki(zi)∈Rn×n (i=0,1,2,…,m) such that(7)limt→+∞⁡e=limt→+∞⁡∑i=0mLiKiziz0-y=0,where · represents the Euclidean norm.Remark 4.
IfL0≠0, Li=0 (i=1,2,…,m), and K0(z0) is a nonzero function matrix, then the CGFPS scheme will be turned into FPS scheme. If L0=0, Li≠0 (i=1,2,…,m), and LiKi(zi)=Kizi (i=1,2,…,m) is a nonzero function matrix, then the CGFPS scheme will be turned into compound synchronization scheme. If Li=0 (i=0,1,2,…,m) in the CGFPS scheme, then the CGFPS scheme will be turned into a chaos control problem.Remark 5.
The scaling function matrixKi(zi)∈Rn×n (i=0,1,2,…,m) in the CGFPS comes from multidrive chaotic systems (2) and (3), and the scaling function matrix K(z0)∈Rn×n in the FPS comes from one drive system (2) (the scaling-drive system). On the other hand, the scaling matrix ki∈Rn×n (i=1,2,…,m) in compound synchronization scheme is a real constant matrix. Therefore, the CGFPS scheme is more general than the FPS scheme and the compound synchronization scheme.Remark 6.
The scaling function matrixKi(zi)∈Rn×n (i=0,1,2,…,m) in CGFPS comes from different chaotic systems (m+1 chaotic systems), and the term ∑i=0mLiKi(zi) in CGFPS may be nonlinear. On the other hand, the scaling function matrix K(z0)∈Rn×n in FPS comes from one drive system, and the term ∑i=1mKizi in compound synchronization is linear. Since additional secret codes can come from Ki(zi)∈Rn×n (i=0,1,2,…,m) and ∑i=0mLiKi(zi) in the secure communication, the CGFPS scheme should be more secure than the FPS and compound synchronization schemes.Remark 7.
The scaling-drive system (2) and the base-drive systems (3) in CGFPS scheme are maybe integer order systems. Therefore, the CGFPS between integer order chaotic systems and fractional-order systems can be achieved. Moreover, the CGFPS can be applied for integer order chaotic systems; that is, the scaling-drive system (2), the base-drive systems (3), and response system (4) are integer order chaotic systems.Next, we will discuss how to achieve CGFPS between the drive systems (2) and (3) and the response system (4). Now, let the CGFPS error vectors between the drive systems (2) and (3) and the response system (4) be(8)e=y-∑i=0mLiKiziz0,where e=(e1,e2,…,en)T, el=yl-∑j=1n{[∑i=0mLiKi(zi)]lj[(z0)j1]} (l,j=1,2,…,n), [∑i=0mLiKi(zi)]lj (l,j=1,2,…,n) denotes the element of matrix ∑i=0mLiKi(zi), and (z0)j1 (j=1,2,…,n) indicates the element of matrix z0.Now, the controllerσ(z0,z1,…,zm,y)∈Rn×1 in system (4) can be chosen as(9)σz0,z1,…,zm,y=Dqr∑i=0mLiKiziz0-r∑i=0mLiKiziz0+Ω1z0,z1,…,zm,ye,where Ω1(z0,z1,…,zm,y)∈Rn×n is a feedback controller which will be designed later. Li∈Rn×n (i=0,1,2,…,m) is a bounded nonzero constant matrix, and Ki(zi)∈Rn×n (i=0,1,2,…,m) is a bounded nonzero function matrix.Using the above controller (9), system (4) can be rewritten as(10)Dqry=Dqr∑i=0mLiKiziz0-r∑i=0mLiKiziz0+Ω1z0,z1,…,zm,ye+ry.Using e=y-[∑i=0mLiKi(zi)]z0, system (10) can be changed as(11)Dqre=ry-r∑i=0mLiKiziz0+Ω1z0,z1,…,zm,ye.For response system (4), we assume that r(y)-r(y′)=Ω2(y,y′)(y-y′) in this paper, where y′∈Rn and Ω2(y,y′)∈Rn×n. In fact, this assumption can be satisfied by many fractional-order chaotic/hyperchaotic systems, such as fractional-order Lorenz chaotic/hyperchaotic systems, fractional-order Chen chaotic/hyperchaotic systems, fractional-order Lü chaotic/hyperchaotic systems, and the fractional-order chaotic systems which will be described in Section 3.Combiningr(y)-r(y′)=Ω2(y,y′)(y-y′), system (11) can be rewritten as(12)Dqre=Ω2z0,z1,…,zm,ye+Ω1z0,z1,…,zm,ye,where Ω2(z0,z1,…,zm,y)∈Rn×n.Therefore, we can yield the following fractional-order system about errors:(13)Dqre=Ω1z0,z1,…,zm,y+Ω2z0,z1,…,zm,ye.According to system (13), we know that the CGFPS between systems (2) and (3) and system (4) is turned into the following problem: choose suitable vectors Ω1(z0,z1,…,zm,y)∈Rn×n such that fractional-order error system (13) is asymptotically converged to zero.Here, let us recall one result for the stability of nonlinear fractional-order system [23, 27, 28]. Given the following n-dimension nonlinear fractional-order systems,(14)Dqx=hx,where 0<q<1, x∈Rn, h:Rn→Rn is a nonlinear continuous vector function. The equilibrium point x∗ (i.e., h(x∗)=0) of system (14) is asymptotically stable if all the eigenvalues λ of the Jacobean matrix ∂h/∂xx=x∗,h(x∗)=0 satisfy arg⁡(λ)>0.5πq. Then, we have the main theorem of this paper as follows.Theorem 8.
If one chooses the feedback controllerΩ1(z0,z1,…,zm,y) and real symmetric positive definite matrix P, Q such that(15)PΩ1z0,z1,…,zm,y+Ω2z0,z1,…,zm,y+Ω1z0,z1,…,zm,y+Ω2z0,z1,…,zm,yHP=-Q,then the compound generalized function projective synchronization (CGFPS) between drive systems (2) and (3) and response system (4) can be achieved, where H denotes the conjugate transpose of a matrix.Proof.
Use(16)PΩ1z0,z1,…,zm,y+Ω2z0,z1,…,zm,y+Ω1z0,z1,…,zm,y+Ω2z0,z1,…,zm,yHP=-Q.According to the results mentioned in [23, 29], we can yield the following:(17)arg⁡λΩ1z0,z1,…,zm,y+Ω2z0,z1,…,zm,y>0.5πqr.Here, λ[Ω1(z0,z1,…,zm,y)+Ω2(z0,z1,…,zm,y)] are the eigenvalues of the matrix Ω1(z0,z1,…,zm,y)+Ω2(z0,z1,…,zm,y).
According to the stability for nonlinear fractional-order systems [23, 27, 28], inequality (17) implies that the equilibrium point e=(0,0,…,0)T of fractional-order nonlinear system (13) is asymptotically stable; that is, (18)limt→+∞⁡e=limt→+∞⁡y-∑i=0mLiKiziz0=0.
This result implies that the compound generalized function projective synchronization (CGFPS) between drive systems (2) and (3) and response system (4) can be achieved. The proof is completed.


## 3. Illustrative Examples

In this section, numerical experiments are used to verify the effectiveness of the CGFPS scheme in our paper. Some examples are given and the numerical simulations are performed.Now, choose the fractional-order Arneodo chaotic system as the scaling-drive system, which is given by [30](19)Dqd0z01=z02,Dqd0z02=z03,Dqd0z03=5.5z01-3.5z02-z03-z013.The chaotic attractor of system (19) with qd0=0.995 is shown in Figure 1.Figure 1
The chaotic attractor of system (19) for qd0=0.995.
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(b)Then, choose the fractional-order Chen chaotic system and the fractional-order Lu chaotic system as the base-drive systems. The fractional-order Chen chaotic system [22] is defined as(20)Dqd1z11=35z12-z11,Dqd1z12=-7z11+28z12-z11z13,Dqd1z13=z11z12-3z13and the chaotic attractor of system (20) with qd1=0.9 is shown as in Figure 2. The fractional-order Lü chaotic system [31] is defined as(21)Dqd2z21=36z22-z21,Dqd2z22=20z22-z21z23,Dqd2z23=z21z22-3z23and the chaotic attractor of system (21) with qd2=0.96 is shown in Figure 3.Figure 2
The chaotic attractor of system (20) for qd1=0.9.
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(b)Figure 3
The chaotic attractor of system (21) for qd2=0.96.
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(b)Next, we choose the fractional-order Loren chaotic system as response system, which is given by [22](22)Dqry1=10y2-y1,Dqry2=28y1-y2-y1y3,Dqry3=y1y2-8y33.The chaotic attractor of system (22) with qr=0.997 is shown in Figure 4.Figure 4
The chaotic attractor of system (22) for qr=0.997.
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(b)Next, we discuss how to realize the CGFPS between drive systems (19)–(21) and response system (22). Now, let Li∈R3×3 (i=0,1,2) be a constant real matrix and let Ki(zi)∈R3×3 (i=0,1,2) be a scaling function matrix, where zi=(zi1,zi2,zi3)T (i=0,1,2).According to Section2, we can yield that(23)Ω2z0,z1,z2,y=-1010028-y3-1-Λy2Λ-83,where Λ=z01[∑i=02LiKi(zi)]11+z02[∑i=02LiKi(zi)]12+z03[∑i=02LiKi(zi)]13, [∑i=02LiKi(zi)]1j (j=1,2,3) are the elements of matrix [∑i=02LiKi(zi)], and y=(y1,y2,y3)T.If we choose the feedback controllerΩ1(z0,z1,z2,y) and the real symmetric positive definite matrix P as follows(24)Ω1z0,z1,z2,y=0-38+y3-y2000000,P=100010001,then, it follows that(25)PΩ1z0,z1,z2,y+Ω2z0,z1,z2,y+Ω1z0,z1,z2,y+Ω2z0,z1,z2,yHP=-Q,where Q=diag⁡(20,3,16/3), which is a real symmetric positive definite matrix. According to (17) in Section 2, we obtain(26)arg⁡λΩ1z0,z1,z2,y+Ω2z0,z1,z2,y>0.5πqr.Then, according to the stability for nonlinear fractional-order systems [23, 27, 28], we have(27)limt→+∞⁡e=limt→+∞⁡y-∑i=02LiKiziz0=0,which implies(28)limt→+∞⁡e1=limt→+∞⁡y1-z01∑i=02LiKizi11+z02∑i=02LiKizi12+z03∑i=02LiKizi13=0,limt→+∞⁡e2=limt→+∞⁡y2-z01∑i=02LiKizi21+z02∑i=02LiKizi22+z03∑i=02LiKizi23=0,limt→+∞⁡e3=limt→+∞⁡y3-z01∑i=02LiKizi31+z02∑i=02LiKizi32+z03∑i=02LiKizi33=0,where [∑i=02LiKi(zi)]lj (l,j=1,2,3) denotes the ljth element of matrix [∑i=02LiKi(zi)].Equations (27)–(28) indicate that the compound generalized function projective synchronization (CGFPS) between drive systems (19)–(21) and response system (22) can be achieved.Now, two cases with numerical simulations are given to verify the effectiveness of proposed scheme.Case 1.
ChooseLi=diag⁡(1,1,1) (i=0,1,2), K0(z0)=diag(z02,z03,5.5z01-3.5z02-z03-z013), K1(z1)=diag[35(z12-z11),-7z11-z11z13,z11z12-3z13], and K2(z2)=diag[36(z22-z21),-z21z23,z21z22-3z23]. The numerical results are shown in Figure 5, where the initial conditions are (z01,z02,z03)=(1,1,1), (z11,z12,z13)=(2,2,2), (z21,z22,z23)=(3,3,3), and (y1,y2,y3)=(6,-22,4), respectively.Figure 5
The CGFPS errors between the multidrive systems (19)–(21) and the response system (22).
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(c)Case 2.
ChooseL0=0, Li=diag(1,1) (i=1,2), K1(z1)=diag(z12+z11,-z11z13,z11z12), and K2(z2)=diag(z22+z21,z21z23,-z21z22). The corresponding numerical results are shown in Figure 6, where the initial conditions are (z01,z02,z03)=(1,1,1), (z11,z12,z13)=(2,2,2), (z21,z22,z23)=(3,3,3), and (y1,y2,y3)=(15,9,1), respectively.Figure 6
The CGFPS errors between the multidrive systems (19)–(21) and the response system (22).
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## 4. Conclusions

In this paper, we present a new synchronization scheme for fractional-order chaotic system and called this type of synchronization compound generalized function projective synchronization, or briefly denoted it by CGFPS. In this scheme, there are one scaling-drive system, more than one base-drive system, and one response system, and the scaling function matrix comes from multidrive systems. So, the CGFPS in this paper is different from all the previous synchronization reported before. To verify its effectiveness, we achieve the CGFPS between three drive systems and one response system, where the scaling-drive system is the fractional-order Arneodo chaotic system, the base-drive systems are the fractional-order Chen chaotic system and the fractional-order Lu chaotic system, and the response system is the fractional-order Loren chaotic system. Numerical simulations suggest that the presented CGFPS scheme works well. It is worth mentioning that the CGFPS scheme in our paper can also be used for other fractional-order chaotic systems or integer order chaotic systems.



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