# Synchronization in Tempered Fractional Complex Networks via Auxiliary System Approach

Authors: Weiyuan Ma; Changpin Li; Jingwei Deng
Journal: Complexity
 (2019)

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## Abstract

In the famous continuous time random walk (CTRW) model, because of the finite lifetime of biological particles, it is sometimes necessary to temper the power law measure such that the waiting time measure has a convergent first moment. The CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. In this article, we introduce the tempered fractional derivative into complex networks to describe the finite life span or bounded physical space of nodes. Some properties of the tempered fractional derivative and tempered fractional systems are discussed. Generalized synchronization in two-layer tempered fractional complex networks via pinning control is addressed based on the auxiliary system approach. The results of the proposed theory are used to derive a sufficient condition for achieving generalized synchronization of tempered fractional networks. Numerical simulations are presented to illustrate the effectiveness of the methods.


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## Body

## 1. Introduction

In the past few decades, the study of fractional calculus has attracted substantial attention. Fractional calculus is a powerful mathematical tool for modeling systems in the fields of secure communication, biological science, chemical reactors, laser systems, and so on [1–5]. Recently, tempered fractional derivatives have drawn the wide interests of researchers. A tempered fractional derivative is closer to reality in the sense of the finite life span or bounded physical space of particles. As a generalization of fractional calculus, tempered fractional calculus does not simply have the properties of fractional calculus but can describe some other complex dynamics [6, 7]. Tempered fractional derivatives and the corresponding tempered fractional differential equations have played key roles in poroelasticity [8], finance [9], ground-water hydrology [10], geophysical flows [11], and so on. Therefore, complex network models with tempered fractional dynamics can become more realistic.Synchronization, as one of the vital dynamical phenomena, exists not only in integer systems but also in fractional systems. Its applications range from electrology to biology, from physics to engineering, and even from economics to nervous system, communication system, and control processing [12–16]. Many studies have focused on the outer synchronization between two fractional complex networks [17–19]. Yu et al. investigated α-synchronization for fractional neural networks [20]. Wu et al. constructed outer synchronization for two different fractional complex networks with nonlinear controllers [21]. Yang and Jiang considered adaptive synchronization for fractional complex networks with uncertain parameters [22]. Chai et al. proposed a pinning synchronization for general fractional complex networks [23]. Dadras et al. studied fractional integral sliding-mode control for uncertain fractional nonlinear systems [24]. Shao et al. discussed adaptive sliding-mode synchronization for a class of fractional chaotic systems [25]. However, in most of this work, all individuals in two networks are assumed to have completely identical dynamics, which is not the case in real practice, for example, predator-prey interactions in ecological communities. Clearly, groups of predators and prey have their own dynamical behavior. Therefore, outer synchronization between two different tempered fractional complex networks is a practical and significant problem worth investigating.The scenario in which two networks reach harmonious coexistence is regarded as generalized synchronization, which is weaker than complete synchronization [26]. Generalized synchronization exists widely in nature and society. For example, predators and preys influence each other’s behaviors. Predators cannot live without preys, and too many predators would bring the preys into extinction. The complex systems of predators and preys will finally reach harmonious coexistence without man-made sabotage. The auxiliary system approach [27] was proposed to realize generalized synchronization between two complex networks. In addition to the drive and response systems, this method constructs an auxiliary system that has an identical dynamical system to that of the response system, as described by(1)DC0,tα1,λ1xt=Gxt,drive system,DC0,tα2,λ2ytHxt,yt,response system,DC0,tα2,λ2zt=Hxt,zt,auxiliary system,where xt,yt,zt∈ℝn are the states of drive, response, and auxiliary system, respectively. DC0,tα,λ is the tempered fractional Caputo derivative operator which is defined later on. If the response system and auxiliary system reach complete synchronization, that is, limt⟶∞yt−zt=0 for any initial conditions yt0≠zt0, then generalized synchronization between the drive system and response systems is achieved. Figure 1 gives the schematic diagram of generalized synchronization based on the auxiliary system approach. Specifically, layer I is the drive system, and layer II is the response system which is driven by signals from layer I. Layer III is the auxiliary system, which is an identical duplication of layer II driven by the same signals from the drive layer. This technique is a very effective method to realize outer synchronization between two complex networks [29]. However, as far as we know, no one has discussed the auxiliary system approach for fractional complex networks.Figure 1
A schematic diagram [28] of generalized synchronization via the auxiliary system approach.Synchronization of tempered fractional complex networks in a generalized sense leads to richer behavior than identical node dynamics in coupled networks. It may disclose a more complicated connection between the synchronized trajectories in the state spaces of coupled networks. In this paper, we first study the tempered fractional complex network and its generalized synchronization. Additionally, we derive some properties of tempered fractional calculus to construct the synchronization criteria of tempered fractional complex networks. Tempered Mittag-Leffler stable is proposed, which can better describe stability feature of tempered fractional systems. Third, tempered fractional chaotic systems have more alterable dynamical behaviors than fractional ones. It may be more useful in secure communication and control processing. Finally, an auxiliary system approach is used to consider the generalized synchronization for fractional complex networks. This is another difference between the fractional complex network synchronization in this paper and those of previous papers.The rest of this paper is organized as follows. Some necessary preliminaries are given in Section2. Generalized synchronization is discussed in Section 3. Simulation results are given in Section 4. Finally, the paper is concluded in Section 5.


## 2. Properties of the Tempered Fractional Derivative

Some definitions and properties are introduced in this section.Definition 1 (see [9]).
Forn−1≤α<n,n∈ℕ,andλ≥0, the tempered fractional Caputo derivative is defined as(2)DCt0,tα,λxt=e−λtΓn−α∫t0t1t−sα−n+1dneλsxsdsnds,where Γ⋅ denotes the Gamma function defined by(3)Γz=∫0∞e−ttz−1dt.Remark 1.
If the tempered parameterλ=0, the tempered fractional Caputo derivative (2) reduces to the corresponding fractional Caputo derivative.
Consider the tempered Caputo fractional nonautonomous system(4)DC0,tα,λxt=ft,x,with initial condition x0, where α∈0,1,λ≥0, f:0,+∞ is piecewise continuous in t and locally Lipschitz in x, and Ω⊂ℝn is a domain that contains the origin x=0.Definition 2.
The constantx0 is an equilibrium point of the tempered Caputo fractional dynamic system (4) if and only if(5)DC0,tα,λx0=ft,x0.
For convenience, assume the equilibrium point in all definitions and theorems isx0=0. Because any equilibrium point can be shifted to the origin via a change of variables, there is no loss of generality in doing so. Suppose the equilibrium point for (5) is x¯≠0, and consider the change of variable y=x−x¯. The tempered Caputo fractional derivative of y is given by(6)DC0,tα,λy=DC0,tα,λx−x¯=ft,x−x¯e−λtDC0,tα,λeλt=ft,y+x¯−x¯e−λtDC0,tα,λeλt=g¯t,y,where g¯t,0=0, and in the new variable y, the system has equilibrium at the origin.Definition 3.
The solution of (4) is said to be tempered Mittag-Leffler stable if(7)xt≤mx0e−λtEα−ktαb,where t≥0,λ≥0,α∈0,1,b>0,k≥0,m0=0,mx≥0, and mx is locally Lipschitz on x∈B⊂ℝn with Lipschitz constant m0.Remark 2.
Ifλ=0, tempered Mittag-Leffler stability reduces to Mittag-Leffler stability.Remark 3.
Mittag-Leffler stability and tempered Mittag-Leffler stability imply asymptotic stability, that is,limt⟶∞xt=0.Definition 4 (see [30]).
The Mittag-Leffler function is defined as(8)Eαz=∑k=1∞zkΓkα+1,where α>0. The Mittag-Leffler function with two parameters is defined as(9)Eα,βz=∑k=1∞zkΓkα+β,where α>0 and β>0.
Forβ=1, we have Eα,1z=Eαz and E1,1z=ez. The Laplace transform of the Mittag-Leffler function in two parameters is(10)ℒtβ−1Eα,β−λtα=sα−βsα+λ,ℛs>λ1/α,where t and s are variables in the time domain and Laplace domain, respectively. ℛs is the real part of s, λ∈ℝ, and ℒ⋅ is the Laplace transform.
To obtain the main results, several lemmas are given below.Lemma 1 (see [31]).
Assume thatQ=qijN×N is symmetric. Let M∗=diagm1∗,m2∗,…,ml∗,︸l 0,…,0︸N−l, 1≤l≤N, and mi∗>0 (i=1,2,…,l), Q−M∗=E−M˜∗SSTQl, M˜∗=diagm1∗,…,ml∗, and m∗=min1≤i≤lmi∗. Ql is a block matrix of Q obtained by removing its first l1≤l≤N row-column pairs, and E and S are matrices with appropriate dimensions. When m∗>λmaxE−SQl−1ST, Q−M∗<0 is equivalent to Ql<0.Lemma 2 (see [32]).
The Laplace transform of tempered fractional Caputo derivative (2) is given as(11)ℒDC0,tα,λxt=s+λαXs−∑k=0n−1s+λα−k−1dkdtkeλtxtt=0,where Xs=ℒxt denotes the Laplace transform of xt.
We can obtain the following theorems for the tempered fractional derivative.Theorem 1.
Assumext∈ℝ is a continuous and differentiable function; then, for any time instant t≥0,(12)DC0,tα,λx2t≤2xtDC0,tα,λxt,where 0<α<1 and λ≥0.Proof.
Letys=eλsxt−xs2, s∈0,t. According to Definition 1, for any t≥0, we have(13)DC0,tα,λx2t=e−λtΓ1−α∫0t1t−sαλeλsx2s+2eλsxsx˙sds=e−λtΓ1−α∫0teλst−sαλx2s+2xsx˙s−2λxtxs−2xtx˙s+2λxtxs+2xtx˙sds=e−λtΓ1−α∫0teλst−sαλx2s+2xsx˙s−2λxtxs−2xtx˙sds+2xtDC0,tα,λxt≤e−λtΓ1−α∫0teλst−sαλxs−2λxtxs+λx2t+2xsx˙s−2xtx˙sds+2xtDC0,tα,λxt=e−λtΓ1−α∫0teλst−sαλxt−xs2+2xsx˙s−2xtx˙sds+2xtDC0,tα,λxt=e−λtΓ1−α∫0t1t−sαy˙sds+2xtDC0,tα,λxt=e−λtΓ1−αyst−sα0t−α∫0tyst−sα+1ds+2xtDC0,tα,λxt≤e−λtΓ1−αlims⟶tyst−sα+2xtDC0,tα,λxt.
Because the functionys is differentiable, by L’Hôpital’s rule, we obtain(14)e−λtΓ1−αlims⟶tyst−sα=−e−λtΓ1−αlims⟶tλeλsxt−xs2αt−sα−1+2eλsx˙sxs−xtαt−sα−1=−e−λtαΓ1−αlims⟶teλsλxt−xs2+2x˙sxs−xtt−s1−α=0.
Combining (13) and (14), we obtain (15)DC0,tα,λx2t≤2xtDC0,tα,λxt.
The proof is completed.Theorem 2.
Letxt∈ℝn be a differentiable vector function. Then, for any time instant t≥0, the following inequality holds:(16)DC0,tα,λxTtPxt≤2xTtPDC0,tα,λxt,where 0<α<1,λ≥0, and P is a symmetric and positive definite matrix.Proof.
SinceP is a symmetric and positive definite matrix, there exists an orthogonal matrix B∈ℝn×n and a diagonal matrix Λ∈ℝn×n, such that(17)P=BΛBT,where Λ=diagλ11,λ22,…,λnn, λii>0i=1,2,…,n. Then,(18)xTtPxt=xTtBΛBTxt=BTxtTΛBTxt.
Define an auxiliary variableyt=BTxt. One can then write(19)xTtPxt=yTtΛyt=∑i=1nλiiyi2t.
Using Theorem 1, one has(20)DC0,tα,λxTtPxt=DC0,tα,λ∑i=1nλiiyi2t=∑i=1nDC0,tα,λλiiyi2t≤2∑i=1nλiiyitDC0,tα,λyit=2yTtΛDC0,tα,λyt=2BTxtTΛDC0,tα,λBTxt=2xTtPDC0,tα,λxt,which completes the proof.Theorem 3.
Letxt∈ℝn be a differentiable vector function. If a continuous function V:0,+∞ satisfies(21)DC0,tα,λVt,xt≤−θVt,xt,then(22)Vt,xt≤V0,x0e−λtEα−θtα,where 0<α<1,λ≥0, and θ is a positive constant.Proof.
From inequality (21), there exists a nonnegative function Mt satisfying(23)DC0,tα,λVt,xt+θVt,xt+Mt=0.
Taking the Laplace transform (11) of (23), we obtain(24)s+λαVs−s+λα−1V0,x0+θVs+Ms=0,where Vs=ℒVt,xtandMs=ℒMt. Therefore,(25)Vs=s+λα−1V0,x0−Mss+λα+θ.
Taking the inverse Laplace transform of (25), we have(26)Vt,xt=V0,x0e−λtEα−θtα−Mt∗tα−1e−λtEα,α−θtα,where ∗ means convolution. Because both Mt and tα−1e−λtEα,α−θtα are nonnegative functions, we can obtain(27)Vt,xt≤V0,x0e−λtEα−θtα.
This completes the proof.
Actually, an appropriate functionVt,xt that satisfies the inequality (21) is difficult to find. Therefore, we present a second Lyapunov method to weaken the conditions in Theorem 3.Theorem 4.
Let domainD⊂ℝn and system (4) all contain equilibrium point x=0, and assume Vt,xt:0,+∞ satisfies(28)α1xta≤Vt,xt≤α2xtab,(29)DC0,tα,λVt+,xt+≤−α3xtabholding almost everywhere,and that Vt,xt is locally Lipschitz with respect to x, Vt,xt is piecewise continuous, limτ⟶t+V˙τ,xτ exists, and Vt+,xt+≜limτ⟶t+Vτ,xτ, where t≥0,x∈D,α∈0,1,λ≥0,α1,α2,α3,a, and b are given positive constants. Then, x=0 is tempered Mittag-Leffler stable. If the assumptions hold globally on ℝn, then x=0 is globally tempered Mittag-Leffler stable.Proof.
From equations (28) and (29), we obtain(30)DC0,tα,λVt+,xt+≤−α3α2Vt,xtholding almost everywhere.
There exists a nonnegative functionMt satisfying(31)DC0,tα,λVt+,xt++Mt=−α3α2Vt,xtDC0,tα,λVt+,xt+≤−α3α2Vt,xtholding almost everywhere.
Taking the Laplace transform of (31), we have(32)s+λαV+s−s+λα−1V0++Ms=−α3α2Vs,where V0+=limτ⟶0+Vτ,xτ≥0 and V+s=ℒVt+,xt+. Due to continuity of function Vt,xt and (32), we obtain Vt+,xt+=Vt,xt,V+s=Vs and(33)Vs=V0s+λα−1−Mss+λα+α3/α2.
Applying the inverse Laplace transform, the unique solution of (33) is(34)Vt=V0e−λtEα−α3α2tα−Mt∗e−λttα−1Eα,α−α3α2tα.
Becausetα−1,e−λt, and Eα,α−α3/α2tα are nonnegative functions, we obtain(35)Vt≤V0e−λtEα−α3α2tα.
Substituting (35) into (28) gives(36)xt≤V0α1e−λtEα−α3α2tα1/a.
Letm=V0/α1≥0. It follows from (26) that(37)xt≤me−λtEα−α3α2tα1/a,where x0=0 if and only if V0/α1=0.
SinceVt,x about x satisfies the local Lipschitz condition, we can obtain V0,x0/α1 near x0 which satisfies the local Lipschitz condition. This result also means system (4) is tempered Mittag-Leffler stable, which completes the proof.Remark 4.
In Theorems 1–4, ifλ=0, one has some properties about fractional derivative and fractional system, which have been discussed [33–36]. So, the results are more general.Remark 5.
It should be noted that we pay special attention to the tempered Mittag-Leffler stability for the following reasons. On one hand, the tempered Mittag-Leffler stability is essential in the synchronization analysis of tempered fractional complex networks. On the other hand, the tempered Mittag-Leffler stability is an important character when evaluating the tempered fractional system.


## 3. Generalized Synchronization Criteria for Two-Layer Tempered Fractional Networks via Pinning Control

In this section, a pinning control method for achieving the generalized synchronization between two-layer tempered fractional complex networks is presented.Consider the following tempered fractional complex network composed ofN unidirectionally coupled nodes that is described by(38)DC0,tα1,λ1xit=Hxit+fxit+ε1∑j=1NaijPxjt,i=1,2,…,N,where 0<α1<1,λ1≥0, and xit=xi1t,xi2t,…,xintT∈ℝn is the state vector of the i-th node. Hxit and fxit=f1xit,f2xit,…,fnxitT:ℝn⟶ℝn are the linear and nonlinear parts of the isolated i-th node, respectively. ε1 is the coupling strength. P∈ℝn×n is the positive definite diagonal inner-coupling matrix, and A=aijN×N is the symmetric coupling configuration matrix, in which aij is defined as follows: if there is a connection from node j to node ii≠j, aij=1; otherwise, aij=0. The diagonal elements of matrix A are defined as aii=−∑j=1,j≠iNaij,i=1,2,…,N.Equation (38) is regarded as the drive network, and the response network with N coupling nodes is chosen as follows:(39)DC0,tα2,λ2yit=Myit+gyit+ε2∑j=1NbijQyjt,i=1,2,…,N,where yit=yi1t,yi2t,…,yintT∈ℝn is the state vector. Myit and gyit are the linear and nonlinear parts of the isolated i-th node, respectively. ε2 is the coupling strength. Q and B=bijN×N have the same meanings as those of P and A in network (38), respectively.Without loss of generality, the firstl nodes are selected as the pinned nodes in the complex network (39). Therefore, the pinned network with pinning controllers uixi,yi,wherei=1,2,…,l, is described as follows:(40)DC0,tα2,λ2yit=Myit+gyit+ε2∑j=1NbijQyjt+uixi,yi,i=1,2,…,l,DC0,tα2,λ2yit=Myit+gyit+ε2∑j=1NbijQyjt,i=l+1,l+2,…,N.Output signals from the drive network (39) are taken as input for the response network (40), so the latter will reach harmonious coexistence with the former.To realize the generalized synchronization between the drive layer (39) and the response layer (40), an auxiliary network layer is selected as the following form:(41)DC0,tα2,λ2zit=Mzit+gzit+ε2∑j=1NbijQzjt+uixi,zi,i=1,2,…,l,DC0,tα2,λ2zit=Mzit+gzit+ε2∑j=1NbijQzjt,i=l+1,l+2,…,N,where zi is the state vector of the i-th node and uixi,zii=1,2,…,l are controllers of the same form as that of uixi,yi.According to the auxiliary system approach, the drive layer (38) and response layer (40) are said to achieve generalized synchronization if the response network (40) and the auxiliary network (41) reach complete outer synchronization. If limt⟶∞yit−zit=0 for any initial conditions yi0≠zi0i=1,2,…,N, the generalized synchronization between the two-layer tempered fractional networks (38) and (40) is realized.Hereafter, suppose that the nonlinear functiongxt satisfies the Lipschitz condition. Then, there exists a nonnegative constant L such that(42)gxt−gyt≤Lxt−yt,for any time-varying vectors xt,yt∈ℝn with the norm x=xTx.Define the synchronization error between (40) and (41) as eit=zit−yiti=1,2,…,N. Then, the error dynamical network is described by(43)DC0,tα2,λ2eit=Meit+gzi−gyi+ε2∑j=1NbijQejt+uixi,zi−uixi,yi,i=1,2,…,l,DC0,tα2,λ2eit=Meit+gzi−gyi+ε2∑j=1NbijQejt,i=l+1,l+2,…,N.Theorem 5.
The pinning controllers are chosen as(44)uixi,zi=−kizi−xi,i=1,2,…,l,(45)uixi,yi=−kiyi−xi,i=1,2,…,l.
Letq=Q, K=diagk1,k2,…,kl,0,…,0︸N−l, IN be an N×N identity matrix, B˜ be a modified matrix of matrix B by replacing the diagonal elements bii with ρminbii/q, and k=min1≤i≤lki. If Lipschitz condition (42) and the condition(46)L+λMIN+ε2qB˜−K<0,are satisfied, then the drive network (38) and the response network (40) can reach generalized synchronization with controllers (44) and (45).Specifically, for a suitablek, if there exists a natural number 1≤l<N such that the largest eigenvalue of B˜ satisfies L+λM+ε2qλmaxB˜l+1<0, then the tempered fractional response network (40) asymptotically synchronizes to the drive network (38) with the pinning scheme (44) and (45).Proof.
According to (44) and (45), the error network (43) can be rewritten as(47)DC0,tα2,λ2eit=Meit+gzi−gyi+ε2∑j=1NbijQejt−kizi−yi,i=1,2,…,l,DC0,tα2,λ2eit=Meit+gzi−gyi+ε2∑j=1NbijQejt,i=l+1,l+2,…,N.
Consider the Lyapunov function candidate(48)Vt=12∑i=1NeiTteit.
According to Theorem 2, its tempered fractional calculus along the trajectory of (47) is(49)DC0,tα2,λ2Vt≤∑i=1NeiTtDC0,tα2,λ2eit≤∑i=1NeiTtMeit+∑i=1NeiTtgzi−gyi+ε2∑i=1N∑j=1NbijeiTtQejt−∑i=1lkieiTteit≤∑i=1NeiTtM+MT2eit+L∑i=1NeiTteit+ε2∑i=1N∑j=1,j≠iNbijeiTtQejt+ε2∑i=1NbiieiTtQ+QT2eit−∑i=1lkieiTteit≤L+λM∑i=1NeiTteit−∑i=1lkieiTteit+ε2q∑i=1N∑j=1,j≠iNbijeit⋅ejt+ε2q∑i=1N∑j=1,j≠iNbijeit⋅ejt=eTL+λMIN+ε2qB˜−Ke,where λM is the maximum eigenvalue of matrix M+MT/2, ρmin is the minimum eigenvalue of matrix Q+QT/2, and e=e1,e2,…,eNT.
If the conditionH=L+λMIN+ε2qB˜−K<0 holds, we have η=−λmaxH>0 and(50)DC0,tα2,λ2Vt≤−ηeTe=−ηVt.
According to Theorem 3, we have(51)Vt,xt≤V0,x0e−λ2tEα2−ηtα2,that is, eit⟶0i=1,2,…,N as t⟶∞, which means that the generalized synchronization between the drive network (38) and the response network (40) is realized.
Specifically, letk=min1≤i≤lki and B˜l+1 be the reduced matrix of B˜ obtained by removing the first l row-column pairs. According to Lemma 1, H<0 is equivalent to L+λMIN−l+cqB˜l+1<0 for suitable k. Since Hl+1=L+λMIN−l+ε2qB˜l+1 is a real symmetric matrix, the largest eigenvalue λmaxHl+1 of Hl+1 is real and λmaxHl+1=L+λM+ε2qλmaxB˜l+1. Thus, if there exists a positive constant 1≤l≤N such that L+λM+ε2qλmaxB˜l+1<0, one has L+λMIN−l+ε2qB˜l+1<0. Therefore, H<0, and we have obtained the desired results.Remark 6.
In Theorem 5, ifλ=0, the results hold too, that is, the generalized synchronization in two-layer fractional complex networks via pinning control can be realized.


## 4. Simulation Results

In this section, a predictor–corrector algorithm and an example are provided to illustrate the effectiveness of the proposed method.


### 4.1. Predictor–Corrector Algorithm

The numerical calculation for tempered fractional systems is not as simple as that of an ordinary differential equation. Here, we use the generalized Adams–Bashforth–Moulton method [37, 38]. A brief introduction of this algorithm is given.Consider the differential equation(52)DC0,tα,λxt=ft,xt,0<t<T,with initial conditions(53)dkdtkeλtxtt=0=ck,k=0,1,…,α−1,where n≤α<n,n∈ℕ+,λ>0, and ck are arbitrary real numbers.The tempered fractional equations (52) and (53) are equivalent to the following Volterra integral equation [32]:(54)xt=x0t+1Γα∫0te−λt−τt−τα−1fτ,xτdτ,where x0t=∑k=0α−1cke−λttk/k!. For uniform nodes tn+1=n+1hn=0,1,…,N, with h=T/N being the step size of the numerical computation. Then, the corrector formula for (54) to compute xn+1≈xtn+1 is(55)xn+1=x0t1+hαΓα+2ft1,x1P+αe−λhf0,x0,ifn=0,x0tn+1+hαΓα+2∑i=0nai,n+1fti,xi+ftn+1,xn+1P,ifn≥1,where the predicted value xn+1P is determined by(57)xn+1P=x0t1+hαΓα+1e−λhf0,x0,ifn=0,x0tn+1+hαΓα+2∑i=0n−1ai,n+1fti,xi+e−λh2α+1−1⋅ftn,xn,ifn≥1,ai,n+1=e−λn+1hnα+1−n+1αn−α,ifi=0,e−λn+1−ihn−iα+1−2n+1−iα+1+n+2−iα+1,if1≤i≤n.The estimation error of this approximation is(58)max0≤n≤Nxtn−xn=Oh2,ifα≥0.5,Oh1+2α,if0<α<0.5.The numerical solution of a tempered fractional system can be determined by applying the above method.


### 4.2. Numerical Simulations

In this section, an example is provided to illustrate the effectiveness of the pinning strategies for the generalized synchronization in two-layer tempered fractional complex networks.Now, we obtain a new tempered fractional chaotic system, that is, a generalization of the corresponding integer order system [39]. This chaotic system is adopted as the node dynamics in the layer of drive network (38), which is described as(59)DC0,tα1,λ1xi1=−xi2xi3+a1xi1,DC0,tα1,λ1xi2=xi1xi3−b1xi2,DC0,tα1,λ1xi3=13xi1xi2−c1xi3,where i=1,2,…,10,α1=0.99, a1=5,b1=10, and c1=3.8. One can obtain the attractor of tempered fractional chaotic systems with different λ, as shown in Figure 2.Figure 2
The attractor of the new tempered fractional system with differentλ1. (a) λ1=0. (b) λ1=1. (c) λ1=2. (d) λ1=3.
(a)
(b)
(c)
(d)In the layer of the response network (39), tempered fractional Chen system [33] is chosen as the node dynamics:(60)DC0,tα2,λ2yi1=a2yi2−yi1,DC0,tα2,λ2yi2=c2−a2yi1−yi1yi3+c2yi2,DC0,tα2,λ2yi3=yi1yi2−b2yi3,where i=1,2,…,10,α2=0.995, a2=35,b2=3, and c2=28. One can obtain the attractor of the fractional chaotic Chen system, as shown in Figure 3. From Figures 2 and 3, we can see that the tempered fractional system has more abundant dynamical behavior than the corresponding fractional one. Additionally, in the layer of the response network (41), the dynamics of each node are in the same form as those in the response layer.Figure 3
The attractor of the tempered fractional Chen system with differentλ2. (a) λ2=0. (b) λ2=2. (c) λ2=4. (d) λ2=6.
(a)
(b)
(c)
(d)For convenience, let inner-coupling matricesP=Q=I. The coupling configuration matrices A and B are given as(61)A=B=−21010000000−21010000010−30011000001−10000000101−30001000101−20000000001−31010101000−31010100010−41001100000−2.Fractional Chen systems are bounded. Actually, yi1≤60,yi2≤40,yi3≤40,zi1≤60,zi2≤40,zi3≤40,i=1,2,…,10, and fxi−fyi≤−yi1yi3+zi1zi32+yi1yi2−zi1zi22≤84.85ei, that is, L=84.85. The out-degrees of nodes 2, 3, and 4 are larger than their in-degrees, so we select them as the pinned nodes [31]. After rearranging the network nodes, the new order is 4, 3, 2, 1, 6, 10, 5, 7, 8, and 9.Letki=120,ε1=0.45, and ε2=5; we can obtain λmaxL+λMIN+ε2qB˜−K=−26.2501<0. From Theorem 5, the drive network (38) and the response network (40) can achieve generalized synchronization. The simulation results are shown in Figure 4, which illustrates the time waveforms of errors ei1,ei2,andei3i=1,2,…,10.Figure 4
Time evolution of the error statesei1,ei2, and ei3 with λ1=1andλ2=2. (a) ei1i=1,2,…,10. (b) ei2i=1,2,…,10. (c) ei3i=1,2,…,10.
(a)
(b)
(c)The temper parameterλ has a direct effect on the chaotic behavior of nonlinear dynamical systems. For convenience, we define the average error norm as(62)ea=1N∑i=1Nei=1N∑i=1Nyi−zi.We let the parameters depict as above, exceptλ2. From Figure 5, the synchronization behaviors of fractional complex networks are realized with α2=0.995 and different tempered parameters λ2. Furthermore, from Figure 6, fractional complex networks can reach synchronization for λ2=2 and different fractional orders α2. The figures show that fractional complex networks (38) and (40) are synchronized, which demonstrate the effectiveness and robustness of the proposed method.Figure 5
Time evolution of the average error normsea with different tempered parameters λ2.Figure 6
Time evolution of the average error normsea with different fractional orders α2.


## 4.1. Predictor–Corrector Algorithm

The numerical calculation for tempered fractional systems is not as simple as that of an ordinary differential equation. Here, we use the generalized Adams–Bashforth–Moulton method [37, 38]. A brief introduction of this algorithm is given.Consider the differential equation(52)DC0,tα,λxt=ft,xt,0<t<T,with initial conditions(53)dkdtkeλtxtt=0=ck,k=0,1,…,α−1,where n≤α<n,n∈ℕ+,λ>0, and ck are arbitrary real numbers.The tempered fractional equations (52) and (53) are equivalent to the following Volterra integral equation [32]:(54)xt=x0t+1Γα∫0te−λt−τt−τα−1fτ,xτdτ,where x0t=∑k=0α−1cke−λttk/k!. For uniform nodes tn+1=n+1hn=0,1,…,N, with h=T/N being the step size of the numerical computation. Then, the corrector formula for (54) to compute xn+1≈xtn+1 is(55)xn+1=x0t1+hαΓα+2ft1,x1P+αe−λhf0,x0,ifn=0,x0tn+1+hαΓα+2∑i=0nai,n+1fti,xi+ftn+1,xn+1P,ifn≥1,where the predicted value xn+1P is determined by(57)xn+1P=x0t1+hαΓα+1e−λhf0,x0,ifn=0,x0tn+1+hαΓα+2∑i=0n−1ai,n+1fti,xi+e−λh2α+1−1⋅ftn,xn,ifn≥1,ai,n+1=e−λn+1hnα+1−n+1αn−α,ifi=0,e−λn+1−ihn−iα+1−2n+1−iα+1+n+2−iα+1,if1≤i≤n.The estimation error of this approximation is(58)max0≤n≤Nxtn−xn=Oh2,ifα≥0.5,Oh1+2α,if0<α<0.5.The numerical solution of a tempered fractional system can be determined by applying the above method.


## 4.2. Numerical Simulations

In this section, an example is provided to illustrate the effectiveness of the pinning strategies for the generalized synchronization in two-layer tempered fractional complex networks.Now, we obtain a new tempered fractional chaotic system, that is, a generalization of the corresponding integer order system [39]. This chaotic system is adopted as the node dynamics in the layer of drive network (38), which is described as(59)DC0,tα1,λ1xi1=−xi2xi3+a1xi1,DC0,tα1,λ1xi2=xi1xi3−b1xi2,DC0,tα1,λ1xi3=13xi1xi2−c1xi3,where i=1,2,…,10,α1=0.99, a1=5,b1=10, and c1=3.8. One can obtain the attractor of tempered fractional chaotic systems with different λ, as shown in Figure 2.Figure 2
The attractor of the new tempered fractional system with differentλ1. (a) λ1=0. (b) λ1=1. (c) λ1=2. (d) λ1=3.
(a)
(b)
(c)
(d)In the layer of the response network (39), tempered fractional Chen system [33] is chosen as the node dynamics:(60)DC0,tα2,λ2yi1=a2yi2−yi1,DC0,tα2,λ2yi2=c2−a2yi1−yi1yi3+c2yi2,DC0,tα2,λ2yi3=yi1yi2−b2yi3,where i=1,2,…,10,α2=0.995, a2=35,b2=3, and c2=28. One can obtain the attractor of the fractional chaotic Chen system, as shown in Figure 3. From Figures 2 and 3, we can see that the tempered fractional system has more abundant dynamical behavior than the corresponding fractional one. Additionally, in the layer of the response network (41), the dynamics of each node are in the same form as those in the response layer.Figure 3
The attractor of the tempered fractional Chen system with differentλ2. (a) λ2=0. (b) λ2=2. (c) λ2=4. (d) λ2=6.
(a)
(b)
(c)
(d)For convenience, let inner-coupling matricesP=Q=I. The coupling configuration matrices A and B are given as(61)A=B=−21010000000−21010000010−30011000001−10000000101−30001000101−20000000001−31010101000−31010100010−41001100000−2.Fractional Chen systems are bounded. Actually, yi1≤60,yi2≤40,yi3≤40,zi1≤60,zi2≤40,zi3≤40,i=1,2,…,10, and fxi−fyi≤−yi1yi3+zi1zi32+yi1yi2−zi1zi22≤84.85ei, that is, L=84.85. The out-degrees of nodes 2, 3, and 4 are larger than their in-degrees, so we select them as the pinned nodes [31]. After rearranging the network nodes, the new order is 4, 3, 2, 1, 6, 10, 5, 7, 8, and 9.Letki=120,ε1=0.45, and ε2=5; we can obtain λmaxL+λMIN+ε2qB˜−K=−26.2501<0. From Theorem 5, the drive network (38) and the response network (40) can achieve generalized synchronization. The simulation results are shown in Figure 4, which illustrates the time waveforms of errors ei1,ei2,andei3i=1,2,…,10.Figure 4
Time evolution of the error statesei1,ei2, and ei3 with λ1=1andλ2=2. (a) ei1i=1,2,…,10. (b) ei2i=1,2,…,10. (c) ei3i=1,2,…,10.
(a)
(b)
(c)The temper parameterλ has a direct effect on the chaotic behavior of nonlinear dynamical systems. For convenience, we define the average error norm as(62)ea=1N∑i=1Nei=1N∑i=1Nyi−zi.We let the parameters depict as above, exceptλ2. From Figure 5, the synchronization behaviors of fractional complex networks are realized with α2=0.995 and different tempered parameters λ2. Furthermore, from Figure 6, fractional complex networks can reach synchronization for λ2=2 and different fractional orders α2. The figures show that fractional complex networks (38) and (40) are synchronized, which demonstrate the effectiveness and robustness of the proposed method.Figure 5
Time evolution of the average error normsea with different tempered parameters λ2.Figure 6
Time evolution of the average error normsea with different fractional orders α2.


## 5. Conclusion

In this paper, the tempered fractional derivative is introduced to chaotic systems and complex networks. Synchronization of tempered fractional complex networks may be more useful in secure communication and control processing due to the addition of the tempered parameter. The pinning control scheme and auxiliary system approach are used to verify the existence of generalized synchronization in tempered fractional complex networks. Some properties of the tempered fractional derivative and tempered fractional system are discussed. Generalized synchronization criteria are established based on the proposed theory. The numerical results demonstrate the effectiveness of the proposed method. Furthermore, generalized synchronization can be achieved with differentε2. Future work will study synchronization-based parameter identification [29] of tempered fractional delayed complex networks.



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