# Fixed-Time Synchronization for Dynamical Complex Networks with Nonidentical Discontinuous Nodes

Authors: Xiaoliang Qian; Qian Liu; Qingbo Li; Qi Yang; Yuanyuan Wu; Wei Wang
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

This article investigates the fixed-time synchronization issue for linearly coupled complex networks with discontinuous nonidentical nodes by employing state-feedback discontinuous controllers. Based on the fixed-time stability theorem and linear matrix inequality techniques, novel conditions are proposed for concerned complex networks, under which the fixed-time synchronization can be realized onto any target node by using a set of newly designed state-feedback discontinuous controllers. To some extent, this article extends and improves some existing results on the synchronization of complex networks. In the final numerical example section, the Chua circuit network is introduced to indicate the effectiveness of our method by showing its fixed-timely synchronization results with the proposed control scheme.


---

## Body

## 1. Introduction

As we know, in the last few decades, complex networks have been widely presented in our real world, for example, electrical power grids, metabolic pathways, neural networks, food webs, and World Wide Web [1–5]. Synchronization is a well-known crucial collective behavior for complex networks, so the synchronization of complex networks has received more attention due to many crucial applications [6–8] in information processing, secure communication, and biological systems [9–12]. Up to now, there are many research studies on complex network synchronization, most of which focus on asymptotic synchronization, mainly on asymptotic synchronization behavior [13–15] and exponential synchronization results [16], but the two kinds of synchronization belong to the infinite-time category [17–19].Since it has been found that finite-time control ways will further enhance the rate of convergence greatly and synchronization will be performed in a settling time by designing appropriate finite-time synchronization controllers, the finite-time synchronization research [20–25] in complex networks has been carried out one after another [26–30]. In [20], the issue of the finite-time synchronization is studied between complex networks with nondelay and delay coupling by using pulse control and periodic intermittent control. By use of aperiodically intermittent control, Liu et al. [21] considered the finite-time synchronization problem in dynamic networks with time delay. The global random finite-time synchronization issue is investigated in [22] for discontinuous semi-Markov switched neural networks with time delay and noise interference. The finite-time synchronization analysis of linear coupled complex networks is discussed [23] with discontinuous nonidentical nodes.The convergence rate of classical finite-time synchronization is relatively fast in contrast to asymptotic synchronization and exponential synchronization. However, it has an obvious disadvantage that the synchronization convergence rate of complex networks depends on the initial states of all nodes. Unfortunately, it is very difficult or even impossible for some chaotic systems to know their state previously. In these results, the finite-time control methods may be ineffective. Taking advantage of the benefits of finite-time control, a special finite-time synchronization is proposed in [20]. As for the novel fixed-time synchronization, the settling time has no relation with the initial conditions of the network system and only depends on the control parameters of the system controller, see [21, 22]. Thus, synchronization can be accomplished by using the fixed-time controller within a specified time. This remarkable characteristic makes fixed-time synchronization control more desirable than other synchronization controls and improves its practical application range. Therefore, fixed-time synchronization control of complex networks has received more attention [23–25, 31–34].Moreover, if the dynamics of the nodes are different, then the synchronization issue will be more complex and challenging than the same node condition. By using the free matrix, the equilibrium solution synchronization is concerned on all alone nodes together with the average state trajectory synchronization of different nodes in [35]. The intermittent controller is employed to fix the complex network with different nodes in [36]. In [37], the cluster synchronization problem is investigated for complex dynamic networks with time-delay coupling and nonidentical nodes by the pinning control method. Furthermore, the finite-time synchronization issue is considered for coupled complex networks with discontinuous nonidentical nodes in [23].Recently, the complex networks with perturbations have attracted more attention for their wide applications [38–43]. In [40], the global exponential synchronization issue is studied for linear coupled neural networks with impulsive disturbance and time-varying delay. The clustering synchronization scheme is deeply concerned with regard to uncertain delayed complex networks in [41]. The adaptive pinning control design is proposed in [42] for the clustering synchronization problem of coupled complex networks with uncertain disturbances.Until now, there are several research results on the finite-time synchronization of complex networks with different nodes or uncertain disturbances, mostly about the asymptotic or exponential synchronization. However, it has not been fully investigated for the fixed-time synchronization analysis of heterogeneous networks with uncertain disturbances, and the relevant research results are rarely covered. In a word, it is indispensable and significant to consider the fixed-time synchronization problem of complex networks with different nodes and uncertain disturbances, which has profound theoretical and practical significance. From the above analysis, we face two difficulties: (i) what conditions are applicable and easy to verify for general complex networks with different nodes and uncertain disturbances? (ii) How to design the controller to overcome heterogeneity and uncertain disturbance of network nodes? This paper tries to conquer these two difficulties and realize the fixed-time synchronization of a certain kind of complex linear coupled networks with different nodes and uncertain disturbances, and then the theoretical results of network synchronization can be further enriched.Applying the discontinuous control scheme, the fixed-time synchronization problem is analysed for complex networks with uncertain disturbances and nonidentical nodes. Our main contributions here can be concluded as follows: (1) for a class of heterogeneous networks with uncertain disturbances, a novel state-feedback discontinuous controller is designed to get over the influence on the fixed-time synchronization from heterogeneous nodes and uncertain disturbances simultaneously; (2) several criteria are proposed to deduce the fixed-time synchronization for the considered networks. Unlike most existing results, the obtained fixed-time synchronization conditions are expressed by linear matrix inequality, which is easy to be verified; (3) as special cases, the fixed-time synchronization of complex networks without uncertain disturbances is also considered by employing some existing controllers, respectively, and the corresponding results are given in some corollaries.The rest of the paper is arranged as follows. A network model is established with uncertain disturbances and nonidentical nodes, and then the problem of the fixed-time synchronization is described; meanwhile, some necessary definitions and assumptions are given in Section2. The fixed-time synchronization conditions are achieved in Section 3. Several numerical examples are introduced in Section 4 to indicate the effectiveness of the proposed results. Section 5 summarizes the research conclusions of this paper and puts forward the future research directions.


## 2. Problem Formulation and Preliminaries

A kind of nonlinear system includingN nonidentical nodes with diffusion linear coupling is considered, in which each node can be regarded as an n-dimensional dynamic system, as shown in the following:(1)x˙it=Aixit+fit,xit+hit,xit+c∑j=1NGijΓxjt,where xit=xi1t,…,xintT∈ℝn denotes the state vector of the ith dynamical node; the dynamics of the ith uncoupled node is x˙it=Aixit+fit,xit+hit,xit in which Ai∈ℝn×n, and fit,xit=fi1t,xit,fi2t,xit,…,fint,xitT:ℝ+×ℝn⟶ℝn represents a smooth nonlinear vector showing the node self-dynamics; moreover, hit,xit=hi1t,xit,hi2t,xit,…,hint,xitT:ℝ+×ℝn⟶ℝn is a uncertain vector and represents the disturbance. The constant c>0 can be considered as the coupling strength of the concerned networks, and Γ=γijij∈ℝn×n is a matrix and denotes the inner coupling relation between the network nodes and indicates how the components of each pair of nodes are connected with each other, and γij≥0; G=GijN×N is a coupling configuration constant matrix, which describes the topological structure and can be exhibited as the diffusion structure, i.e., Gij≥0 and Gii=−∑j=1,j≠iNGij. In this paper, the driven dynamical node of (1) satisfies(2)x˙0t=A0x0t+f0t,x0t+h0t,x0t,where A0∈ℝn×n, f0t,x0t∈ℝn, and h0t,x0t∈ℝn.In fact, most of the well-known chaotic systems can be described by the above dynamical equation, such as Sprott circuit, Chua circuit, Rössler’s systems, and Chen system [42].Definition 1 (see [44]).
Complex network (1) is said to be synchronized onto (2) in finite time if there exist a designed feedback controller to system (1) and a constant t∗>0 such that(3)limt⟶t∗xit−x0t=0,xit−x0t≡0,i=1,2…,N,where t∗>0 is called the settling time and often depends on the initial state vector value X0=x1T0,…,xNT0T.Definition 2 (see [44]).
Complex network (1) is said to be synchronized onto (2) in fixed time if there exists a fixed settling time T∗>0 such that(4)limt⟶T∗xit−x0t=0,xit−x0t≡0,i=1,2…,N,t>T∗,where T∗>0 is called the settling time and is independent of the initial synchronization error X0=x1T0,…,xNT0T.
In this paper, the goal is to fixed-timely synchronize the state of network (1) onto the driven one (2) by designing feedback controllers.
Obviously, controlled complex network (1) can be rewritten as follows:(5)x˙it=Aixit+fit,xit+hit,xit+c∑j=1NGijΓxjt+uit.
Introduce the synchronization errorseit, which are defined as eit=xit−x0t,i=1,2,…,N, Fit=fit,xi−f0t,x0+Ai−A0xit, and Hit,eit=hit,xit−h0t,x0t. Subtracting (2) from (5), the error dynamical network model can be given by(6)e˙it=A0eit+Fit,eit+Hit,eit+c∑j=1NGijΓej+uit,i=1,…,N.
In order to obtain our main results, some necessary assumptions are listed as follows.Assumption 1.
There exist constantMi>0 and uniformly symmetric positive definite matrix Li, i=1,2,…,n, such that fit,x satisfies(7)y−xTfit,y−fit,x≤y−xTLiy−x+MiΣj=1nyj−xj,i=1,…,n,for all t≥0, x=x1,x2,…,xnT∈ℝn, and y=y1,y2,…,ynT∈ℝn.Assumption 2.
There exist constantM0>0 and uniformly symmetric positive definite matrix L0 such that f0t,x satisfies(8)y−xTf0t,y−f0t,x≤y−xTL0y−x+M0Σi=1nyj−xj,for all t≥0, x=x1,x2,…,xnT∈ℝn, and y=y1,y2,…,ynT∈ℝn.Assumption 3 (see [43]).
There exists a time-varying functionμt≥0 such that(9)fit,x−f0t,x≤μt,i,j=1,…,N.Assumption 4.
For anyi, i==1,2,…,n, the uncertain function vector hit,xit is assumed to be continuous at t,xit≥0 and bounded. Moreover, there is a known nonnegative number hmax such that(10)hit,xit≤hmax,i=0,1,…,N.Remark 1.
Assumptions1 and 2 are general and satisfied with most of the well-known chaotic systems, for instance, Chua circuit [43], Rössler’s systems, and discontinuous Chen system. In fact, the above systems meet the following conditions: there exist some positive constants kij>0,βj,i,j=1,2,…,n, satisfying(11)fit,y−fit,x≤Σj=1nkijyjt−xjt+βi,i=1,2,…,n,for any t≥0, x=x1,x2,…,xnT∈ℝn, and y=y1,y2,…,ynT∈ℝn. Using condition (11), we have(12)yt−xtTfit,y−fit,x≤Σi=1nΣj=1nkijyit−xityjt−xjt+Σj=1nβiyjt−xjt=Σi=1nΣj=1nkij1/2yit−xitkij1/2yjt−xjt+Σj=1nβiyjt−xjt≤Σi=1nΣj=1n12kij+kjiyit−xit2+Σj=1nβiyjt−xjt≤yt−xitLiyt−xt+MiΣj=1nyjt−xjt,where Li=diagl11i,l22i,…,lnni, ljji=Σj=1n1/2kij+kji, and Mi=βi,1≤i≤n. Clearly, if α=max1≤i≤nljji,j=1,2,…,n and M=max1≤i≤nβi, then Assumptions 1 and 2 involve conditions (H2) and (H3) in [25]. Moreover, the continuous chaotic system is also a special case by setting Mi=0 in Assumption 1 or M0=0 in Assumption 2, for instance, the continuous Rössler system, Chua’s circuit, Chen system, Lorenz system, and logistic differential system. Hence, Assumptions 1 and 2 are more general, and most popular chaotic systems are applicable. Assumptions 3 and 4 take advantage of conditions on the activation function, and it is seen that they are diffusely imposed in the literature [23, 35–37].
According to Definition2, it is clear that the fixed-time synchronization of dynamical network (5) onto (2) can be degenerated into the fixed-time stabilization of error dynamic system (6).


## 3. Fixed-Time Synchronization Analysis

In this part, the controllers are designed for the fixed-time synchronization problem of complex network (1), and concerned complex network (5) can realize the fixed-timely synchronization under the appropriate designed controllers. Firstly, we give the synchronization controller design of complex network (1), and then fixed-time synchronization criteria can be obtained based on error system (6). Several corollaries are also obtained for (5) and (2) with identical nodes. For concerned complex network (1), the control input uit∈Rn, i=1,…,N, is designed as follows:(13)uit=A0−Aix0−dieit−ηitsigneit−sign¯eitaeitp+beitq,where d1,…,dN are positive constants, ηit is a function to be determined, a,b are positive constants, signeit=signei1t,…,signeintT, eitp=ei1tp,…,eintpT, eitq=ei1tq,…,eintqT, sign¯eit=diagsignei1t,…,signeint, and the real numbers p,q follow 0<q<1,p>1.The following lemmas are necessary and given to derive the subsequent main results.Lemma 1 (see [2, 45, 46]).
Suppose that functionVt:ℝn⟶ℝ is C-regular and xt:0,+∞⟶ℝn is absolutely continuous on any compact interval 0,+∞. Denote υt=Vxt if there exists a continuous function γ:0,+∞⟶ℝ with γσ>0 for σ∈0,+∞ such that(14)υ˙t≤−γυ,for any t>0 that υt>0, and υt is differentiable at t and satisfies(15)∫0υ01γυ=t1<+∞.Then, we haveυt=0 for t≥t1. In particular, if γυ=Qυμ, where μ∈0,1 and Q>0, then the setting time is estimated by(16)t1=υ1−μQ1−μ.Lemma 2 (see [47]).
For matricesA,B,C,andD with appropriate dimensions and a scalar α, the following assertions hold:(1)
αA⊗B=A⊗αB,(2)
A+B⊗C=A⊗C+B⊗C,(3)
A⊗BC⊗D=AC⊗BD, and(4)
A⊗BT=AT⊗BT,where ⊗ is the Kronecker product.Lemma 3 (see [48]).
Suppose there exists a continuous, positive-definite functionV (t) satisfying(17)dVtdt≤IVt−kVαt,∀t≥t0,V1−αt0≥kI,where k>0,I>0,and0<α<1 are three constants. Then, the following inequality is true:(18)V1−αt≥V1−αt0−k1−αt−t0,t0<t<t1,Vt≡0,∀t>t1, and the settling time t1 is estimated by(19)t1=t0+ln1−I/kV1−αt0Iα−1.Lemma 4.
Suppose there exists a continuous radially unbounded functionVet:RnN⟶0,+∞ satisfying the following two conditions:(i)
Ifet≠0, then Vet>0 and Vet=0⇔et=0.(ii)
Any solutionet of system (6) satisfies(20)dVetdt≤−aVpet−bVqet,where a>0,b>0,p>1,and0<q<1 are all constants. Then, Vet satisfies Vet≡0,∀t>T∗, and the fixed settling time is estimated by(21)T∗=1ap−1+1b1−q.Lemma 5 (see [48]).
Suppose thatai≥0 (i=1,…,n), 0<p≤1, and 0<q<2; it follows that(22)∑i=1naip≤∑i=1naip,∑i=1naiq≥∑i=1nai2q/2,1n∑i=1naiqq≥1n∑i=1nai21/2.Theorem 1.
For concerned complex network (1) with the control input, if Assumptions 1–4 hold and(23)μt+2hmax+Mi−ηit<0,i=1,2,…,N,(24)ℒ+A+cGs⊗Γ−D⊗In<0,with D=diagd1,…,dN>0, Gs=G+GT/2, ℒ=diagL1s,L2s,…LNs, and A=diagA1s,A2s,…,ANs, then driven-response complex networks (1) and (2) can achieve fixed-time synchronization under controller (13), with the settling time(25)T∗=121+p/2aNn1−p/2p−1+121+q/2b1−q,where V0=1/2∑i=1NeiT0ei0 and ei0 is the initial value of eit=xit−x0t for i=1,…,N. Proof.For error dynamical system (6), a Lyapunov function is listed by(26)Vt=12∑i=1NeiTteit,and the derivative of the above Lyapunov function along the trajectory of system (6) can be computed as(27)V˙t=∑i=1NeiTte˙it=∑i=1NeiTtA0eit+Fit,eit+Hit,eit+c∑j=1NGijΓejt+uit=∑i=1NeiTt−dieit−ηitsigneit−sign¯eitaeitp+beitq+I1t+I2t+I3t,with I1t=∑i=1NeiTtA0eit+c∑i=1NeiTt∑j=1NGijΓejt, I2t=∑i=1NeiTtHit,eit, and I3t=∑i=1NeiTtFit,eit−Ai−A0x0t.According to Lemma2, we know the following equation is true:(28)I1t=eTtIN⊗A0et+ceTtG⊗Γet=eTtIN⊗A0set+ceTtG⊗Γ+G⊗ΓT2et=eTtIN⊗A0s+cGs⊗Γet.For the termI2t, using condition (10) in Assumption 4, we get the following inequality:(29)I2t=∑i=1NeiTthit,x1t−h0t,x0t≤∑i=1Neithit,x1t−h0t,x0t≤∑i=1Neit2hmax=∑i=1N∑j=1Neijt2hmax.Also, it can be seen that(30)I3t=∑i=1NeiTtfit,xit−f0t,x0t+Ai−A0xit−Ai−A0x0t=∑i=1NeiTtfit,xit−fit,x0t+fit,x0t−f0t,x0t+Ai−A0eit=I31t+I32t+etTA−IN⊗A0set,with I31t=∑i=1NeiTtfit,xit−fit,x0t, I32t=∑i=1NeiTtfit,x0t−f0t,x0t, and A=diagA1s,A2s,…,ANs.From (7) in Assumption 1, it is known that(31)I31t≤∑i=1NeiTtLieit+∑i=1NMi∑j=1Neijt=eTtℒet+∑i=1NMi∑j=1Neijt,where ℒ=diagL1s,L2s,…,LNs.By (9) in Assumption 3, one can get(32)I32t≤∑i=1Neitfit,x0t−f0t,x0t≤∑i=1N∑j=1Neijtμt.Submitting (28)–(32) into (27) yields that(33)V˙t≤eTtℒet+eTtIN⊗A0s+cGs⊗Γet+μt∑i=1N∑j=1Neijt+∑i=1N∑j=1Neijt2hmax+etTA−IN⊗A0set+∑i=1NMi∑j=1Neijt+∑i=1NeiTt−dieit−ηitsigneit−sign¯eitaeitp+beitq=W1t+W2t+W3t,where(34)W1t=eTtℒet+eTtIN⊗A0s+cGs⊗Γet+etTA−IN⊗A0set−∑i=1NeiTtdieit,W2t=∑i=1N∑j=1Neijtμt+∑i=1N∑j=1Neijt2hmax+∑i=1NMi∑j=1Neijt−∑i=1NeiTtηitsigneit,(35)W3t=−∑i=1NeiTtsign¯eitaeitp+beitq.By use of Lemma2 and (24), it gives that(36)W1t=eTtℒ+IN⊗A0s+cGs⊗Γ+Aeit−eTtIN⊗A0set−eTtD⊗Ineit=eTtℒ+A+cGs⊗Γ−D⊗Inet≤0,where D=diagd1,d2,…,dN.Obviously,eiTtsigneit=ei1t,…,eintsignei1t,…,signeintT=∑j=1neijt and eit2−eiTtsign¯eit=∑j=1neij2t1/2−∑j=1neijt≤0, and applying Lemma 5, it gives that(37)W2t=∑i=1N∑j=1Neijtμt+∑i=1N∑j=1Neijt2hmax+∑i=1NMi∑j=1Neijt−∑i=1NeiTtηitsign¯eit=∑i=1Nμt+2hmax+Mi−ηit∑j=1Neijt≤0.Considering thateit=|ei1t|,…,eintT and eitβ=ei1tβ,…,eintβT, we have(38)W3t=−∑i=1NeiTtsign¯eitaeitp+beitq−∑i=1NeiTtksigneiteitβ=−a∑i=1NeitTeitp+b∑i=1NeitTeitq=−a∑i=1N∑j=1neijt1+p+b∑i=1N∑j=1neijt1+q.By Lemma5, it can be obtained that(39)∑i=1N∑j=1neijt1+q1/1+q≥∑i=1N∑j=1neijt21/2,and then(40)b∑i=1N∑j=1neijt1+q≥b∑i=1N∑j=1neijt21+q/2.Similarly, using Lemma5, we have(41)∑i=1N∑j=1neijt1+p1/1+p≥Nn1/2−1/1+p∑i=1N∑j=1neijt21/2.Therefore,(42)a∑i=1N∑j=1neijt1+p≥aNn1−p/2∑i=1N∑j=1neijt21+p/2,which together with (38)–(42) implies that(43)W3t≤−aNn1−p/2∑i=1N∑j=1neijt21+p/2−b∑i=1N∑j=1neijt21+p/2=−aNn1−p/2∑i=1NeitTeit1+p/2−b∑i=1NeitTeit1+p/2=−aNn1−p/221+p/2Vt1+p/2−b21+p/2Vt1+p/2.Submitting (36), (37), and (43) to (33), we can get(44)V˙t≤−21+p/2Nn1−p/2aVt1+p/2−21+p/2bVt1+p/2.According to Lemma4, Vt converges to zero within a settling time T∗, which is defined in Definition 2, and one can obtain that, by use of controller (13), the considered complex network (1) is fixed-timely synchronized onto driven node (2) within the fixed time T∗, which is given by(45)T∗=121+p/2aNn1−p/2p−1+121+p/2b1−q.Therefore, it can be concluded that the error vectoreit converges to zero within T∗, and driven-response complex networks (1) and (2) are fixed-timely synchronized under controller (13) within the fixed time T∗. The proof is completed.Remark 2.
In recent years, a lot of extensive research has been conducted on the finite-time synchronization and fixed-time synchronization of complex networks, and many breakthroughs have been made. However, as far as we know, there are few published papers that deal with the fixed-time synchronization of heterogeneous complex networks. Theorem1 suggests a way to choose the controller to realize the fixed-time synchronization for the heterogeneous complex network. The controller consists of three sections: the first two terms are used to overcome the influence from the linear condition of the nonlinear function, the second one −ηitsigneit is introduced to compensate the influence of disturbance hit,xit, and finally, the last section sign¯eitaeitp+beitq is employed to force the considered networks achieve the fixed-time synchronization.
Now, ifM=max1≤i≤nMi and ηt=max1≤i≤nηit, then the controllers uit∈Rn can be designed as follows i=1,…,N:(46)uit=−Ai−A0x0−dieit−ηtsigneit−sign¯eitaeitp+beitq,where the parameters di,a,b, p,andq are defined as the same as in (13).
Therefore, by using the same analysis method in Theorem1, we can obtain Corollary 1 that is a similar conclusion with [25].Corollary 1.
For concerned complex networks (1) and (2) under controller (46), if Assumptions 1–4 hold and the control parameters ηit and di in (46) satisfy the following inequalities,(47)μt+2hmax+M−ηt<0,ℒ+A+cGs⊗Γ−D⊗In<0,where D=diagd1,…,dN>0, Gs=G+GT/2, ℒ=diagL1s,L2s,…LNs,andA=diagA1s,A2s,…,ANs, then driven-response complex networks (1) and (2) can achieve fixed-time synchronization under controller (46), with the settling time(48)T∗=121+p/2aNn1−p/2p−1+121+q/2b1−q.If the uncertain disturbance is not considered in the complex network model, i.e.,h1=h2=⋯=hN=0, then the following network model is degenerated as(49)x˙it=Aixit+fit,xit+c∑j=1NGijΓxjt,i=1,…,N.Letx0|t=0=x00, and then the driven network node is governed by(50)x0.t=A0x0t+f0t,x0t.Then, the corresponding error dynamical system can be rewritten as follows:(51)e˙it=Fit,eit+c∑j=1NGijΓej+uit,where Fit=fit,xit−f0t,x0t+Ai−A0xit.The controllers are the same as before, and then a criterion can be obtained on the fixed-time synchronization of the concerned complex networks with nonidentical nodes. By takinghmax=0 in Theorem 1, one can easily get the following corollary, and its proof is omitted here.Corollary 2.
Consider complex network (49) with drive node (50) under the set of controllers (46). If Assumptions 1–3 hold and the controller parameters satisfy the following matrix inequalities,(52)μt+M−ηt<0,ℒ+A+cGs⊗Γ−D⊗In<0,where D=diagd1,…,dN>0,Gs=G+GT/2,ℒ=diagL1s,L2s,…LNs,andA=diagA1s,A2s,…,ANs, then (49) can be synchronized to the state of drive node (50) within a fixed time T∗ and the settling time(53)T∗=121+p/2aNn1−p/2p−1+121+q/2b1−q.Furthermore, ifhi=0,Ai=0,andfi=f for i=0,1,…,N in (49), then complex network (1) is further reduced to(54)x˙it=ft,xit+c∑j=1NGijΓxjt,and driven network node (2) is changed correspondingly into the following form:(55)x0.t=ft,x0t.In the issue, the proposed fixed-time synchronization scheme can be applied to the corresponding complex networks with identical nodes here, and the criteria are given in the following corollary.Corollary 3.
For concerned complex networks (54) and (55) under the controllers(56)uit=−dieit−ηtsigneit−sign¯eitaeitp+beitq,i=1,…,N,where di≥0,i=1,2,…,N, 0<q<1,p>1, eitβ=ei1tβ,…,eintβT, and signeit=diagsignei1t,…,signeint, if Assumptions 1 and 2 hold and the controller parameters satisfy the following matrix inequalities,(57)M−ηt<0,IN⊗L+cGs⊗Γ−D⊗In<0,with D=diagd1,d2,…,dN>0andL1=L2=⋯∈=LN=L, then complex network (54) can be synchronized within a fixed time T∗:(58)T∗=121+p/2aNn1−p/2p−1+121+q/2b1−q.


## 4. Numerical Example

In this section, numerical simulation results are given to show that the proposed synchronization criterion is feasible. Consider the following discontinuous chaotic Chua circuit with linear and diffusive coupling, where the dynamics of theith node is described as follows:(59)x˙it=Aixit+fit,xi+hit,xi+k∑j=15GijΓxjt,i=1,…,5,with xit=xi1t,xi2t,xi3tT and initial values xi0T=0,2,0T+−1ii2/5,sini,−i2/4T. The inner coupling matrix Γ, activation matrix Ai, and Laplacian matrix G are defined as(60)Γ=100010001,Ai=a1+i10b1−i1001−110c1+i5−1,G=−110000−220000−220000−111110−3.Its activation functionfit,xit=1+i/10θsignxi1t,0,0T, and its uncertain disturbance hit,xi=sin50t/100cos50t/100−sin50t/100T, where a1=−2.75,b1=9.0,c1=−17,θ=3.86, and the coupling strength k=2.0. The drive dynamical node is(61)x˙0t=A0x0t+f0t,x0+h0t,x0,with initial values x00=0,2,0T, where f0t,x0=θsignx01t,0,0Tandh0t,x0=0,0,0T.By analyzing the state and error response trajectory of uncontrolled heterogeneous complex dynamic network (59) in Figures 1 and 2, we can easily draw the conclusion that the state of the nodes has not been synchronized, and the synchronization errors cannot tend to zero without the control input.Figure 1
State responses without control. (a) State responsexi1. (b) State response xi2. (c) State response xi3.
(a)(b)(c)Figure 2
Synchronization error without control. (a) Synchronization errorei1. (b) Synchronization error ei2. (c) Synchronization error ei3.
(a)(b)(c)From the analysis of [13], systems (59) and (61) have chaotic behavior, and there exist positive constants Li=0andMi=21+i/10θ (i = 0, 1, …, 5) satisfying Assumptions 1 and 2. Similarly, for μ=2+5/10θ=2.5θ=9.65, it is easy to verify that systems (59) and (61) satisfy Assumption 3. Let hmax=33/100; then, hit,xi≤hmax=33/100 is right for i=1,2,…N. So, systems (59) and (61) satisfy Assumption 4. With D=diag9,6,9,3,3 and ηit=20>μt+2hmax+maxMi=2.5θ+233/100+21+5/10θ=5.5θ+233/100,k=1,β=0.6, it is known that λmaxℒ+A+cGs⊗Γ−D⊗In=−3.6480, which shows that condition (22) can be satisfied. Applying Theorem 1, network (59) under the set of controllers (13) with a=2,b=3,p=3,q=0.5,andηit=20 can realize the synchronization within the fixed time t∗=1.3339. As a matter of fact, the real time of the synchronization is 0.0609 seconds in the numerical simulation, and the synchronization results of controlled networks are shown in Figures 3 and 4.Figure 3
State responses of fixed-time synchronization. (a) State responsexi1. (b) State response xi2. (c) State response xi3.
(a)(b)(c)Figure 4
Synchronization error of fixed-time synchronization. (a) Synchronization errorei1. (b) Synchronization error ei2. (c) Synchronization error ei3.
(a)(b)(c)


## 5. Conclusions

The fixed-time synchronization problem is studied for a type of dynamic complex networks with nonidentical nodes and uncertain disturbances. By employing the Lyapunov function theory, some novel sufficient conditions are provided and further applied to some special cases, such as the identical node issue. Future work may be centered on synchronous applications of complex networks with nonidentical nodes and uncertain disturbances.



---

*Source: 6654193-2021-03-19.xml*