# Finite-Time Synchronization of Complex Dynamical Networks with Time-Varying Delays and Nonidentical Nodes

Authors: Ahmadjan Muhammadhaji; Abdujelil Abdurahman; Haijun Jiang
Journal: Journal of Control Science and Engineering
 (2017)

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## Abstract

In this paper, we investigated the finite-time synchronization (FTS) problem for a class of time-delayed complex networks with nonidentical nodes onto any uniformly smooth state. By employing the finite-time stability theorem and designing two types of novel controllers, we obtained some simple sufficient conditions for the FTS of addressed complex networks. Furthermore, we also analyzed the effects of control variables on synchronization performance. Finally, we showed the effectiveness and feasibility of our methods by giving two numerical examples.


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## Body

## 1. Introduction

In the last decades, complex networks have been deeply investigated in various types of subjects, such as biology, engineering, physics, and mathematics [1–3]. In general, a complex network can be depicted as a large set of nodes connected by edges, where every node is a basic unit with specific dynamics. In fact, the structure of many types of natural and artificial systems, such as genetic networks, metabolic pathways, social networks, electrical power distribution networks, and World Wide Web (WWW), can be modeled via complex networks.Synchronization is unique in nature and can play an extremely vital role in many fields of science including biology, climatology, sociology, and ecology [4, 5]. Over the past two decades, synchronization of complex networks with identical dynamical topology has been broadly studied in different fields of engineering and sciences owing to the fact that they not only can well depict a great number of natural phenomena, but also have many useful applications in biological systems [6], secure communication [7], image processing [8], and so on. For this reason, many useful methods have been introduced for the synchronization of complex networks without control [9–12]. However, sometimes we cannot achieve the synchronization of network without adding any controller to the dynamics of individual node. Thus, how to synchronize the complex networks by designing a suitable controller is seen to be a most significant topic in both theory and application. As a result, many useful approaches have been developed to achieve chaos stabilization and chaos synchronization, such as adaptive control [13, 14], pinning control [15], impulsive control [16, 17], sliding mode control [18], and intermittent control [19, 20].Time delays inevitably exist in natural and man-made systems and cannot be neglected; for instance, delay effects cannot be ignored in the communication systems due to the limited switching speed of the hardware [21]. So far, in most of the existing works the networks with coupling time delays were considered. However, the time delays in the dynamical nodes [22–28], which can be more complicated, are still neglected in most of the existing works. In addition, it maybe unpractical to always use the hypothesis that all of the network nodes are the same since some real-world complex networks can be modeled by using different dynamical nodes [25]. Taking a software community network or metabolic network as an example, the dynamics of any two nodes in different communities are totally different, while the dynamics of each individual node in every community can be described by the same functional units. When the dynamics of nodes in a complex network are allowed to be nonidentical, the synchronization approaches for networks consisting of identical nodes will not work anymore. Thus, it is of great importance to develop new synchronization approaches for time-delayed complex networks with nonidentical nodes [25–30]. In [26], some exponential synchronization criteria for a class of complex networks with nonidentical nodes were established via combining the local intermittent controller with the open-loop controller. In [27], the outer synchronization of two complex networks with nonidentical dynamical nodes and coupling time delays was investigated by employing the adaptive control technique. In [28], by using pinning control scheme, the cluster synchronization of complex dynamical networks with time-delayed coupling and dynamic nodes was studied.Many of the results mentioned above have been employed to ensure the asymptotical or exponential stability of error dynamics, which means that the synchronization between the controlled system and the desired system can only be achieved in the infinite horizon. In practical application, however, it is more desirable that the synchronization aim is achieved in a finite-time. As is known to all, the finite-time control technique has been used as an efficient approach to realize the synchronization in a given time due to the fact that the finite-time control schemes may lead to better system performance such as better robustness and disturbance rejection properties [31–35]. In [36], the authors investigated the FTS problem for a class of complex networks with stochastic noise perturbations. In [37], the FTS between two complex networks with general coupling was studied by employing two different types of controllers, that is, periodically intermittent control and impulsive control. In [38], based on the finite-time stability theory and nonlinear control theory, the authors were concerned with FTS synchronization of class of nonidentical drive-response chaotic systems with time-varying delay. In [39], FTS problem of a class of fractional-order memristor-based neural networks with time delays was studied by using Laplace transform method and generalized Gronwall’s inequality technique. However, it is worth pointing out that most of the existing results on the FTS of complex networks are concerned with those identical nodes and delay-independent or constant delays, and very little attention has been paid to solving the problem which emerged from complex networks with nonidentical dynamical nodes and time-varying delays.Inspired by the above analysis, in the paper, we deal with the problem of FTS for a class of complex networks with nonidentical nodes and time-varying delays. By employing the well-known finite-time stability theorem, designing two types of novel controllers, and using some inequality techniques, we develop some simple but useful sufficient criteria for the FTS of the addressed network.Notations. In this paper, IN denotes the N×N identity matrix and sgn represents the sign function. For a vector x=(x1,x2,…,xn)T, |x| is the vector of the form |x|=(|x1|,|x2|,…,|xn|)T, and its Euclidean norm ·2 is defined as x22=xTx, where T denotes transposition. A=(aij)n×n denotes a matrix of n-dimension, As=(1/2)(A+AT), and A2=λmax(ATA), where λmax(·) denotes the maximum eigenvalue of a symmetric matrix. The notation A≤0 represents A is symmetric and seminegative definite matrix.The rest of the paper is organized as follows. In Section2, the FTS problem of a complex network with nonidentical nodes and time-varying delays is introduced, and some related definitions and preliminary lemmas are given. Next section is devoted to investigating the FTS of the addressed networks. In Section 4, two numerical examples with their simulations are given to demonstrate the effectiveness of the obtained theoretical results. Finally, we conclude the paper with some general conclusions in Section 5.


## 2. Preliminaries

In the paper, we consider the following controlled complex networks model consisting ofN nonidentical nodes, in which every node is an n-dimensional dynamical system described by (1)x˙it=fit,xit,xit-τit+c∑j=1NaijΓxjt+uit,i∈I≜1,2,…,N,where xi(t)=(xi1(t),xi2(t),…,xin(t))T is state variable of the ith individual node, fi:R+×Rn×Rn→Rn is a vector-valued continuous function, τi(t) is a time-varying delay and 0≤τi(t)≤τi for each i∈I, the positive constant c is the coupling strength, the Γ∈Rn×n is the inner connecting matrix between nodes, and the constant matrix A=(aij)N×N denotes the linear coupling configuration of the network, where aij is given as follows: if there is a connection between node i and node j(i≠j), then aij>0; otherwise, aij=0, and the diagonal elements of matrix A are defined by aii=-∑j=1,j≠iNaij, and ui(t) is the control input.The initial values of system (1) are given by (2)xis=ϕis,s∈-τ,0,where τ=maxi∈Iτi, ϕi(s)=(ϕi1(s),ϕi2(s),…,ϕin(s))T∈C([-τ,0],Rn), which denotes the Banach space of all continuous functions mapping [-τ,0] into Rn with norm defined by (3)φ2=sups∈-τ,0∑i=1nφis21/2,forφ∈C-τ,0,Rn.The main aim of this paper is to finite-timely synchronize the states of networks (1) onto any smooth dynamics s(t), where s(t)∈Rn can be any desired state: equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit. That is, by designing a suitable feedback controller in system (1), there exists time t∗ such that (4)x1t=x2t=⋯=x2t=st,for  anyt≥t∗.Throughout this paper, for system (1), we assume that the following hypothesis is satisfied.(H1) For every vector-valued function fi, there exist positive constants Li1 and Li2 such that(5)xit-yitTfit,xit,xit-τit-fit,yit,yit-τit≤Li1xit-yitTxit-τit-yit-τit+Li2xit-yitTxit-yit,∀xit,yit∈Rn,i∈I.Definition 1.
Assume thats(t)∈Rn is any smooth dynamics. The complex network (1) is said to be synchronized onto the homogeneous state s(t) in a finite-time if, for a suitable designed feedback controller ui(t), there exists a constant t∗>0 such that (6)limt→t∗xit-st2=0,and xit-st2=0 for t>t∗, i∈I. The constant t∗ is called the settling time of system (1).Lemma 2 (see [36]).
Assume thatV(t) is a positive definite continuous function. If V(t) satisfies the differential inequality (7)V˙t≤-ϖVϱt,∀t≥t0,Vt0≥0,where ϖ>0, 0<ϱ<1 are two constants, then, for any given t0, V(t) satisfies the following inequality: (8)V1-ϱt≤V1-ϱt0-ϖ1-ϱt-t0,t0≤t≤t∗,Vt≡0,∀t≥t∗,with t∗ given by (9)t∗=t0+V1-ϱt0ϖ1-ϱ.Lemma 3 (Jesen inequality [34]).
Ifb1,b2,…,bp, p∈N+ are positive numbers and 0<m<n, then (10)∑i=1pbin1/n≤∑i=1pbim1/m.


## 3. Main Results

In this section, some effective control schemes are developed to finite-timely synchronize the complex network (1) to any smooth dynamics s(t). First, we design controller ui(t) of system (1) as the following form: (11)uit=Fit-ηieit-Li1sgneiteit-τit-ksgn⁡eiteitβ,where Fi(t)=s˙(t)-fi(t,s(t),s(t-τi(t))), |ei(t)|β=(|ei1(t)|β,|ei2(t)|β,…,|ein(t)|β)T, sgn(ei(t))=diag(sgn(ei1(t)),sgn(ei2(t)),…,sgn(ein)(t)), ηi>0 is a positive constant determined later, k is a tunable constant, and the real number β satisfies 0<β<1.Let synchronization errorei(t)=xi(t)-s(t) for i∈I. Then, according to system (1) and the control law (11), the error dynamical system can be derived as (12)e˙it=f¯it,eit,eit-τit+c∑j=1NaijΓejt+u¯it,i∈I,where (13)f¯it,eit,eit-τit=fit,xit,xit-τit-fit,st,st-τit,u¯it=-ηieit-Li1sgneiteit-τit-ksgneiteitβ,i∈I.It is not difficult to see that the global FTS of the controlled complex network (1) is realized when the zero solution of the error system (12) is globally finite-time stable and this can be ensured by the following theorems.Theorem 4.
Under hypothesis(H1), assume that the following inequality is satisfied: (14)λL+αλA^s-η≤0,where λL=λmax(L), λA^s=λmax(A^s), α=cΓ2, η=mini∈Iηi, L=diag(L12,L22,…,LN2), A^=(a^ij)N×N,a^ij=aij for i≠j and a^ii=(ρmin/Γ2)aii,ρmin is minimum eigenvalue of Γs. Then, the controlled complex network (1) is synchronized to the given smooth dynamics s(t) in a finite-time (15)t∗=V01-δk1-δ,where V(0)=(1/2)∑i=1NeiT(0)ei(0) and δ=(1+β)/2.Proof.
Lete(t)=(e1T(t),e2T(t),…,eNT(t))T and construct the following Lyapunov function: (16)Vt=12∑i=1NeiTteit.Calculating the time derivative of V(t) along the trajectory of system (12) leads to(17)dVtdt=∑i=1NeiTte˙it=∑i=1NeiTtf¯it,eit,eit-τit+c∑j=1NaijΓejt+u¯it=∑i=1NeiTtf¯it,eit,eit-τit+c∑i=1N∑j=1NaijeiTtΓejt+∑i=1NeiTtu¯it.In view of the definition of u¯i(t), we have(18)dVtdt=∑i=1NeiTtf¯it,eit,eit-τit+c∑i=1N∑j=1NaijeiTtΓejt-∑i=1NηieiTteit-∑i=1NLi1eiTtsgneiteit-τit-k∑i=1NeiTtsgneiteitβ.From the hypothesis (H1), we get(19)dVtdt≤∑i=1NLi1eiTteit-τit+∑i=1NLi2eiTteit+c∑i=1N∑j=1NaijeiTtΓejt-∑i=1NηieiTteit-∑i=1NLi1eiTteit-τit-k∑i=1N∑j=1Neijt1+β≤∑i=1NLi2eiTteit+c∑i=1NρminaiieiTteit+α∑i,j=1,i≠jNaijeit2ejt2-∑i=1NηieiTteit-k∑i=1N∑j=1Neijt1+β=e¯TtL+αA^s-Ξe¯t-k∑i=1N∑j=1Neijt1+β,where Ξ=diag(η1,η2,…,ηN) and e¯(t)=(e1t2,e2t2,…,eNt2)T. Since A^s is the symmetric matrix, from inequality (14), we have(20)dVtdt≤e¯TtλL+αλA^s-ηe¯t-k∑i=1N∑j=1Neijt1+β≤-k∑i=1N∑j=1Neijt1+β.Using Lemma 3, one has (21)∑i=1N∑j=1Neijt1+β1/1+β≥∑i=1N∑j=1Neijt21/2.Hence,(22)∑i=1N∑j=1Neijt1+β≥∑i=1N∑j=1Neijt21+β/2=∑i=1NeiTteit1+β/2.Thus, from (20) and the above inequality, we get(23)dVtdt≤-k∑i=1N∑j=1Neijt1+β≤-k∑i=1NeiTteit1+β/2=-kV1+β/2t.Then, from Lemma 2, V(t) converges to zero in a finite-time t∗, where t∗ is given by (24)t∗=V01-δk1-δ,where δ=(1+β)/2. Therefore, according to Definition 1, the controlled complex network (1) is finite-timely synchronized in the settling time t∗. The proof of Theorem 4 is now completed.Theorem 5.
Suppose that the hypothesis(H1) holds and Γ is a positive definite diagonal matrix with diagonal elements γ1,γ2,…,γN; if the inequality is satisfied (25)Lj2-ηjIN+cγjA≤0,∀j∈I,then the controlled complex network (1) is synchronized to the given smooth dynamics s(t) in a finite-time (26)t∗=V01-δk1-δ,where V(0) and δ are the same as defined in Theorem 4.Proof.
Lete(t)=(e1T(t),e2T(t),…,eNT(t))T and consider the following Lyapunov function: (27)Vt=12∑i=1NeiTteit.The time derivative of V(t) along the trajectory of (12) is (28)dVtdt=∑i=1NeiTte˙it=∑i=1NeiTtf¯it,eit,eit-τit+c∑i=1N∑j=1NaijeiTtΓejt+∑i=1NeiTtu¯it.From the hypothesis (H1) and the definition of u¯i(t), we get (29)dVtdt≤∑i=1NLi1eiTteit-τit+∑i=1NLi2eiTteit+c∑i=1N∑j=1NaijeiTtΓejt-∑i=1NηieiTteit-∑i=1NLi1eiTteit-τit-k∑i=1N∑j=1Neijt1+β≤∑j=1NLj2e~jTte~jt+c∑j=1Nγje~jTtAe~jt-∑j=1Nηje~jTte~jt-k∑i=1N∑j=1Neijt1+β=∑j=1Ne~jTtLj2-ηjIN+cγjAe~jt-k∑i=1N∑j=1Neijt1+β,where IN is the identity matrix and e~j(t)=[e~1j(t),e~2j(t),…,e~Nj(t)]T is the jth column vector of the e(t). From inequality (25), we have (30)dVtdt≤-k∑i=1N∑j=1Neijt1+β.From the analysis in Theorem 4, we have (31)∑i=1N∑j=1Neijt1+β≥∑i=1NeiTteit1+β/2.Thus, together with (20), Lemma 3, and the above inequality, we get (32)dVtdt≤-k∑i=1NeiTteit1+β/2=-kV1+β/2t.By Lemma 2, V(t) converges to zero in a finite-time t∗, where t∗ is given by (33)t∗=V01-δk1-δ.Therefore, according to Definition 1, the controlled complex network (1) is finite-timely synchronized in the settling time t∗. The proof of Theorem 5 is now completed.In the controlled network (1), if the evolution function fi=fi(t,xi(t)) for i∈I, then system (1) is reduced to the following complex network: (34)x˙it=fit,xit+c∑j=1NaijΓxjt+uit,i∈I.Simultaneously assumption (H1) can be replaced by the following inequality.(H1′) For all xi,yi∈Rn and i∈I, there exist constants Li such that(35)xit-yitTfit,xit-fit,yit≤Lixit-yitTxit-yit.If we let(36)uit=s˙t-fit,st-Rit,where Ri(t)=ηiei(t)-ksgn(ei(t))|ei(t)|β, then, the error system is the following form: (37)e˙it=f¯it,eit+c∑j=1NaijΓejt+Rit,i∈I.Thus, for the FTS of controlled complex network (34), we have following corollaries.Corollary 6.
Assume that hypothesis(H1′) holds. If η1,η2,…,ηN satisfy inequality (14), then the controlled complex network (34) is finite-timely synchronized to the given smooth dynamics s(t) under controller (36). Moreover, the synchronization time is estimated by (38)t∗=V01-δk1-δ,where V(0) and δ are the same as defined in Theorem 4.Corollary 7.
Under the hypothesis(H1′), assume that Γ is a positive definite diagonal matrix. If η1,η2,…,ηN satisfy inequality (25), then the controlled complex network (34) is finite-timely synchronized to the given smooth dynamics s(t) under controller (36). Moreover, the synchronization time is estimated by (39)t∗=V01-δk1-δ.Remark 8.
In Theorems4 and 5, by using special finite-time controller, we achieved the FTS of nonidentical complex network (1) onto any smooth goal dynamics s(t). However, the used control law ui(t) is somehow expensive and not easily applicable, especially if the governing functions fi of system (1) satisfy some special conditions. Below, we will modify the adaptive laws ui(t) to improve the applicability of our results.
Suppose that the evolution functionsfi are bounded and let(40)u¯it=-ηieit-sgneitMi-ksgneiteitβ,i∈I.Then, for the FTS of complex system (1), we have following results.Theorem 9.
Assume that evolution functionsfi in system (1) satisfy(41)fit,ut,ut-τit≤Mi,∀ut∈Rn,i∈I,where Mi=[Mi1,Mi2,…,Min]T. If the error system (12) is controlled with controller (40), and its control strengths ηi satisfy(42)αA^s-Ξ≤0,where α,A^s and Ξ are the same as in Theorem 4, then the controlled complex network (1) is finite-timely synchronized to the given smooth dynamics s(t) in settling time t∗, where t∗ is the same as in (15).Proof.
Consider the following Lyapunov function:(43)Vt=12∑i=1NeiTteit.Taking the time derivative of V(t) along the solutions of (12), from (40) and (42), we have(44)dVtdt=∑i=1NeiTtf¯it,eit,eit-τit+c∑i=1N∑j=1NaijeiTtΓejt+∑i=1NeiTtu¯it≤∑i=1NeiTtMi+c∑i=1N∑j=1NaijeiTtΓejt-∑i=1NηieiTteit-∑i=1NeiTtsgneitMi-k∑i=1NeiTtsgneiteitβ≤c∑i=1NρminaiieiTteit+α∑i,j=1,i≠jNaijeit2ejt2-∑i=1NηieiTteit-k∑i=1NeiTtsgneiteitβ=e¯TtαA^s-Ξe¯t-k∑i=1N∑j=1Neijt1+β≤-kV1+β/2t.Thus, from Lemma 2, V(t) converges to zero in a finite-time t∗, where t∗ is given by (45)t∗=V01-δk1-δ.Therefore, according to Definition 1, the controlled complex network (1) is finite-timely synchronized in the settling time t∗. The proof of Theorem 9 is now completed.Theorem 10.
Suppose thatΓ is a positive definite diagonal matrix and the evolution functions fi in system (1) satisfy inequality (41). If the error system (12) is controlled with controller (40), and its control strengths ηj satisfy (46)cγjA-ηjIN≤0,j∈I,then the controlled complex network (1) is finite-timely synchronized to the given smooth dynamics s(t). Moreover, the synchronization time is estimated by (47)t∗=V01-δk1-δ,where V(0) and δ are the same as defined in Theorem 4.Proof.
Consider the following Lyapunov function:(48)Vt=12∑i=1NeiTteit.Calculating the time derivative of V(t) along the trajectory of system (12) and using (40) and (46) lead to(49)dVtdt≤∑i=1NeiTtMi+c∑i=1N∑j=1NaijeiTtΓejt-∑i=1NηieiTteit-∑i=1NeiTtsgneitMi-k∑i=1NeiTtsgneiteitβ≤c∑j=1Nγje~jTtAe~jt-∑j=1Nηje~jTte~jt-k∑i=1N∑j=1Neijt1+β=∑j=1Ne~jTtcγjA-ηjINe~jt-k∑i=1N∑j=1Neijt1+β≤-kV1+β/2t.Thus, from Lemma 2, V(t) converges to zero in a finite-time t∗, where t∗ is given by (50)t∗=V01-δk1-δ.Therefore, according to Definition 1, the controlled complex network (1) is finite-timely synchronized in the settling time t∗. The proof of Theorem 10 is now completed.In the nonidentical complex network (1), if all the functions fi are the same, that is, fi=f for all i∈I, then system (1) is degenerated to the following identical complex network:(51)x˙it=ft,xit,xit-τit+c∑j=1NaijΓxjt+uit,i∈I.Also, the synchronization state s(t) is chosen as a solution of a decoupled state satisfying (52)s˙t=fs,st,st-τit;then the FTS problem of the complex network (1) becomes the FTS problem of a complex network with identical nodes studied in much of the literatures, such as [34–37]. Additionally, the finite-time control scheme (11) is degenerated to(53)uit=-ηieit-Li1sgneiteit-τit-ksgneiteitβ.Evidently, the following criteria can be directly derived to ensure the FTS of the complex network with identical nodes (51).Corollary 11.
Assume that hypothesis(H1) holds and η1,η2,…,ηN satisfy inequality (14); then the controlled complex network (51) is globally finite-timely synchronized to the decoupled state (52) under controller (53). Moreover, the synchronization time is estimated by (54)t∗=V01-δk1-δ,where V(0) and δ are the same as defined in Theorem 4.In the controlled network (51), if the evolution function f is not relevant to delay, that is, f=f(t,xi(t)) for i∈I, then system (51) is reduced to following complex network: (55)x˙it=ft,xit+c∑j=1NaijΓxjt+uit,i∈I.In addition, the synchronization state s(t) is chosen as a solution of a decoupled state satisfying (56)s˙it=fs,st.Let (57)uit=ηieit-ksgneiteitβ.Then, the error system is the following form: (58)e˙it=f¯it,eit+c∑j=1NaijΓejt+uit,i∈I.Thus, for the FTS of controlled complex network (55), we have following result.Corollary 12.
Suppose that hypothesis(H1′) holds. If η1,η2,…,ηN satisfy inequality (14), then the controlled complex network (55) is finite-timely synchronized to the given decoupled state (56) under controller (57). Moreover, the synchronization time is estimated by (59)t∗=V01-δk1-δ,where V(0) and δ are the same as defined in Theorem 4.Remark 13.
From the proofs of Theorems4, 5, 9, and 10, we can see that the parameters η1,η2,…,ηN, k, and β in controllers (11), (40), (53), and (57) play a central role in FTS of controlled complex networks. Inequality (20) indicates that the synchronization rate increases when η1,η2,…,ηN, k, or β increases. On the other hand, whether network (1) can be synchronized or not relies on the value of η1,η2,…,ηN, whereas the synchronization time depends on values of k and β and has nothing to do with the value of η1,η2,…,ηN.Remark 14.
Since control inputs (11), (40), (53), and (57) contain the discontinuous sign function, an undesirable chattering may appear as a hard switcher. To avoid the chattering, the continuous tanh function can be used for discontinuous sign function to remove discontinuity. For example, controller (11) can be modified to(60)uit=Fit-ηieit-Li1tanh⁡θieiteit-τit-ktanh⁡θieiteitβ,where θi>0 for i∈I.Remark 15.
In [37], the FTS between two identical complex networks with delayed and nondelayed coupling was investigated by using two different types of controller, that is, periodically intermittent controller and impulsive controller. In [40], the FTS of complex dynamical networks with or without coupling delay was studied by employing the Lyapunov function method and well-known finite-time stability theorem. However, the proposed controllers ui(t) in [37, 40] are very complex and ui(t)→∞ as ei(t)→0. Thus, the designed control laws in [37, 40] are too expensive and not easily applicable. Evidently, the improper technique somewhat affects the validity of results obtained in [37, 40]. To avoid this, a different technique is used in this paper to achieve the FTS of considered nonidentical complex system.Remark 16.
In the past few years, FTS has drawn an increasing attention due to the fact that it requires considered chaotic networks to be synchronized in finite-time rather than merely asymptotically. In the literature, so far, there are many excellent results on the TNS of complex networks, neural networks, and multiagent systems based on the continuous or discontinuous control approaches [41–44]. However, there are very few results to address the problem of the FTS for complex networks with delays and nonidentical nodes. Generally, different nodes in a complex network have different features including dynamic behaviors and initial conditions, which may result in their different control protocols to optimize synchronization time. In this paper, we employ the finite-time control technique to achieve the synchronization of a class of complex networks with nonidentical dynamical nodes and time-varying delays. Obviously, the results obtained in this paper can fasten the synchronization speed in great extent than the methods of asymptotical or exponential synchronization [22, 28]. Therefore, our results are more conducive and the extension and improvement of the existing results can be seen.


## 4. Numerical Simulations

In this section, two numerical examples and their simulations are given to demonstrate the feasibility of the proposed FTS schemes.Example 1.
Consider the nonidentical complex dynamical network described by following equation:(61)x˙it=fixit,xit-τt+c∑j=14aijΓxjt+uit,i∈I=1,2,3,4where xi(t)=(xi1(t),xi2(t),xi3(t))T and(62)f1xit,xit-τt=f2xit,xit-τt=Bxit+g11xit+g12xit-τ1t,f3xit,xit-τt=f4xit,xit-τt=Dxit+g21xit+g22xit-τ2t.Here, g11(xi(t))=(0,-xi1xi3,xi1xi2)T, g12(xi(t))=(0,0.2xi2,0)T, g21(xi(t))=(-2.95(|xi1+1|-|xi1-1|),0,0)T, g22(xi(t))=(0,0,-0.5sin(0.5xi1))T, τi(t)=1,i∈I, and(63)B=-1010028-1000-83,D=-16.81001-110-14.870.The coupling configuration matric A is chosen as (64)A=-10.30.20.50.3-0.80.40.10.20.4-1.20.60.50.10.6-1.2.The time evolutions of delayed chaotic Lorenz system and chaotic Chua oscillator with initial conditionsx1(θ)=[0,1,0]T and x3(θ)=[0.7,0,0]T for θ∈[-1,0] are shown in Figures 1 and 2.Figure 1
The chaotic behavior of time-delayed Lorenz system.Figure 2
The chaotic behavior of time-delayed Chua oscillator.Now we select Rössler system as the synchronized aim; it is given by(65)s˙1t=-s2t+s3t,s˙2t=s1t+0.2s2t,s˙3t=s3ts1t-5.7+0.2.The chaotic behavior of Rössler system (65) with initial values s(0)=[0,2,0]T is depicted in Figure 3.Figure 3
The chaotic behavior of Rössler system.By easy computation, we getλmax(Bs)=14.0256, λmax(Ds)=7.3184, and λmax(As)=0, |x11|<m1=27.87, |x12|<m2=38.39, and |x13|<m3=73.85 for Lorenz system and |x31|<r1=2.32, |x32|<r2=0.49, and |x33|<r3=3.91 for Chua oscillator. Letting M=[0,m3,m2;m3,0,0;m2,0,0] and R=[0,r3,r2;r3,0,0;r2,0,0] (see [40]), then we have λmax(M)=83.2323 and λmax(R)=3.9406. Thus, take L12=L22=λmax(Bs)+λmax(M)=97.2579, Ł32=Ł42=λmax(Ds)+λmax(R)=11.2590, Ł11=Ł21=0.2, and L31=L41=0.5. Then, the hypothesis (H1) is satisfied for system (61). Using finite-time controller (11) and letting c=1,k=5, and β=0.5, it is not difficult to check that inequality (14) is satisfied with η1=η2=97.3 and η3=η4=11.3. Therefore, according to Theorem 4, the controlled coupled network (61) is synchronized to the chosen orbit (65) in a finite-time. The synchronization errors are given in Figure 4.Figure 4
Time evolution of synchronization errors for (61) with different initial values.Example 2.
Consider a complex dynamic network consisting of two communities, in which each dynamical node is a chaotic neural network given by(66)x˙it=-A1xit+B1gxit+D1gxit-τ1t,i=1,2,x˙it=-A2xit+B2hxit+D2hxit-τ2t,i=3,4,where xi(t)=(xi1,xi2)T, g(xi)=(0.5(|xi1+1|-|xi1-1|))T,0.5(|xi2+1|-|xi2-1))T, h(xi)=(tanh(xi1),tanh(xi2))T, τ1(t)=τ2(t)=et/(1+et), A1=A2=diag(1.2,1.2), and(67)B1=1+π4200.11+π4,B2=2-0.1-53,D1=1.32π40.10.11.32π4,D2=1.50.10.22.5.The time evolutions of each dynamical node in (66) are depicted in Figures 5 and 6.Figure 5
The chaotic behavior of each node in the first cluster in (66).Figure 6
The chaotic behavior of each node in the second cluster in (66).Now, we consider the following controlled complex networks given by(68)x˙it=-A1xit+B1gxit+D1gxit-τ1t+c∑j=14aijΓxjt+uit,i=1,2,x˙it=-A2xit+B2hxit+D2hxit-τ2t+c∑j=14aijΓxjt+uit,i=3,4.The inner connecting matrixΓ=diag(1.5,2)T and the matrix A is selected as (69)-0.600.30.30.2-0.80.40.20.40.5-1.20.30.30.40.2-0.9.The following periodic orbit is chosen as a synchronization aim:(70)st=4sin⁡0.5t,3cos⁡tT.It is not difficult to check that the hypothesis(H1) is satisfied for system (66) with L12=L22=13.04, L32=L42=6.30, L11=L21=1.56, and L31=L41=2.53. Choosing c=1,k=8, η1=η2=13.1, η3=η4=6.40, and β=0.6, then inequality (14) also holds. Use the following finite-time controller (see Remark 14):(71)uit=Fit-ηieit-Li1tanh⁡θieiteit-τit-ktanh⁡θieiteitβ,where θi=0.1, Fi(t)=s(t)-fi(t,s(t),s(t-τi(t))), i=1,2,3,4. Then, from Theorem 4, the controlled coupled network (68) is synchronized to the chosen periodic orbit (70) in a finite-time. The synchronization errors are given in Figure 7, and the synchronization curves are depicted in Figure 8.Figure 7
Time evolution of synchronization errors for (26) with different initial values.Figure 8
Finite-time synchronization curves ofxij for (68) with different initial values.


## 5. Conclusion

In this paper, we investigate the FTS of nonidentical complex networks with time-varying delays. By using finite-time stability theorem, inequality techniques, and designing suitable controllers, we develop some simple but useful sufficient criteria on the FTS of the addressed model. Our results of the FTS technique have optimal convergence time than the asymptotical and exponential synchronization methods. Finally, we gave two examples to demonstrate the effectiveness and feasibility of the developed FTS schemes.



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