# Cluster Synchronization of Impulsive Complex Networks with Stochastic Perturbations and Time-Varying Delays

Authors: Yi Zhao; Jianwen Feng; Jingyi Wang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper investigates the cluster synchronization of impulsive complex networks with stochastic perturbation and time-varying delays. Besides, the nodes in the complex networks are nonidentical. By utilizing the Lyapunov stability theory, stochastic analysis theory, and linear matrix inequalities (LMI), sufficient conditions are derived to guarantee the cluster synchronization. The numerical simulation is provided to show the effectiveness of the theoretical results.


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## Body

## 1. Introduction

Over the past few years, the study of complex networks has become an important issue. The complex networks are constructed with individual units called nodes, connected by links that exhibit complex topological properties, such as coupled biological and chemical system, neural networks, social interacting species, the Internet, and the World Wide Web [1]. Thereafter, the problem of control and synchronization in complex networks has attracted increasing attention for its wide applications in real world.From the literature, there are three common phenomena in many evolving networks: delay effects, stochastic effects, and impulsive effects. First of all, considering finite switching speed of amplifiers and finite signal propagation time, time delay is ubiquitous in the implementation of electronic networks. There are two kinds of time delays in complex dynamical networks. The inner delay time causes chaos, such as delayed neural networks and delayed Chua’s circuit system and the outer coupling delay time also exists widely, such as in communication and traffic congestion. Therefore, time delays cannot be ignored in order to simulate realistic networks; see [2–5]. Second, the nodes in dynamical networks are often subject to various types of noise and uncertainty, which can have a great influence on the behavior of dynamical networks. For instance, signals transmitted between nodes of complex networks are unavoidably subject to stochastic perturbations from environment, which may cause information contained in these signals to be lost; see [6, 7]. Therefore, stochastic perturbations should be included in the complex network model. Third, impulsive exchange is a common phenomenon in many evolving networks. The state of nodes is often subject to instantaneous perturbations and experience abrupt change at certain instants which may be caused by switching phenomena, frequency change, or sudden noise. For example, the states of electronic networks and biological networks are often subject to instantaneous disturbances and experience abrupt changes at certain instants, which may be caused by switching phenomenon, frequency change, or other sudden noises, which could be expressed as impulsive effects. Thus, impulsive dynamical networks, which involve sudden changes at certain discrete times, are receiving more and more attention of researchers for various fields, that is, [8–10] and the references therein. Since time delays, stochastic perturbations, and impulsive effects can heavily affect the dynamical behaviors of the networks, it is necessary to investigate the time delay, stochastic perturbations, and impulsive effects on the synchronization of dynamical networks. Actually, impulsive stochastic perturbed complex networks have attracted the interest of many researchers for their various applications in information science, economic systems, automated control systems, and so forth.Synchronization is one of the hot topics in the investigation of complex networks. Generally speaking, synchronization is the process in which two or more dynamical systems seek to adjust a certain prescribed property of their motion to a common behavior in the limit as time tends to infinity either by virtue of coupling or by forcing [11]. Synchronization of complex network has observed wide potential applications in many different areas such as population dynamic, power system, chemical process simulation, automatic control, and many others [12–18]. Synchronization patterns are classified as complete synchronization [19], lag synchronization [20], cluster synchronization [21], phase synchronization [22], and partial synchronization [23]. Among them, the cluster synchronization has received more and more attention. The cluster synchronization requires that the coupled oscillators split into subgroups called clusters, such that the oscillators synchronize with one another in the same cluster, but there is no synchronization among different clusters. The clusters can be classified by different function communities contained in the networks of real world, and the local dynamics between two function communities are different. For instances, in metabolic, neural, or software community networks, the individual nodes in each community can be viewed as the identical functional units, whereas the nodes in different communities are different since they have different functions [24]. There have been already some papers focused on sufficient conditions for the global stability of cluster synchronization of some related networks; see [25, 26] and the references therein. In [27], Wang et al. investigated the exponential synchronization of stochastic perturbed complex networks with time-varying delays via periodically intermittent pinning. Compared with the work in [27], in this paper, we would focus on the cluster synchronization of complex networks with stochastic perturbations, time-varying delays, and impulsive effects. It should also be mentioned that synchronization problems could be seen as an application of stability. There are many papers that are concerned with the stability of complex systems with time delays, such as [28–31], which inspire the studies of synchronization problems.As everyone knows, the real-world networks normally have a large number of nodes, and it is usually impractical to control a complex network by adding the controllers to all nodes. Pinning control, in which controllers are only applied to a small fraction of nodes, is an effective way to reduce the number of controlled nodes. Chen et al. [32] proved that the complex networks with symmetric or asymmetric coupling matrix achieved synchronization by a single pinning controller. In [33], Zhao et al. proposed a pinning scheme to solve the problems how to stabilize a network onto a homogeneous state globally exponentially and how to select an optimal combination between the number of pinned nodes and the feedback control gain.Based on the above analysis, in this paper, we study the cluster synchronization of impulsive complex networks with time-varying delays and stochastic perturbations by adding feedback controllers and impulsive controllers on a fraction of selected nodes. To obtain our main results, we first formulate a new complex network nondelayed and time-varying delayed linear coupling and vector-form stochastic perturbations. By using the Lyapunov functional method, the stochastic stability analysis theory, and linear matrix inequality technique (LMI), some novel sufficient conditions are derived to guarantee cluster synchronization of the complex networks.Remark 1.
The main tools to derive the results in this paper are Lyapunov functional method, Itô formula, and linear matrix inequality technique (LMI). It is well known that Lyapunov functional method is one of the most useful tools to handle the stability problems. Since stochastic perturbations are considered in the model of this paper, Itô formula is employed to deal with the stochastic differential equations arising in the analysis of synchronization. Linear matrix inequality technique also plays an important role in the proof of our main theorem.The paper is organized as follows. In Section2, a general model of impulsive complex network with both time-varying delays and stochastic perturbations is given, and some preliminaries are stated. In Section 3, some cluster synchronization criteria for such complex dynamical network are established. In Section 4, a numerical example for verifying the effectiveness of the theoretical results is provided. We conclude the paper in Section 5.


## 2. Preliminaries

For the facility of statements, we give some definitions of notations and lemmas, which would be used in the analysis of the next sections. Similar definitions can also be found in [27]. Suppose thatℝnandℝn×ndenote then-dimensional Euclidean space and the set of alln×nreal matrices, respectively. The superscriptTdenotes the transpose of a matrix or vector,Tr(·)denotes the trace of the corresponding matrix, andIndenotes the identity matrix withndimensions. For square matricesM, the notationM>0(<0) denotes thatMis positive-definite (negative-definite) matrix. Suppose thatAis a symmetric matrix. Denoteλmax(A)as the largest eigenvalue ofA, and denoteλmin(A)as the smallest eigenvalue ofA. Let(Ω,ℱ,{ℱt}t≥0,𝒫)be a completed probability space with a filtration{ℱt}t≥0which is right continuous andℱ0contains all𝒫-null sets. Letw(t)=(w1(t),w2(t),…,wn(t))Tbe anm-dimensional Brownian motion defined on probability space. Denote byC([-τ,0];Rn)the family of continuous functionϕfrom[-τ,0]toℝnwith the uniform norm∥ϕ∥2=sup-τ≤s≤0ϕ(s)Tϕ(s). Denote byCℱ02([-τ,0];ℝn)the family of allℱ0measurable,C([-τ,0];ℝn)-valued stochastic variablesξ={ξ(θ):-τ≤θ≤0}such that∫-τ0‍𝔼∥ξ(s)∥2ds≤∞, where𝔼stands for the correspondent expectation operator with respect to the given probability measure𝒫.Consider an impulsive complex network consisting ofNnodes andmcommunities (N>m≥2) with nondelayed and time-varying delayed linear coupling and vector-form stochastic perturbations, which is described as
(1)dxi(t)={∑j=1,i≠jN‍fμi(t,xi(t),xi(t-τ(t)))+∑j=1,i≠jN‍aijΓ(xj(t)-xi(t))+∑j=1,i≠jN‍bijΓ(xj(t-τc(t))-xi(t))}dt+σi(t,x(t),x(t-τ(t)),x(t-τc(t)))dwi(t),pppppppppppi=1,2,…,N,t≥0,t≠tk,k∈ℕxj(tk+)-xi(tk+)=ν(xj(tk-)-xi(tk-))pppppppppppfork∈ℕ(i,j):aij>0,
wherexi(t)=(xi1(t),xi2(t),…,xin(t))T∈ℝnrepresents the state vector of theith node of the network;fμi(t,xi(t),xi(t-τ(t)))=[fμi1(t,xi(t),xi(t-τ(t))),fμi2(t,xi(t),xi(t-τ(t))),…,fμin(t,xi(t),xi(t-τ(t)))]Tis a continuous vector-form function. If nodeibelongs to thejth community, then we letμi=j. We denote byUithe set of all nodes in theith community andU~iA, which is the subset ofUi, is the index set of all nodes in theith community having direct connections to other communities inA. And by the similar way, we can defineU~iB. The functionfμi(·)describes the local dynamics of nodes in theμith community, which is differentiable and capable of performing abundant dynamical behaviors. For any pair of indicesiandj, ifμi≠μj, which means nodeiand nodejbelong to different communities, thenfμi≠fμj.Γ=diag(γ1,γ2,…,γn)is an inner coupling of the networks, satisfyingγj>0,j=1,2,…,n.A=[aij]∈ℝn×nandB=[bij]∈ℝn×nare outer coupling matrices of the networks at timetandt-τc(t), respectively, which are both symmetric, satisfyingaij≥0fori≠j,aii=-∑j=i,j≠iN‍aij, andbij≥0fori≠j,bii=-∑j=i,j≠iN‍bij;τ(t)is the inner time-varying delay satisfyingτ≥τ(t)≥0, andτc(t)is the coupling time-varying delay satisfyingτc≥τc(t)≥0;σi(t,x(t),x(t-τ(t)),x(t-τc(t)))=σi(t,x1(t),…,xn(t),x1(t-τ(t)),…,xn(t-τ(t)),x1(t-τc(t)),…,xn(t-τc(t)))∈ℝn×nis the noise intensity matrix, andwi(t)=(wi1(t),wi2(t),…,win(t))T∈ℝnis a bounded vector-form Weiner process, satisfying
(2)𝔼wij(t)=0,𝔼wij2(t)=1,𝔼wij(t)wij(s)=0(s≠t).
In this paper, we always assume thatAis irreducible in the sense that there is no isolated node and|ν|<1, which means that corresponding impulsive effects are synchronizing.The initial conditions associated with (1) are
(3)xi(s)=ξi(s),-τˇ≤s≤0,i=1,2,…,N,
whereτˇ=max{τ,τc},ξi∈Cℱ0b([-τˇ,0],ℝn)with the norm∥ξi∥2=sup-τˇ≤s≤0ξi(s)Tξi(s). We definesμi=(sμi1,sμi2,…,sμin)T∈Rnthat satisfies
(4)dsμi(t)={fμi(t,sμi(t),sμi(t-τ(t)))-biiΓ(sμi(t-τ(t))-sμi(t))fμi(t,sμi(t),sμi(t-τ(t)))}dt,
which describes the identical local dynamics for the nodes in theμith community. Obviously, the synchronization state in the same communitysμi(t)is uniform. Therefore, we assume thatb11=b22=⋯=bNN.In order to achieve the cluster synchronization objective, the feedback controllers as well as impulsive controllers are added to part of its nodes. Whent≠tkfork∈ℕ(5)ui(t)={-εi(xi(t)-sμi(t))-∑j=1N‍aijΓsμj(t)i∈U~μiA∖U~μiB,-εi(xi(t)-sμi(t))-∑j=1N‍bijΓsμj(t)i∈U~μiB∖U~μiA,-εi(xi(t)-sμi(t))-∑j=1N‍aijΓsμj(t)-∑j=1N‍bijΓsμj(t)i∈U~μiA∩U~μiB,0,iotherwise,
and whent=tkfork∈ℕ(6)ei(tk+)=νei(tk-)fori∈U~μiA∪U~μiB,
whereεi>0(i∈U~μiA∪U~μiB)represents control gain and by definingεi=0foriotherwise, we can denoteΞ=diag{ε1,ε2,…,εN}. Define the synchronization error asei(t)=xi(t)-sμi(t)(i=1,2,…,N), according to the controllers (5) and (6), the error system is derived as follows:
(7)dei(t)={∑j=1N‍fμi(t,xi(t),xi(t-τ(t)))-fμi(t,sμi(t),sμi(t-τ(t)))+∑j=1N‍aijΓej(t)+∑j=1N‍bijΓej(t-τc(t))-biiΓ(ei(t-τc(t))-ei(t))-εiei(t)∑j=1N‍}dt+σi(x(t-τc(t)))t,x(t),x(t-τ(t)),ppppppppx(t-τc(t)))dwi(t),fori=1,2,…,N,t≥0,t≠tk,k∈ℕei(tk+)-ej(tk+)=ν(ei(tk-)-ej(tk-))for(i,j):aij>0,t=tk,k∈ℕei(tk+)=νei(tk-)fori∈U~μiA∪U~μiB,t=tk,k∈ℕ.Remark 2.
Since both state coupling and impulsive coupling are considered in (1), it is reasonable to expect that both the state-feedback controller (5) and the impulsive controller (6) will play important roles in the stabilizing process of the complex network (1). Thus, the controller designed in this paper consists of both the state-feedback controller (5) and the impulsive controller (6).Definition 3.
The complex network (4) is said to be exponentially synchronized if the trivial solution of system (9) satisfies
(8)∑i=1N‍𝔼∥ei(t,t0,ξi)∥2≤Ke-κt,
whereK>0andκ>0, for any initial dataξi∈𝒞ℱ0b([-τ,0];ℝn).Definition 4 (see [6]).
A continuous functionf(t,x,y):[0,+∞]×ℝn×ℝn→ℝnis said to be in theQUADfunction class, denoted asf∈QUAD(P,Δ,η,ζ), for given matrixΓ=diag{γ1,γ2,…,γn}, if there exist positive definite diagonal matricesP=diag{p1,p2,…,pn}, diagonal matricesΔ=diag{δ1,δ2,…,δn},and constantsη>0,ζ>0, such thatfsatisfies the following condition:
(9)(x-y)TP((f(t,x,z)-f(t,y,w))-ΔΓ(x-y))≤-η(x-y)T(x-y)+ζ(z-w)T(z-w)
for allx,y,z,w∈ℝn.We will simply introduce the following notations:(10)pˇ=max{p1,p2,…,pn},p^=min{p1,p2,…,pn},δˇ=max{δ1,δ2,…,δn}.Definition 5 (see [34]).
The average impulsive interval of the impulsive sequenceζ={t1,t2,…}is less thanTa, if there exist a positive integerN0and a positive numberTa, such that
(11)Nζ(T,t)≥T-tTa-N0,∀T≥t≥0,
whereNζ(T,t)denotes the number of impulsive times of the impulsive sequenceζin the time interval(t,T).The following assumptions will be used throughout this paper in establishing our synchronization condition.(H1)
b11=b22=⋯=bNN=b-.(H2)
τ(t)andτc(t)are abounded and continuously differentiable functions, satisfying0<τ(t)≤τ,τ˙(t)<τ-<1,0<τc(t)≤τc, andτ˙c(t)<τ-c<1. Denoteτ-ˇ=max{τ-,τ-c}.(H3)
Denoteσ(t,e(t),e(t-τ(t)),e(t-τc(t)))=σ(t,e1(t),…,eN(t),e1(t-τ(t)),…,eN(t-τ(t)),e1(t-τc(t)),…,eN(t-τc(t))). There exist appropriate dimensional positive definite constant matricesΥi1,Υi2, andΥi3fori=1,2,…,Nsuch that
(12)Tr[σi(t,e(t),e(t-τ(t)),e(t-τc(t)))T×σi(t,e(t),e(t-τ(t)),e(t-τc(t))(t-τc(t)))T)]≤∑j=1N‍ej(t)TΥi1ej(t)+∑j=1N‍ej(t-τ(t))TΥi2ej(t-τ(t))+∑j=1N‍ej(t-τc(t))TΥi3ej(t-τc(t)).Lemma 6 (see [35] (Itô formula)).
Consider ann-dimensional stochastic differential equation
(13)dx(t)=f(t,x(t),x(t-τ))dt+σ(t,x(t),x(t-τ))dω(t).
LetC2,1(ℝ+×ℝn;ℝ+)denote the family of all nonnegative functionsV(t,x)onℝ+×ℝn, which are twice continuously differentiable inxand once differentiable int. If   V∈C2,1(ℝ+×ℝn;ℝ+), define an operatorℒVfromℝ+×ℝntoℝby
(14)ℒV(t,x)=Vt(t,x)+Vx(t,x)f(t,x,y)+12Tr[σ(t,x,y)TVxxσ(t,x,y)],
whereVt(t,x)=∂V(t,x)/∂t,Vx(t,x)=(∂V(t,x)/∂x1,…,∂V(t,x)/∂xn),Vxx(t,x)=(∂2V(t,x)/∂xixj)n×n.IfV∈C2,1(ℝ+×ℝn;ℝ+); then for any∞>t>t0≥0,
(15)𝔼V(t,x(t))=𝔼V(t0,x(t0))+𝔼∫t0t‍ℒV(s,x(s))ds
as long as the expectations of the integrals exist.


## 3. Main Results

We would give some sufficient conditions for the cluster synchronization of impulsive complex networks with time-varying delays and stochastic perturbations in this section.Theorem 7.
Suppose the assumptions (H1), (H2), and (H3) hold andfμi∈QUAD(P,Δ,η,ζ). If there exist positive constantsαandβ, such that
(16)[A+(δˇ+b-)IN-Ξ+αINB-b-IN2BT-b-IN2-βIN]≤0,(17)0≤τ-ˇ≤1-b+ca,
whereγ>0is the smallest root of the equation
(18)γ-a+b1-τ-eγτ+c1-τ-ceγτc=0,a=λmin(2ηIn-pˇ∑i=1NΥi1+2αPΓ)pˇ,(19)b=λmax(∑i=1NPΥi2+2ζIN)p^,c=λmax(∑i=1NPΥi3+2βPΓ)p^,
then the solutionx1(t),x2(t),…,xN(t)of system (4) could reach cluster synchronization under controller (5) when the average impulsive interval is less thanTa, and0<ν2+(bτ/(1-τ-))eγτ+(cτc/(1-τ-c))eγτc<1.Proof.
Lete~k(t)=(e1k(t),e2k(t),…,eNk(t))T,k=1,2,…,n. Define a Lyapunov function
(20)V(t,e(t))≡V(t)=12∑i=1N‍ei(t)TPei(t).
Whent∈[tk,tk+1), according to Lemma 6, we have
(21)ℒV(t,e(t))=∑i=1N‍ei(t)TP{∑j=1N‍fμi(t,xi(t),xi(t-τ(t)))ppppppppppppp-fμi(t,sμi(t),sμi(t-τ(t)))ppppppppppppp+∑j=1N‍aijΓej(t)+∑j=1N‍bijΓej(t-τc(t))ppppppppppppp-b-Γ(ei(t-τc(t))-ei(t))+εiei(t)∑j=1N‍}+12Tr{σi(t,x(t),x(t-τ(t)),x(t-τc(t)))Tpppppppppp×Pσi(t,x(t),x(t-τ(t)),x(t-τc(t)))((t-τc(t)))T}=∑i=1N‍ei(t)TP{fμi(t,xi(t),xi(t-τ(t)))ppppppppppppp-fμi(t,s(t),s(t-τ(t)))-ΔΓei(t)}+∑i=1N‍ei(t)TPΔΓei(t)+∑i=1N‍∑j=1N‍aijei(t)TPΓej(t)+∑i=1N‍∑j=1N‍bijei(t)TPΓej(t-τc(t))-∑i=1N‍b-ei(t)TPΓei(t-τc(t))+∑i=1N‍b-ei(t)TPΓei(t)-∑i=1l‍εiei(t)TPΓei(t)+12∑i=1N‍Tr{σi(t,x(t),x(t-τ(t)),x(t-τc(t)))Tpppppppppppp×Pσi(t,x(t),x(t-τ(t)),x(t-τc(t)))((t-τc(t)))T}≤-η∑i=1N‍ei(t)Tei(t)+ζ∑i=1N‍ei(t-τ(t))Tei(t-τ(t))+∑k=1n‍pkγk{e~k(t)T(A+(δˇ+b-)IN-Ξ)e~k(t)ppppppppppppp+e~k(t)T(B-b-IN)e~k(t-τc(t))}+pˇ2∑j=1N‍∑i=1N‍{ej(t)TΥi1ej(t)+ej(t-τc(t))TΥi3ej(t-τc(t))ppppppppppppp+ej(t-τ(t))TΥi2ej(t-τ(t))ppppppppppppp+ej(t-τc(t))TΥi3ej(t-τc(t))}≤∑j=1N‍ej(t)T(-ηIn+pˇ2∑i=1N‍Υi1-α1PΓ)ej(t)+∑j=1N‍ej(t-τ(t))T(ζIN+pˇ2∑i=1N‍Υi2)ej(t-τ(t))+∑j=1N‍ej(t-τc(t))T(pˇ2∑i=1N‍Υi3+β1PΓ)ej(t-τc(t))+∑k=1n‍pkγk{e~k(t-τc(t))Te~k(t)T(A+(δˇ+b-)IN-Ξ+α1IN)e~k(t)ppppppppppppp+e~k(t)T(B-b-IN)e~k(t-τc(t))ppppppppppppp-β1e~k(t-τc(t))Te~k(t-τc(t))}.
From condition (16), we obtain
(22)ℒV(t,e(t))≤-aV(t)+bV(t-τ(t))+cV(t-τc(t)).
Define(23)W(t)≡W(t,e(t))=eγtV(t,e(t)),
whereγis the largest root of (18). Fort∈[tk,tk+1), we calculate the operator
(24)ℒW(t,e(t))=eγt[γV(t)+ℒV(t)]≤eγt[γV(t)-a1V(t)+bV(t-τ(t))ppppp+cV(t-τc(t))].
Therefore, by the generalized Itô formula, we have that, for anyt>t0≥0,
(25)W(t)=W(t0)+𝔼∫t0t‍ℒW(s)ds.
Fort∈[0,t1), we have
(26)eγt𝔼V(t)≤𝔼V(0)+𝔼∫0t‍eγs[γV(s)-aV(s)+bV(s-τ(s))ppppppppppp+cV(s-τc(s))]ds≤pˇ2∑i=1N‍𝔼∥ξi∥2+(γ-a)∫0t‍eγs𝔼V(s)ds+beγτ∫0t‍eγ(s-τ(s))𝔼V(s-τ(s))ds+ceγτc∫0t‍eγ(s-τc(s))𝔼V(s-τc(s))ds.
By changing variables-τ(s)=u, we have
(27)∫0t‍eγ(s-τ(s))𝔼V(s-τ(s))ds=∫-τ(0)t-τ(t)‍eγu𝔼V(u)du1-τ˙(s)≤∫-τt‍eγu𝔼V(u)du1-τ-≤pˇ2τ1-τ-∑i=1N‍𝔼∥ξi∥2+11-τ-∫0t‍eγu𝔼V(u)du.
Substituting variables-τc(s)=ugives
(28)∫0t‍eγ(s-τc(s))𝔼V(s-τc(s))ds=∫-τc(0)t-τc(t)‍eγu𝔼V(u)du1-τ˙c(t)≤pˇ2τc1-τ-c∑i=1N‍𝔼∥ξi∥2+11-τ-c∫0t‍eγu𝔼V(u)du.
Taking (27) and (28) into (26) and by condition (18), we obtain
(29)𝔼V(t)≤pˇ2(1+bτ1-τ-eγτ+cτc1-τ-ceγτc)∑i=1N‍𝔼∥ξi∥2e-γt.
Whent∈[t1,t2), we have𝔼V(t1)=ν2𝔼V(t1-)and
(30)eγt𝔼V(t)≤eγt1𝔼V(t1)+𝔼∫t1t‍eγs[γV(s)-aV(s)+bV(s-τ(s))ppppppipppp+cV(s-τc(s))]ds≤eγt1𝔼V(t1)+b1-τ-eγτ∫t1-τt1‍eγs𝔼V(s)ds+c1-τ-ceγτc∫t1-τct1‍eγs𝔼V(s)ds≤{pˇ2(1+bτ1-τ-eγτ+cτc1-τ-ceγτc)×(ν2+bτ1-τ-eγτ+cτc1-τ-ceγτc)}∑i=1N‍𝔼∥ξi∥2.
Similarly, we can prove that, whent∈[tk,tk+1), we have
(31)eγt𝔼V(t)≤{(ν2+bτ1-τ-eγτ+cτc1-τ-ceγτc)kpˇ2(1+bτ1-τ-eγτ+cτc1-τ-ceγτc)×(ν2+bτ1-τ-eγτ+cτc1-τ-ceγτc)k}∑i=1N‍𝔼∥ξi∥2.
If0<ν2+(bτ/(1-τ-))eγτ+(cτc/(1-τ-c))eγτc<1, we get
(32)𝔼V(t)≤{(bτ1-τ-eγτ)kpˇ2(1+bτ1-τ-eγτ+cτc1-τ-ceγτc)×(ν2+bτ1-τ-eγτ+cτc1-τ-ceγτc)k}∑i=1N‍𝔼∥ξi∥2e-γt≤{(bτ1-τ-eγτ)kpˇ2(1+bτ1-τ-eγτ+cτc1-τ-ceγτc)×(ν2+bτ1-τ-eγτ+cτc1-τ-ceγτc)t/Tb-N0}×∑i=1N‍𝔼∥ξi∥2e-γt={(bτ1-τ-eγτ)kpˇ2(1+bτ1-τ-eγτ+cτc1-τ-ceγτc)×(ν2+bτ1-τ-eγτ+cτc1-τ-ceγτc)-N0}×e((ln(ν2+(bτ/(1-τ-))eγτ+(cτc/(1-τ-c))eγτc))/Tb)t∑i=1N‍𝔼∥ξi∥2e-γt≤Ke(((ln(ν2+(bτ/(1-τ-))eγτ+(cτc/(1-τ-c))eγτc))/Ta)-γ)t,
where
(33)K={pˇ2(1+bτ1-τ-eγτ+cτc1-τ-ceγτc)(ν2+bτ1-τ-eγτ+cτc1-τ-ceγτc)-N0×(ν2+bτ1-τ-eγτ+cτc1-τ-ceγτc)-N0}∑i=1N‍𝔼∥ξi∥2e-γt.
The proof is then completed.Remark 8.
In Theorem7, we can see that the speed of stabilizing process depends on the dynamical structure and stochastic perturbations of the complex network, as well as the impulsive effects|ν|and the average impulsive intervalTa. The smaller|ν|andTaare, the higher the speed of the stabilizing process will be.When there is no time-varying delay coupling, the network (1) can be changed as
(34)dxi(t)={fμi(t,xi(t),xi(t-τ(t)))+∑j=1,i≠jNaijΓ(xj(t)-xi(t))fμi(t,xi(t),xi(t-τ(t)))}dt+σi(t,x(t),x(t-τ(t)))dwi(t),i=1,2,…,N,t≥0,t≠tk,k∈ℕ,xj(tk+)-xi(tk+)=ν(xj(tk-)-xi(tk-))fork∈ℕ(i,j):aij>0.Corollary 9.
Suppose that the assumptions (H1), (H2), and (H3) hold andfμi∈QUAD(P,Δ,η,ζ). If there exist positive constantsαsuch that
(35)A+δˇIN+αIN-Ξ≤0,0≤τ-≤1-ba,
whereγ>0is the smallest root of the equation
(36)γ-a+b1-τ-eγτ=0,a=λmin(2ηIn-pˇ∑i=1NΥi1+2αPΓ)pˇ,b=λmax(∑i=1NPΥi2+2ζIN)p^,
then the solutionx1(t),x2(t),…,xN(t)of system (34) could reach cluster synchronization under the controller (5) when the average impulsive interval is less thanTaand0<ν2+(bτ/(1-τ-))eγτ<1.When the time-varying delays are constant (i.e.,τ(t)=τ,τc(t)=τc), we get the following corollary.Corollary 10.
Suppose the assumptions (H1), (H2), and (H3) hold andfμi∈QUAD(P,Δ,η,ζ). If there exist positive constantsαandβ, such that
(37)[A+(δˇ+b-)IN-Ξ+αINB-b-IN2BT-b-IN2-βIN]≤0,
whereγ>0is the smallest root of the equation
(38)γ-a+beγτ+ceγτc=0,a=λmin(2ηIn-pˇ∑i=1NΥi1+2αPΓ)pˇ,b=λmax(∑i=1NPΥi2+2ζIN)p^,c=λmax(∑i=1NPΥi3+2βPΓ)p^,
then the solutionx1(t),x2(t),…,xN(t)of system (4) could reach cluster synchronization under controller (5) when the average impulsive interval is less thanTaand0<ν2+bτeγτ+cτceγτc<1.


## 4. Numerical Simulation

In this section, we give numerical simulation to verify the theorem given in the previous section.Consider the following chaotic delayed neural networks:(39)fi(t,x(t),x(t-τ(t)))=-Cx(t)+Aif(x(t))+Big(x(t-τ(t))),
wheref(x)=g(x)=tanh(x),τ(t)=1,
(40)C=[1001],A1=[2-0.1-54.5],B1=[-1.5-0.1-0.2-4],A2=[2-0.2-54.5],B2=[-1.2-0.1-0.1-4].
TakeP=diag{1,2}andΔ=diag{5,11,5}and we haveη=0.15,ζ=3.25. Hence the condition (9) can be satisfied fori=1,2(see [6]).In order to verify our results, we consider the following complex network:(41)dxi(t)={∑j=1,i≠j10‍fμi(t,xi(t),xi(t-τ(t)))+∑j=1,i≠j10‍aijΓ(xj(t)-xi(t))+∑j=1,i≠j10‍bijΓ(xj(t-τc(t))-xi(t))}dt+σi(t,x(t),x(t-τ(t)),x(t-τc(t)))dwi(t),i=1,2,…,20,
where
(42)Γ=[1001],τc(t)=0.1et1+et,σi(t,x(t),x(t-τ(t)),x(t-τc(t)))=0.1×diag{xi1(t),xi2(t)}.
In this simulation, we consider a network with the topological structure of Figure 1 for 20 nodes with 2 communities. The nodes numberedi=1,2,…,10belong to the first community, while the rest of the nodes belong to the second community. By the pinning scheme above, we pin the nodes1,7,8,9,10,11,12,13,14, and20.The controlled complex network (41). (a) The topological structure of matrix A. (b) The topological structure of matrix B.
(a)
(b)Through computation, we getτ=1,τ-=0,τc=0.1,τ-c=0.1, andΥij=0.01I2fori=1,2,…,Nandj=1,2. Let the control strengthεi=90. By using the Matlab LMI toolbox, we can obtain the following solution for the conditions of Theorem 7:a=67.5,b=5.3,c=0.7,α=45.5,β=0.2,T=0.8, andν=0.7.The initial conditions of the numerical simulations are as follows:xij(t)=ξij,i=1,2,…,10,j=1,2, for allt∈[-1,0], whereξijare chosen randomly in[-10,10]. By using the Euler-Maruyama method, the simulation is carried out withΔ=0.01. The trajectories of the state variables ofxi1andxi2(i=1,2,…,10)in system (41) under pinning control are shown in Figure 2. And Figure 3 exhibits the time evolutions of synchronization errors under pinning control.(a) The trajectories of the state variables ofxi1(i=1,2,…,10)in system (41) under pinning control; (b) The trajectories of the state variables ofxi2(i=1,2,…,10)in system (41) under pinning control.
(a)
(b)(a) The time evolution ofei1(i=1,2,…,10) in system (41) under pinning control. (b) The time evolution ofei2(i=1,2,…,10) in system (41) under pinning control.
(a)
(b)


## 5. Conclusion

In this paper, we investigated the cluster synchronization of impulsive complex networks with time-varying delay coupling and stochastic perturbations. Specifically, we achieved global exponential synchronization by applying pinning control scheme to a small fraction of nodes and derived sufficient conditions for the global exponential stability of synchronization. Finally, for clarity of exposition, a numerical example was considered to illustrate the theoretical analysis by using Matlab.



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*Source: 678543-2013-12-02.xml*