# The Asymptotic Synchronization Analysis for Two Kinds of Complex Dynamical Networks

Authors: Ze Tang; Jianwen Feng
Journal: Advances in Mathematical Physics
 (2012)

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## Abstract

We consider a class of complex networks with both delayed and nondelayed coupling. In particular, we consider the situation for both time delay-independent and time delay-dependent complex dynamical networks and obtain sufficient conditions for their asymptotic synchronization by using the Lyapunov-Krasovskii stability theorem and the linear matrix inequality (LMI). We also present some simulation results to support the validity of the theories.


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## Body

## 1. Introduction

A complex dynamical network is a large set of interconnected nodes that represent the individual elements of the system and their mutual relationships. Owing to their immense potential for applications to various fields, complex networks have been intensively investigated in the past decade in areas as diverse as mathematics, physics, biology, engineering, and even the social sciences [1–3]. The synchronization problem for complex networks was first posed by Saber and Murray [4, 5] who also introduced a theoretical framework for their investigation by viewing them as the adjustments of the rhythms of their interaction states [6] and many different kinds of synchronization phenomena and models have also been discovered such as complete synchronization, phase synchronization, lag synchronization, antisynchronization, impulsive synchronization, and projective synchronization.Time delays are an important consideration for complex networks although these were usually ignored in early investigations of synchronization and control problems [6–11]. To make up for this deficiency, uniformly distributed time delays have recently been incorporated into network models [12–25] and Wang et al. [18] even considered networks with both delayed and nondelayed couplings and obtained sufficient conditions for asymptotic stability. Similarly, Wu and Lu [19] investigated the exponential synchronization of general weighted delay and nondelay coupled complex dynamical networks with different topological structures. There remains, however, much room for improvement in both the scope of the systems considered by Wang and Xu as well as in their methods of proofs.The main contributions of this paper are two-fold. Firstly, we present a more general model for networks with both delayed and nondelayed couplings and derive criteria for their asymptotical synchronization. Secondly, we apply the Lyapunov-Krasovskii theorem and the LMIs to ensure the inevitable attainment of the required synchronization.The rest of the paper is organized as follows. In Section2, we present the general complex dynamical network model under consideration and state some preliminary definitions and results. In Section 3, we present the main results of this paper. In particular, we consider the situation for both time delay-independent and time delay-dependent complex dynamical networks and derive sufficient conditions for their asymptotic synchronization by using the Lyapunov-Krasovskii stability theorem and the linear matrix inequality (LMI). In Section 4, we present some numerical simulation results that verify our theoretical results. The paper concludes in Section 5.


## 2. Preliminaries and Model Description

In general, a linearly  coupled ordinary differential equation system (LCODES) can be described as follows:(2.1)dxi(t)dt=f(xi(t))+c1∑j≠i,j=1N‍bijAxj(t)+c2∑j≠i,j=1N‍bij′A′xj(t-τ).
Since xi-xi=0 for all i=1,…,N, we can choose any values for aii in the above equations. Hence, letting bii=-∑j≠i,j=1N‍bij and bii′=-∑j≠i,j=1N‍bij′, the above equations can be rewritten as follows:
(2.2)dxi(t)dt=f(xi(t))+c1∑j=1N‍bijAxj(t)+c2∑j=1N‍bij′A′xj(t-τ),
where N is the number of nodes, xi(t)=(xi1,xi2,…,xiN)T∈Rn are the state variables of the ith node, t∈[0,+∞) and f:Rn→Rn is a continuously differentiable function. The constants c1 and c2 (possibly distinct) are the coupling strengths, bij≥0, bij′>0 (for i,j=1,…,N), A,A′∈Rn×n are inner-coupled matrices, B,B′∈Rn×n are coupled matrices with zero-sum rows with bij,bij′≥0 for i≠j that determines the topological structure of the network. We assume that B and B′ are symmetric and irreducible matrices so that there are no isolated nodes in the system.If all the eigenvalues of a matrixA∈Rn×n are real, then we denote its ith eigenvalue by λi(A) and sort them by λ1(A)≤λ2(A)≤⋯≤λn(A). A symmetric real matrix A is positive definite (semidefinite) if xTAx>0(≥0) for all nonzero x and denoted by A>0(A≥0). Finally, I stands for the identity matrix and the dimensions of all vectors and matrices should be clear in the context.Definition 2.1.
A complex network with delayed and nondelayed coupling (2.2) is said to achieve asymptotic synchronization if
(2.3)x1(t)=x2(t)=⋯=xN(t)=s(t),t→+∞,
where s(t) is a solution of the local dynamics of an isolated node satisfying s(t)˙=f(s(t)).Definition 2.2.
A matrixL=(lij)i,j=1N is said to belong to the class A1, denoted by L∈A1 if(1)
lij≤0, i≠j, lii=-∑j=1,j≠iN‍lij, i=1,2,…,N,(2)
L  is  irreducible.IfL∈A1 is symmetrical, then we say that L belongs to the class A2, denoted by L∈A2.Lemma 2.3 (see [26]).
IfL∈A1, then rank(L)=N-1, that is, 0 is an eigenvalue of L with multiplicity 1, and all the nonzero eigenvalues of L have positive real part.Lemma 2.4 (Wang and Chen [11]).
IfG=(gij)N×N satisfies the above conditions, then there exists a unitary matrix Φ=(ϕ1,…,ϕN) such that
(2.4)GTϕk=λkϕk,k=1,2,…,N,
where λi,i=1,2,…,N, are the eigenvalues of G.Lemma 2.5 (Schur complement [22]).
The linear matrix inequality (LMI)(2.5)(Q(x)S(x)S(x)TR(x))>0,
where Q(x) and R(x) are symmetric matrices and S(x) is a matrix with suitable dimensions is equivalent to one of the following conditions:(i)
Q(x)>0, R(x)-S(x)TQ(x)-1S(x)>0;(ii)
R(x)>0, Q(x)-S(x)R(x)-1S(x)>0.Lemma 2.6 ((the Lyapunov-Krasovskii stability theorem). (Kolmanovskii and Myshkis, Hale and Verduyn Lunel [16])).
Consider the delayed differential equation(2.6)x(t)=f˙(t,x(t)),
where f:R×C→Rn is continuous and takes R× (bounded subsets of C) into bounded subsets of Rn, and let u,v,w:R+→R+ be continuous and strictly monotonically nondecreasing functions with u(s), v(s), w(s) being positive for s>0 and u(0)=v(0)=0. If there exists a continuous functional V:R×C→R such that
(2.7)u(∥x∥)≤V(t,x)≤v(∥x∥),V˙(t,x(t,x(t)))≤-w(∥x(t)∥),
where V˙ is the derivative of V along the solutions of the above delayed differential equation, then the solution x=0 of this equation is uniformly asymptotically stable.Remark 2.7.
The functionalV is called a Lyapunov-Krasovskii functional.Lemma 2.8 (Moon et al. [22]).
Leta(·)∈Rna, b(·)∈Rnb and M(·)∈Rna×nb be defined on an interval Ω. Then, for any matrices X∈Rna×na, Y∈Rna×nb, and Z∈Rnb×nb, one has
(2.8)-2∫Ω‍a(x)TMb(x)dx≤∫Ω‍[a(x)b(x)]T[XY-MYT-MTZ][a(x)b(x)]dx,
where
(2.9)[XYYTZ]≥0.Lemma 2.9.
For all positive-definite matricesP and vectors x and y, one has
(2.10)-2xTy≤infP>0{xTPx+yTP-1y}.Lemma 2.10 (see [16]).
Consider the delayed dynamical network (2.2). Let
(2.11)0=λ1>λ2≥λ3≥⋯≥λN,0=μ1>μ2≥μ3≥⋯≥μN
be the eigenvalues of the outer-coupling matrices B and B', respectively. If the n-dimensional linear time-delayed and nontime delayed system
(2.12)w˙i(t)=J(t)wi(t)+c1λiAwi(t)+c2μiA′wi(t-τ),k=2,3,…,N,
of N-1 differential equations is asymptotically stable about their zero solutions for some Jacobian matrix J(t)∈Rn×n of f(x(t)) at s(t), then the synchronized states (2.3) are asymptotically stable.


## 3. The Criteria for Asymptotic Synchronization

In this section, we derive the conditions for the asymptotic synchronization of time-delayed coupled dynamical networks when they are either time-dependent or time-independent.


### 3.1. Case 1:  The Time Delay-Independent Stability Criterion

Theorem 3.1.
Consider the general time delayed and non-time delayed complex dynamical network (2.2). If there exist two positive definite matrices P and Q>0 such that
(3.1)[J(t)TP+PJ(t)+2c1λiA+Qc2μNPA'c2μNA'P-Q]>0,
then the synchronization manifold (2.3) of network (2.2) can be asymptotically synchronized for all fixed time delay τ>0.Proof.
For each fixedi=1,2,…,N, choose the Lyapunov-Krasovskii functional
(3.2)Vi(t)=wi(t)TPwi(t)+∫t-τt‍wi(s)TQwi(s)ds
for some matrices P>0 and Q>0 to be determined. Then the derivative of Vi(t) along the trajectories of (3.2) is
(3.3)dVi(t)dt=w˙i(t)TPwi(t)+wi(t)TPw˙i(t)+wi(t)TQwi(t)-wi(t-τ)TQwi(t-τ)
which, upon substitution of (2.12), gives
(3.4)V˙i(t)=[J(t)wi(t)+c1λiAwi(t)+c2μiA′wi(t-τ)]TPwi(t)+wi(t)TP×[J(t)wi(t)+c1λiAwi(t)+c2μiA'wi(t-τ)]+wi(t)TQwi(t)-wi(t-τ)TQwi(t-τ)=wi(t)TJ(t)Twi(t)+wi(t)TλiATPwi(t)+wi(t-τ)TμiA′Pwi(t)+wi(t)TPJ(t)wi(t)+wi(t)Tc1λiAwi(t)+wi(t)TPc2μiA′wi(t-τ)+wi(t)TQwi(t)-wi(t-τ)TQwi(t-τ)=wi(t)T[J(t)TP+PJ(t)+2c1λiA+Q]wi(t)+2wi(t)Tc2μiPA′wi(t-τ)-wi(t-τ)TQwi(t-τ).
Now, by using the inequality in Lemma 2.9, we have
(3.5)2wi(t)Tc2μiPA'wi(t-τ)≤wi(t-τ)TQwi(t-τ)+wi(t)Tc22μi2PA′QA′Pwi(t),
which, upon substituting (3.5) into (3.2), gives
(3.6)V˙i(t)≤wi(t)T[J(t)TP+PJ(t)+2c1λiA+c2μi2wi(t)TPA′QA′Aw(t)+Q]wi(t).
It therefore follows from the Schur complement (Lemma 2.5) and the linear matrix inequality (3.1) that V˙i(t)<0 for all the N-1 equations in the general time delayed and non-time delayed system (2.12) and hence the system (2.12) is asymptotically synchronized by the Lyapunov-Krasovskii stability theorem. So, by Theorem 3.1, the synchronization manifold (2.3) of the network(2.2) is asymptotically synchronized. This completes the proof of the theorem.The following corollaries follow immediately from the above theorem.Corollary 3.2.
Consider the general non-time delayed complex dynamical network(3.7)x˙i(t)=f(xi(t))+c1∑j=1N‍bijAxj(t).
If there exists a positive definite matrix P>0 such that
(3.8)J(t)TP+c1λiAP<0,
then the synchronization manifold (2.3) of network (3.7) can be asymptotically synchronized.Proof.
From Lemma2.10, we have
(3.9)w˙i(t)=J(t)wi(t)+c1λiAwi(t)
and the result follows by choosing the Lyapunov functional Vi(t)=(1/2)wi(t)TPwi(t).Corollary 3.3.
Consider the general time delayed complex dynamical network(3.10)x˙i(t)=f(xi(t))+c2∑j=1N‍bij′A′xj(t-τ).
If there exist two positive definite matrices P>0 and Q>0 such that
(3.11)[J(t)TP+PJ(t)+Qc2μNPA′c2μNA'P-Q]<0,
then the synchronization manifold (2.3) of network (3.10) can be asymptotically synchronized.Remark 3.4.
The results of [16] are obtainable as particular cases of Theorem 3.1.Remark 3.5.
The above analysis is applicable to a general system with arbitrary time delays. A simpler synchronization scheme, however, could be applied to systems with time delays that are already known and are small in value.


### 3.2. Case 2:  The Criterion for Time Delay-Dependent Stability

Theorem 3.6.
Consider the general time delayed and non-time delayed complex dynamical network (2.2) with a fixed time delay τ∈(0,h] for some small h. If there exist three positive definite matrices P,Q,Z>0, such that
(3.12)[(1,1)Pc2μiA'-Y+h(J(t)+c1λiA)TZc2μiA′c2μiA'TP-YT+hc2μiA'TZ(J(t)+c1λiA)hc2μi2A'TZA'-Q]<0
with
(3.13)(1,1)=(J(t)+c1λiA+c2μiA′)TP+P(J(t)+c1λiA+c2μiA′)+hX+YT+Y+Q+h(J(t)+c1λiA)T(J(t)+c1λiA)[XYYTZ]≥0,
then the synchronization manifold (2.3) of network (2.2) can be asymptotically synchronized.Proof.
For each fixedi=1,2,…,N, choose the Lyapunov-Krasovskii functional
(3.14)Vi(t)=wi(t)TPwi(t)+∫t-τt‍wi(s)TQwi(s)+∫-τ0‍∫t+βt‍w˙i(s)TZw˙i(s)dsdβ
for some matrices P,Q,Z>0 to be determined and let
(3.15)V1=w˙i(t)TPwi(t),V2=∫t-τt‍wi(s)TQwi(s),V3=∫-τ0‍∫t+βt‍w˙i(s)TZw˙i(s)dsdβ.
Then, Vi(t)=V1+V2+V3 and it follows from the Newton-Leibniz equation that
(3.16)∫t-τt‍w˙i(ξ)dξ=wi(t)-wi(t-τ)
so that (2.12) can be transformed into
(3.17)w˙i(t)=(J(t)+c1λiA+c2μiA′)wi(t)-c2μiA'∫t-τt‍w˙i(s)ds.
Hence
(3.18)V1=w˙i(t)TPwi(t)+wi(t)TPw˙i(t)=wi(t)T[(J(t)+c1λiA+c2μiA′)TP+P(J(t)+c1λiA+c2μiA′)]wi(t)-2wi(t)TPc2μiA'∫t-τt‍w˙i(s)ds
and so, by Lemma 2.9, we have
(3.19)-2wi(t)TPc2μiA'∫t-τt‍w˙i(s)ds=-2∫t-τt‍wi(t)T(Pc2μiA')w˙i(s)ds≤∫t-τt‍[wi(t)w˙i(s)]T[XY-Pc2μiA'YT-A'Tc2μiPZ][wi(t)w˙i(s)]dx=∫t-τt‍wi(t)TXwi(t)ds+∫t-τt‍w˙i(s)TZw˙i(s)ds+2∫t-τt‍wi(t)T(Y-Pc2μiA′)w˙i(s)ds=τwi(t)TXwi(t)+2wi(t)T(Y-Pc2μiA′)∫t-τt‍w˙i(s)ds+∫t-τt‍w˙i(s)TZw˙i(s)=τwi(t)TXwi(t)+2wi(t)T(Y-Pc2μiA′)wi(t)-2wi(t)T(Y-Pc2μiA′)wi(t-τ)+∫t-τt‍w˙i(s)Zw˙i(s)
and so
(3.20)V1≤wi(t)T[(J(t)+c1λiA+c2μiA′)TP+P(J(t)+c1λiA+c2μiA′)]wi(t)+2wi(t)T(Pc2μiA′-Y)wi(t-τ)+∫t-τt‍w˙i(s)TZwi(s)ds.
Similarly, we have
(3.21)V2=wi(t)TQwi(t)-wi(t-τ)TQwi(t-τ),V3=τw˙i(t)TZw˙i(t)-∫t-τt‍w˙i(s)TZw˙i(s)ds≤h[(J(t)+c1λiA)wi(t)+c2μiA′wi(t-τ)]T×Z[(J(t)+c1λiA)wi(t)+c2μiA′wi(t-τ)]-∫t-τt‍w˙i(s)TZw˙i(s)ds=hwi(t)T(J(t)+c1λiA)TZ(J(t)+c1λiA)wi(t)+hwi(t)T×(J(t)t+c1λiA)TZc2μiA'wi(t-τ)+hwi(t-τ)Tc2μiA'TZT×(J(t)+c1λiA)wi(t)+hwi(t-τ)Tμi2A'TZA′wi(t-τ)-∫t-τt‍w˙i(s)TZw˙i(s)ds=hwi(t)T(J(t)+c1λiA)TZ(J(t)+c1λiA)wi(t)+2hwi(t)T×(J(t)+c1λiA)TZc2μiA'wi(t-τ)+hwi(t-τ)Tμi2A'TZA′wi(t-τ)-∫t-τt‍w˙i(s)TZw˙i(s)ds
and so
(3.22)V˙i(t)=V1+V2+V3≤wi(t)T[(J(t)+c1λiA+c2μiA′)TP+P(J(t)+c1λiA+c2μiA′)+hX+YT+Y+Q+h(J(t)+c1λiA)T(J(t)+c1λiA)]wi(t)+wi(t-τ)T[hμi2A'TZA′-Q]wi(t-τ)+2wi(t)T[(Pc2μiA′-Y)+h(J(t)+c1λi)TZc2μiA′]wi(t-τ).
Finally, we have(3.23)V˙i(t)≤[wi(t)wi(t-τ)]T[(1,1)(1,2)(2,1)hc2μ2A'TZAQ′][wi(t)wi(t-τ)],
where
(3.24)(1,1)=(J(t)+c1λiA+c2μiA′)TP+P(J(t)+c1λiA+c2μiA′)+hX+YT+Y+Q+h(J(t)+c1λiA)T(J(t)+c1λiA)(1,2)=Pc2μiA′-Y+h(J(t)+c1λiA)TZc2μiA′(2,1)=c2μiA'TP-YT+hc2μiA'TZ(J(t)+c1λiA)[XYYTZ]≥0.
It now follows from Lemma 2.5 that the conditions of the theorem are equivalent to V˙i(t)<0 and that by the Lyapunov-Krasovskii Stability Theorem all the nodes of the system (2.12) are asymptotically stable when (3.12) and (3.13) hold for i=1,2,…,N. This completes the proof of Theorem 3.6.Corollary 3.7.
Consider the general time delayed complex dynamical network (3.10) with a fixed time delay τ∈(0,h](3.25)x˙i(t)=f(xi(t))+c2∑j=1N‍bij′A′xj(t-τ)
for some h<+∞. If there exist two positive definite matrices, P,Q>0, X, Y, and Z such that
(3.26)[(1,1)c2μiPA′-Y+hJ(t)TZc2μiA′c2μA'TP-YT+hc2μiA'TZJ(t)hc22μi2A'TZA′-Q]<0,
where (1,1)=PJ(t)+J(t)T+hX+YT+Y+Q+hJ(tTZJ(t)), then the synchronization manifold (2.3) of network (3.10) is asymptotic synchronization.Remark 3.8.
The proof can be found in [16]. Those are the two results of general complex dynamical network with fixed time-invariant delay τ∈(0,h] for some h<+∞; the conclusions are less conservative than the time-independent delay. The delay-dependent stability is another method applying to the delayed system. And it could provide a useful and meaningful upper bound of the delay h, which could ensure the delayed system achieves asymptotic synchronization only if the time delay is less than h.


## 3.1. Case 1:  The Time Delay-Independent Stability Criterion

Theorem 3.1.
Consider the general time delayed and non-time delayed complex dynamical network (2.2). If there exist two positive definite matrices P and Q>0 such that
(3.1)[J(t)TP+PJ(t)+2c1λiA+Qc2μNPA'c2μNA'P-Q]>0,
then the synchronization manifold (2.3) of network (2.2) can be asymptotically synchronized for all fixed time delay τ>0.Proof.
For each fixedi=1,2,…,N, choose the Lyapunov-Krasovskii functional
(3.2)Vi(t)=wi(t)TPwi(t)+∫t-τt‍wi(s)TQwi(s)ds
for some matrices P>0 and Q>0 to be determined. Then the derivative of Vi(t) along the trajectories of (3.2) is
(3.3)dVi(t)dt=w˙i(t)TPwi(t)+wi(t)TPw˙i(t)+wi(t)TQwi(t)-wi(t-τ)TQwi(t-τ)
which, upon substitution of (2.12), gives
(3.4)V˙i(t)=[J(t)wi(t)+c1λiAwi(t)+c2μiA′wi(t-τ)]TPwi(t)+wi(t)TP×[J(t)wi(t)+c1λiAwi(t)+c2μiA'wi(t-τ)]+wi(t)TQwi(t)-wi(t-τ)TQwi(t-τ)=wi(t)TJ(t)Twi(t)+wi(t)TλiATPwi(t)+wi(t-τ)TμiA′Pwi(t)+wi(t)TPJ(t)wi(t)+wi(t)Tc1λiAwi(t)+wi(t)TPc2μiA′wi(t-τ)+wi(t)TQwi(t)-wi(t-τ)TQwi(t-τ)=wi(t)T[J(t)TP+PJ(t)+2c1λiA+Q]wi(t)+2wi(t)Tc2μiPA′wi(t-τ)-wi(t-τ)TQwi(t-τ).
Now, by using the inequality in Lemma 2.9, we have
(3.5)2wi(t)Tc2μiPA'wi(t-τ)≤wi(t-τ)TQwi(t-τ)+wi(t)Tc22μi2PA′QA′Pwi(t),
which, upon substituting (3.5) into (3.2), gives
(3.6)V˙i(t)≤wi(t)T[J(t)TP+PJ(t)+2c1λiA+c2μi2wi(t)TPA′QA′Aw(t)+Q]wi(t).
It therefore follows from the Schur complement (Lemma 2.5) and the linear matrix inequality (3.1) that V˙i(t)<0 for all the N-1 equations in the general time delayed and non-time delayed system (2.12) and hence the system (2.12) is asymptotically synchronized by the Lyapunov-Krasovskii stability theorem. So, by Theorem 3.1, the synchronization manifold (2.3) of the network(2.2) is asymptotically synchronized. This completes the proof of the theorem.The following corollaries follow immediately from the above theorem.Corollary 3.2.
Consider the general non-time delayed complex dynamical network(3.7)x˙i(t)=f(xi(t))+c1∑j=1N‍bijAxj(t).
If there exists a positive definite matrix P>0 such that
(3.8)J(t)TP+c1λiAP<0,
then the synchronization manifold (2.3) of network (3.7) can be asymptotically synchronized.Proof.
From Lemma2.10, we have
(3.9)w˙i(t)=J(t)wi(t)+c1λiAwi(t)
and the result follows by choosing the Lyapunov functional Vi(t)=(1/2)wi(t)TPwi(t).Corollary 3.3.
Consider the general time delayed complex dynamical network(3.10)x˙i(t)=f(xi(t))+c2∑j=1N‍bij′A′xj(t-τ).
If there exist two positive definite matrices P>0 and Q>0 such that
(3.11)[J(t)TP+PJ(t)+Qc2μNPA′c2μNA'P-Q]<0,
then the synchronization manifold (2.3) of network (3.10) can be asymptotically synchronized.Remark 3.4.
The results of [16] are obtainable as particular cases of Theorem 3.1.Remark 3.5.
The above analysis is applicable to a general system with arbitrary time delays. A simpler synchronization scheme, however, could be applied to systems with time delays that are already known and are small in value.


## 3.2. Case 2:  The Criterion for Time Delay-Dependent Stability

Theorem 3.6.
Consider the general time delayed and non-time delayed complex dynamical network (2.2) with a fixed time delay τ∈(0,h] for some small h. If there exist three positive definite matrices P,Q,Z>0, such that
(3.12)[(1,1)Pc2μiA'-Y+h(J(t)+c1λiA)TZc2μiA′c2μiA'TP-YT+hc2μiA'TZ(J(t)+c1λiA)hc2μi2A'TZA'-Q]<0
with
(3.13)(1,1)=(J(t)+c1λiA+c2μiA′)TP+P(J(t)+c1λiA+c2μiA′)+hX+YT+Y+Q+h(J(t)+c1λiA)T(J(t)+c1λiA)[XYYTZ]≥0,
then the synchronization manifold (2.3) of network (2.2) can be asymptotically synchronized.Proof.
For each fixedi=1,2,…,N, choose the Lyapunov-Krasovskii functional
(3.14)Vi(t)=wi(t)TPwi(t)+∫t-τt‍wi(s)TQwi(s)+∫-τ0‍∫t+βt‍w˙i(s)TZw˙i(s)dsdβ
for some matrices P,Q,Z>0 to be determined and let
(3.15)V1=w˙i(t)TPwi(t),V2=∫t-τt‍wi(s)TQwi(s),V3=∫-τ0‍∫t+βt‍w˙i(s)TZw˙i(s)dsdβ.
Then, Vi(t)=V1+V2+V3 and it follows from the Newton-Leibniz equation that
(3.16)∫t-τt‍w˙i(ξ)dξ=wi(t)-wi(t-τ)
so that (2.12) can be transformed into
(3.17)w˙i(t)=(J(t)+c1λiA+c2μiA′)wi(t)-c2μiA'∫t-τt‍w˙i(s)ds.
Hence
(3.18)V1=w˙i(t)TPwi(t)+wi(t)TPw˙i(t)=wi(t)T[(J(t)+c1λiA+c2μiA′)TP+P(J(t)+c1λiA+c2μiA′)]wi(t)-2wi(t)TPc2μiA'∫t-τt‍w˙i(s)ds
and so, by Lemma 2.9, we have
(3.19)-2wi(t)TPc2μiA'∫t-τt‍w˙i(s)ds=-2∫t-τt‍wi(t)T(Pc2μiA')w˙i(s)ds≤∫t-τt‍[wi(t)w˙i(s)]T[XY-Pc2μiA'YT-A'Tc2μiPZ][wi(t)w˙i(s)]dx=∫t-τt‍wi(t)TXwi(t)ds+∫t-τt‍w˙i(s)TZw˙i(s)ds+2∫t-τt‍wi(t)T(Y-Pc2μiA′)w˙i(s)ds=τwi(t)TXwi(t)+2wi(t)T(Y-Pc2μiA′)∫t-τt‍w˙i(s)ds+∫t-τt‍w˙i(s)TZw˙i(s)=τwi(t)TXwi(t)+2wi(t)T(Y-Pc2μiA′)wi(t)-2wi(t)T(Y-Pc2μiA′)wi(t-τ)+∫t-τt‍w˙i(s)Zw˙i(s)
and so
(3.20)V1≤wi(t)T[(J(t)+c1λiA+c2μiA′)TP+P(J(t)+c1λiA+c2μiA′)]wi(t)+2wi(t)T(Pc2μiA′-Y)wi(t-τ)+∫t-τt‍w˙i(s)TZwi(s)ds.
Similarly, we have
(3.21)V2=wi(t)TQwi(t)-wi(t-τ)TQwi(t-τ),V3=τw˙i(t)TZw˙i(t)-∫t-τt‍w˙i(s)TZw˙i(s)ds≤h[(J(t)+c1λiA)wi(t)+c2μiA′wi(t-τ)]T×Z[(J(t)+c1λiA)wi(t)+c2μiA′wi(t-τ)]-∫t-τt‍w˙i(s)TZw˙i(s)ds=hwi(t)T(J(t)+c1λiA)TZ(J(t)+c1λiA)wi(t)+hwi(t)T×(J(t)t+c1λiA)TZc2μiA'wi(t-τ)+hwi(t-τ)Tc2μiA'TZT×(J(t)+c1λiA)wi(t)+hwi(t-τ)Tμi2A'TZA′wi(t-τ)-∫t-τt‍w˙i(s)TZw˙i(s)ds=hwi(t)T(J(t)+c1λiA)TZ(J(t)+c1λiA)wi(t)+2hwi(t)T×(J(t)+c1λiA)TZc2μiA'wi(t-τ)+hwi(t-τ)Tμi2A'TZA′wi(t-τ)-∫t-τt‍w˙i(s)TZw˙i(s)ds
and so
(3.22)V˙i(t)=V1+V2+V3≤wi(t)T[(J(t)+c1λiA+c2μiA′)TP+P(J(t)+c1λiA+c2μiA′)+hX+YT+Y+Q+h(J(t)+c1λiA)T(J(t)+c1λiA)]wi(t)+wi(t-τ)T[hμi2A'TZA′-Q]wi(t-τ)+2wi(t)T[(Pc2μiA′-Y)+h(J(t)+c1λi)TZc2μiA′]wi(t-τ).
Finally, we have(3.23)V˙i(t)≤[wi(t)wi(t-τ)]T[(1,1)(1,2)(2,1)hc2μ2A'TZAQ′][wi(t)wi(t-τ)],
where
(3.24)(1,1)=(J(t)+c1λiA+c2μiA′)TP+P(J(t)+c1λiA+c2μiA′)+hX+YT+Y+Q+h(J(t)+c1λiA)T(J(t)+c1λiA)(1,2)=Pc2μiA′-Y+h(J(t)+c1λiA)TZc2μiA′(2,1)=c2μiA'TP-YT+hc2μiA'TZ(J(t)+c1λiA)[XYYTZ]≥0.
It now follows from Lemma 2.5 that the conditions of the theorem are equivalent to V˙i(t)<0 and that by the Lyapunov-Krasovskii Stability Theorem all the nodes of the system (2.12) are asymptotically stable when (3.12) and (3.13) hold for i=1,2,…,N. This completes the proof of Theorem 3.6.Corollary 3.7.
Consider the general time delayed complex dynamical network (3.10) with a fixed time delay τ∈(0,h](3.25)x˙i(t)=f(xi(t))+c2∑j=1N‍bij′A′xj(t-τ)
for some h<+∞. If there exist two positive definite matrices, P,Q>0, X, Y, and Z such that
(3.26)[(1,1)c2μiPA′-Y+hJ(t)TZc2μiA′c2μA'TP-YT+hc2μiA'TZJ(t)hc22μi2A'TZA′-Q]<0,
where (1,1)=PJ(t)+J(t)T+hX+YT+Y+Q+hJ(tTZJ(t)), then the synchronization manifold (2.3) of network (3.10) is asymptotic synchronization.Remark 3.8.
The proof can be found in [16]. Those are the two results of general complex dynamical network with fixed time-invariant delay τ∈(0,h] for some h<+∞; the conclusions are less conservative than the time-independent delay. The delay-dependent stability is another method applying to the delayed system. And it could provide a useful and meaningful upper bound of the delay h, which could ensure the delayed system achieves asymptotic synchronization only if the time delay is less than h.


## 4. Numerical Simulations

The above criteracould be applied to networks with different topologies and different size. We put two examples to illustrate the validity of the theories.Example 4.1.
We use a three-dimensional stable nonlinear system as an example to illustrate the main results, Theorem3.1, of our paper; this is the time delay-independent situation. The model could be described as follows:
(4.1)[x˙i1x˙i2x˙i3]=[-xi1+xi22-2xi2-3xi3+xi2xi3],i=1,2,3.
The solution of the 3-dimensional stable nonlinear system equations can be written as
(4.2)xi1=c1e-t-ce-4t,xi2=c2e-2t,xi3=c3e-3t-c2e2t2,
which is asymptotically stable at the equilibrium point of the system s(t)=0, where c=-c22/3 and c1, c2, c3 are all constants. It is easy to see that the Jacobian matrix is J=diag{-1,-2,-3}. We assume the inner-coupling matrices A, A′ are all identity matrices, namely, A=A'=diag{1,1,1}, and the outer coupling configuration matrices
(4.3)B=B′=[-21011-21001-211012].
The eigenvalues of the coupling matrices are λ(B)=λ(B′)={0,-1.5,-1.5}. We choose the coupling strength c1=0.5, c2=1. By using Theorem 3.1 and the LMI Toolbox in MATLAB, we obtained the following common two positive-definite matrices:
(4.4)P=diag{1.1204,12.3091,10.165},Q=diag{2.3710,22.5713,26.0849}.
According to the conditions in Theorem 3.1, we know the synchronized state s(t) is global asymptotically stable for any fixed delay. The quantity
(4.5)E(t)=(∑i=1N|xi(t)-s(t)|2N)
is used to measure the quality of the synchronization process. We plot the evolution of E(t) in the upper part in Figure 1. For the time delay here we choose τ=0.1. The lower subplot indicates the synchronization results of the network.Synchronization evolutionE(t) for the delay-independent network with c1=0.5, c2=1, and τ=0.1.
(a)
(b)Example 4.2.
We use a 4-nodes networks model as another example to illustrate the Theorem3.6; this is the time delay-dependent situation. The model could be described as follows:
(4.6)[x˙i1x˙i2x˙i3]=[-xi1-2xi2+xi32-3xi3+xi2xi3],i=1,2,3,4.
We choose the same coupling strength c1=0.5, c2=1; the eigenvalues of the coupling matrices are λ(B)=λ(B′)={0,-2,-2,-4}. By using Theorem 3.6 and the LMI Toolbox in MATLAB, we obtained the following matrices:
(4.7)P=diag{1.4078,1.4057,1.4054},Q={1.4157,1.4157,1.4157},Z={-1.4218,-1.4303,-1.4367},X={18.3024,19.6808,21.0815},Y={0.0221,0.0208,0.0196}.By using Theorem3.6 in this paper, it is found that the maximum delay bound for the complex dynamical network to form asymptotic synchronization is h=1.E(t) are defined the same as in the example. We plot the evolution of E(t) in upper part in Figure 2. The lower subplot indicates the synchronization results of the network. It can be seen from the figures that the network in this example can achieve asymptotic synchronization.Synchronization evolutionE(t) for the delay-dependent network with c1=0.5, c2=1, and τ=0.1.
(a)
(b)


## 5. Conclusion

This paper considered a class of complex networks with both time delayed and non-time delayed coupling. We derived, respectively, a sufficient criterion for time delay-dependent and time delay-independent asymptotic synchronization which are more general than those obtained in previous works. These asymptotic synchronization results were obtained by using the Lyapunov-Krasovskii stability theorem and the linear matrix inequality. Two simple examples were also used to validate the theoretical analysis.



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*Source: 309289-2012-10-24.xml*