， ，

（1.School of Mathematics and Systems Science，Shenyang Normal University，Shenyang 110034，China；2.College of Mathematics and Computer Science，Dali University，Dali 671003，China）

0 Introduction

Complex networks have been the subject of extensive study for more than two decades due that many systems in nature can be described by models of complex networks［1－5］.Examples of the well known complex networks include the internet，the world wide web，food webs，biological neural networks，electrical power grids，telephone cell graphs，coauthor ship and citation networks of scientists，cellular and metabolic networks，etc.There are many interesting issues to study in understanding the complex networks，such as the structural complexity，network evolution，connection diversity，dynamical complexity，node diversity， meta－complication，etc.［3］.The synchronization and stabilization of dynamical networks are also such interesting issues.

The complex network models can be extended from static to dynamic by introducing dynamical elements to be the network nodes［6－7］.For the resulting dynamical networks，the synchronization and stabilization of all its dynamical nodes are significant and interesting phenomenon，which have attracted increasing attention from various fields of science and engineering since the beginning of this century.Because the synchronization of a coupled dynamical network can well explain many natural phenomena observed，recently the synchronization of coupled dynamical networks has become a focal point in the study of dynamics［6－29］.Some notable works are as follows.In 2002，Wang and Chen［6］presented a simple scale－free dynamical network model and investigated its synchronization.They have also studied the synchronization issue in small－world dynamical networks［7］.Later，Lüet al［8］introduced a time－varying complex dynamical network model and investigated its synchronization phenomenon.In［9］，Chen et al derived a simple sufficient condition for global synchronization of linearly coupled neural networks.In［10］，Li and Chen presented a complex dynamical network model with coupling delays，and further obtained some synchronization conditions for both delay－independent and delay－dependent asymptotical stabilities.

Note that all the works mentioned above of coupled dynamical networks are devoting to linearly coupled dynamical networks.He and Yang［11］discussed the adaptive synchronization in nonlinearly coupled dynamical networks.In the present work，we continue the study of He and Yang.We study the adaptive asymptotical synchronization and stabilization in one kind of coupled dynamical network with non－uniform coupling strength.Our work riches and improves the contribution of［11］.

The rest of this paper is organized as follows.A coupled dynamical network model with nonuniform coupling strength is introduced and some necessary preliminaries are introduced in Section 2.The adaptive asymptotical synchronization and stabilization of the network model are examined in Section 3and in Section 4，respectively.In Section 5，we propose numerical examples to verify our results of theory.And in Section 6，we conclude the paper.

1 Preliminaries

Letx（t）＝（x1（t），x2（t），…，xn（t））T∈Rnbe the state variable of an isolated node，which is a dynamical system and described by

wheref（x（t））＝（f1（x（t）），f2（x（t）），…，fn（x（t）））T：Rn→Rnis a given continuous vector valued function（or map）.Consider a complex dynamical network ofNcoupled nodesxi，1≤i≤N，with each node being anndimensional dynamical system，which is described by

Here，xi（t）＝（xi1（t），xi2（t），…，xin（t））T（∈Rnfor givent）is the state variable of nodei，i＝1，2，…，N；h（x）＝（h1（x），h2（x），…，hn（x））T：Rn→Rnis a continuous map；σij＝σji（≥0）fori≠j，which represents the coupling strength between the nodesiandj，in particular，σij＝σji＝0if there is no connection between nodeiandj，andFor the coupling dynamical networks，the concepts of asymptotical synchronization and stabilization are defined as follows.

Definition 1The coupling dynamical network（2）is said to achieve asymptotical synchronization if and only iffor anyi，j＝1，2，…，N.

Definition 2The coupling dynamical network（2）is said to achieve asymptotical stabilization for vectorx0∈Rnif and only iffor anyi＝1，2，…，N.

Add a simple controlleruito nodeifor eachi.Then we obtain the following controlled dynamical network：

We call the network （2）to achieve adaptive asymptotical synchronization （stabilization）with controllersui，i＝1，2，…，N，if the network （3）is to achieve asymptotical synchronization（stabilization）.

2 Synchronization

In this section，we discuss the adaptive synchronization of the network （2）.To make （3）synchronizing，we choose the controllersui＝kxi，wherek（＞0）is a constant.Then（3）becomes the following：

Now，we begin to show the network （4）is to achieve asymptotical synchronization under a certain condition，that is，（2）is to achieve adaptive asymptotical synchronization.

We first assume there exist two constantsl＞0andp＞0such that

for anyx＝（x1，x2，…，xn）∈Rnandx′＝（x′1，x′2，…，x′n）∈Rn，wherel＞0andp＞0is called the Lipschitz constants offandhrespectively

Letxi，i＝1，2，…，N，be a set of solution of（4）.Consider the following system：

Then we have

Lemma 1For（6），assumeLetK＝l＋M.Then the network（4）is to achieve asymptotical synchronization ifk＞K，that is，the network（2）is to achieve adaptive asymptotical synchronization.

ProofConstruct the following function：

On the other hand，if（y2，y3，…，yN）＝0，thenxi＝x1，i＝2，3，…，N.This impliesF（yi）＝0，andfor alli＝2，3，…，N.

In terms of the facts above，we can easily know that the vector 0is the equilibrium point of（6）.Thus，from the Lyapunov function methodThat is，Lemma 1holds.

Theorem 1For the network（2），assume：σij＝σfor anyiandj（i≠j），andk＞（l＋Npσ）.Then，（2）is to achieve adaptive asymptotical synchronization.

ProofFori＝2，…，N， we have

Therefore，byk＞（l＋Npσ），we know Theorem 1holds from Lemma1.

3 Stabilization

This section further discuss the adaptive asymptotical stabilization of the network（2）.

Choose the controllersui＝k（xi－x0），wherex0∈Rnis a fixed vector.Then （3）becomes the following

Theorem 2For the network（2），assume：there exist two constantsl＞0andp＞0such that

for anyx＝（x1，x2，…，xn）∈Rn.LetM＝max｛｜σii｜：i＝1，2，…，N｝，andK＝l＋2pNM.Then the network（7）to achieve asymptotical stabilization for the pointx0ifk＞K，that is，the network（2）is to achieve adaptive asymptotical stabilization.

ProofThen（7）becomes

Construct the following function：

Then we have

On the other hand，by（8），the vector 0is the equilibrium point of（9）.Thereforefor anyi＝1，2，…，N.So，Theorem 2holds.

4 Simulation

In this section，we make a simple report on our simulation experiment to verify the theoretical results of the present work.

For the dynamical system of the isolated node（1），takef（x）（＝－10x1＋10x2，28x1－x1x3－x2，i.e.the Lorenz chaotic system described in［30］witha＝10

To verify Theorem 1，takeh（x）＝（sin2（2x1），cos2（2x2），sin2（2x3））T，σ＝0.02，andN＝200in the coupling dynamical networks（2）and （4）；moreover takek＝0.5in the controlled dynamical network（4）.For the constructed networks（2）and（4），it can be easily known that the conditions of Theorem 1hold.For the two networks with the same initial values of the nodes chosen randomly from－10to 10，by simulations through Matlab，we obtain the results such as the showed by the Fig.1and the Fig.2，respectively.One can easily know that the adaptive asymptotical synchronization can be quickly achieved under the controllerui＝kxifrom comparing the Fig.1and the Fig.2.

Fig.1 The varying states on the nodes of uncontrolled network（2）of Theorem 1

Fig.2 The varying states on the nodes of the controlled network（4）of Theorem1

To verify Theorem 2，in the coupling dynamical networks（2）and（7），takeN＝200，h（x）＝（sin2（2x1），cos2（2x2），sin2（2x3））T，randomly determineσijas one of 0and any value of the interval［0.01，0.03］fori＜j；moreover take alsok＝0.5in the controlled dynamical network（7）.For such constructed networks（2）and（7），the conditions of Theorem 2also hold forFor the two networks with the same initial values of the nodes chosen randomly from－10to 10，by simulations through Matlab，we obtain the results such as the showed by the Fig.3and the Fig.4，respectively.Fig.3shows the states on the variation of the nodes of the uncontrolled network，that is，the network（2）；while Fig.4shows the states on the variation of the nodes of the controlled network，that is，the network（7）.Comparing Fig.4with Fig.3，we have the observation that the stable state can quickly reach after controllersui＝k（xi－x0）are imposed to the dynamical network（2）.

The two figure suggest that adaptive synchronization and stabilization be achieved under the conditions of Theorem 1and Theorem 2，respectively.That is，the simulation results support our theoretical derivations and analysis.

Fig.3 The varying states on the nodes of the uncontrolled network（2）of Theorem 2

Fig.4 The varying states on the nodes of controlled network（7）of Theorem 2

5 Conclusion

The adaptive asymptotical synchronization and stabilization in a kind of coupled dynamical network with non－uniform coupling strength have been studied in the present work，respectively.We try to make the network synchronizing and stabilizing by adding suitable simple controllers to each node＇s dynamical equation.Conditions for both the adaptive asymptotical synchronization and the adaptive asymptotical stabilization are derived，respectively.These conditions are applicable to networks with different sizes.Finally，numerical examples are shown to verify our theoretical results.

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