Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article
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Nonlinear Systems: Asymptotic Methods, Stability, Chaos, Control, and Optimization

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Research Article | Open Access

Volume 2011 |Article ID 978612 | 15 pages | https://doi.org/10.1155/2011/978612

Generalized Synchronization between Two Complex Dynamical Networks with Time-Varying Delay and Nonlinear Coupling

Academic Editor: E. E. N. Macau
Received30 Aug 2010
Revised26 Mar 2011
Accepted17 May 2011
Published02 Aug 2011

Abstract

The generalized synchronization between two complex networks with nonlinear coupling and time-varying delay is investigated in this paper. The novel adaptive schemes of constructing controller response network are proposed to realize generalized synchronization with the drive network to a given mapping. Two specific examples show and verify the effectiveness of the proposed method.

1. Introduction

Over the past decade, complex networks have gained a lot of attention in various fields, such as sociology, biology, physical sciences, mathematics, and engineering [1ā€“5]. A complex network is a large number of interconnected nodes, in which each node represents a unit (or element) with certain dynamical system and edge represents the relationship or connection between two units (or elements). Synchronization is one of the most important dynamical properties of dynamical systems, there are different kinds of methods to realize synchronization such as active control [6], feedback control [7], adaptive control [8], impulsive control [9], passive method [10], and so forth. Synchronization of complex networks includes complete synchronization (CS) [11, 12], projective synchronization (PS) [13, 14], phase synchronization [15, 16], generalized synchronization (GS) [17, 18], and so on.

As a sort of synchronous behavior, GS is an extension of CS and PS, so GS is more widespread than CS and PS in nature and in technical applications. GS of chaos system has been widely researched. However, most of theoretical results about synchronization of complex networks focus on CS and PS. Especially, due to the complexity of GS, the theoretical results for GS are lacking, but GS of complex networks is attracting special attention; in [17], the author gives a novel definition of GS on networks and a numerical simulation example. Reference [18] applies the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators.

Recently, GS of drive-response chaos systems is investigated by the nonlinear control theory in [19]. In this letter, we extend this method to investigate GS between two complex networks, and some criterions for GS are deduced.

This letter is organized as follows. In Section 2, the definition of GS between the drive-response complex networks is given and some preliminary knowledge, including three assumptions and one lemma is also introduced. By employing the Lyapunov theory and BarbĒŽlat lemma, some schemes are designed to construct response networks to realize GS with respect to the given nonlinear smooth mapping. In Section 3, two numerical examples are given to demonstrate the effectiveness of the proposed method in Section 2. Finally, conclusions are given in Section 4.

2. GS Theorems between Two Complex Networks with Nonlinear Coupling

2.1. Definition and Assumptions

Definition 2.1. Suppose š‘„š‘–=(š‘„š‘–1,š‘„š‘–2,ā€¦,š‘„š‘–š‘›)š‘‡āˆˆš‘…š‘›,š‘¦š‘–=(š‘¦š‘–1,š‘¦š‘–2,ā€¦,š‘¦š‘–š‘›)š‘‡āˆˆš‘…š‘›,š‘–=1,2,ā€¦,š‘ are the state variables of the drive network and the response network, respectively. Given the smooth vector function Ī¦āˆ¶š‘…š‘›ā†’š‘…š‘›, the drive network and response network are said to achieve GS with respect to Ī¦. If limš‘”ā†’āˆžā€–ā€–š‘’š‘–ā€–ā€–(š‘”)=0,š‘–=1,2,ā€¦,š‘,(2.1) where š‘’š‘–(š‘”)=š‘„š‘–(š‘”)āˆ’Ī¦(š‘¦š‘–(š‘”)),š‘–=1,2,ā€¦,š‘, the norm ||ā‹…|| of a vector š‘„ is defined as ||š‘„||=(š‘„š‘‡š‘„)1/2.

Remark 2.2. If Ī¦(š‘¦š‘–)=š‘¦š‘–, then GS is CS in [20]. If Ī¦(š‘¦š‘–)=šœ†š‘¦š‘–, then GS is PS in [13, 14].

In this paper, we consider a general complex dynamical network with time-varying nonlinear coupling and consisting of N nonidentical nodes:Ģ‡š‘„š‘–(š‘”)=š‘“š‘–ī€·š‘„š‘–ī€ø+(š‘”)š‘ī“š‘—=1š‘š‘–š‘—ā„Žī€·š‘„š‘—ī€ø(š‘”āˆ’šœ(š‘”)),š‘–=1,2,ā€¦,š‘,(2.2) where š‘„š‘–=(š‘„š‘–1,š‘„š‘–2,ā€¦,š‘„š‘–š‘›)š‘‡āˆˆš‘…š‘›,š‘–=1,2,ā€¦,š‘ are the state variables of the drive network, š‘“š‘–āˆ¶š‘…š‘›ā†’š‘…š‘›, ā„Žāˆ¶š‘…š‘›ā†’š‘…š‘›are smooth nonlinear vector functions, and šœ(š‘”) is time-varying delay. š¶=(š‘š‘–š‘—)š‘Ć—š‘ is unknown or uncertain coupling matrix; if there is a connection between node š‘– and node š‘—(š‘—ā‰ š‘–), then š‘š‘–š‘—ā‰ 0, otherwiseš‘š‘–š‘—=0(š‘–ā‰ š‘—),and the diagonal elements of š¶ are defined byš‘š‘–š‘–=āˆ’š‘ī“š‘—=1š‘—ā‰ š‘–š‘š‘–š‘—.(2.3)

It should be noted that the complex dynamical network model (2.2) is quite general. If š‘“š‘–=š‘“, š‘–=1,2,ā€¦,š‘™; š‘”š‘–=š‘”,š‘–=š‘™+1,š‘™+2,ā€¦,š‘, then we can get the following complex dynamical network:Ģ‡š‘„š‘–ī€·š‘„(š‘”)=š‘“š‘–ī€ø+(š‘”)š‘ī“š‘—=1š‘š‘–š‘—ā„Žī€·š‘„š‘—ī€ø(š‘”āˆ’šœ(š‘”)),š‘–=1,ā€¦,š‘™,Ģ‡š‘„š‘–ī€·š‘„(š‘”)=š‘”š‘–ī€ø+(š‘”)š‘ī“š‘—=1š‘š‘–š‘—ā„Žī€·š‘„š‘—ī€ø(š‘”āˆ’šœ(š‘”)),š‘–=š‘™+1,ā€¦,š‘.(2.4) On the other hand, if ā„Ž(š‘„š‘–)=š“š‘„š‘–, with š“=(š‘Žš‘–š‘—)š‘Ć—š‘ being an inner-coupling constant matrix of the network, then the complex network model (2.2) degenerates into the model of linearly and diffusively coupled network with coupling delays:Ģ‡š‘„š‘–(š‘”)=š‘“š‘–ī€·š‘„š‘–ī€ø+(š‘”)š‘ī“š‘—=1š‘š‘–š‘—š“š‘„š‘—(š‘”āˆ’šœ(š‘”)),š‘–=1,2,ā€¦,š‘.(2.5) Letī€·š‘¦š·Ī¦š‘–ī€ø=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽšœ•šœ™1ī€·š‘¦š‘–ī€øšœ•š‘¦š‘–1šœ•šœ™1ī€·š‘¦š‘–ī€øšœ•š‘¦š‘–2ā‹Æšœ•šœ™1ī€·š‘¦š‘–ī€øšœ•š‘¦š‘–š‘›šœ•šœ™2ī€·š‘¦š‘–ī€øšœ•š‘¦š‘–1šœ•šœ™2ī€·š‘¦š‘–ī€øšœ•š‘¦š‘–2ā‹Æšœ•šœ™2ī€·š‘¦š‘–ī€øšœ•š‘¦š‘–š‘›ā‹Æā‹Æā‹Æā‹Æšœ•šœ™š‘›ī€·š‘¦š‘–ī€øšœ•š‘¦š‘–1šœ•šœ™š‘›ī€·š‘¦š‘–ī€øšœ•š‘¦š‘–2ā‹Æšœ•šœ™š‘›ī€·š‘¦š‘–ī€øšœ•š‘¦š‘–š‘›āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ (2.6) be the Jacobian matrix of the mapping Ī¦(š‘¦š‘–)=(šœ™1(š‘¦š‘–),šœ™2(š‘¦š‘–),ā€¦,šœ™š‘›(š‘¦š‘–))š‘‡,šœ™š‘—(š‘¦š‘–)āˆˆš‘…, š‘–=1,2,ā€¦,š‘, š‘—=1,2,ā€¦,š‘›. When matrix š·Ī¦(š‘¦š‘–(š‘”)) is reversible, we can give the following controller response network:Ģ‡š‘¦š‘–ī€·š‘¦(š‘”)=š·Ī¦š‘–ī€ø(š‘”)āˆ’1īƒ¬š‘“š‘–ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1Ģ‚š‘š‘–š‘—ā„Žī€·Ī¦ī€·š‘¦š‘—īƒ­(š‘”āˆ’šœ(š‘”))ī€øī€ø+š‘¢š‘–,š‘–=1,2,ā€¦,š‘,(2.7) where š‘¦š‘–=(š‘¦š‘–1,š‘¦š‘–2,ā€¦,š‘¦š‘–š‘›)š‘‡āˆˆš‘…š‘›,š‘–=1,2,ā€¦,š‘ are the state variables of the response network, š‘¢š‘–āˆˆš‘…š‘›,š‘–=1,2,ā€¦,š‘ are nonlinear controllers to be designed, and īš¶=(Ģ‚š‘š‘–š‘—)š‘Ć—š‘ is the estimate of the unknown coupling matrix š¶=(š‘š‘–š‘—)š‘Ć—š‘.

Let š‘’š‘–(š‘”)=š‘„š‘–(š‘”)āˆ’Ī¦(š‘¦š‘–(š‘”)), with the aid of (2.2) and (2.7), the following error network can be obtained:Ģ‡š‘’š‘–(š‘”)=Ģ‡š‘„š‘–ī€ŗī€·š‘¦(š‘”)āˆ’š·Ī¦š‘–(š‘”)ī€øī€»Ģ‡š‘¦š‘–(š‘”)=š‘“š‘–ī€·š‘„š‘–ī€ø(š‘”)āˆ’š‘“š‘–ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1š‘š‘–š‘—ā„Žī€·š‘„š‘—ī€øāˆ’(š‘”āˆ’šœ(š‘”))š‘ī“š‘—=1Ģ‚š‘š‘–š‘—ā„Žī€·Ī¦ī€·š‘¦š‘—ī€·š‘¦(š‘”āˆ’šœ(š‘”))ī€øī€øāˆ’š·Ī¦š‘–ī€øš‘¢(š‘”)š‘–=š‘“š‘–ī€·š‘„š‘–ī€ø(š‘”)āˆ’š‘“š‘–ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1š‘š‘–š‘—š»ī€·š‘’š‘—ī€øāˆ’(š‘”āˆ’šœ(š‘”))š‘ī“š‘—=1Ģƒš‘š‘–š‘—ā„Žī€·Ī¦ī€·š‘¦š‘—ī€·š‘¦(š‘”āˆ’šœ(š‘”))ī€øī€øāˆ’š·Ī¦š‘–ī€øš‘¢(š‘”)š‘–,(2.8) where š»ī€·š‘’š‘—ī€øī€·š‘„(š‘”)=ā„Žš‘—ī€øī€·Ī¦ī€·š‘¦(š‘”)āˆ’ā„Žš‘—(š‘”)ī€øī€ø,(2.9)Ģƒš‘š‘–š‘—=Ģ‚š‘š‘–š‘—āˆ’š‘š‘–š‘—.(2.10)

The following conditions are needed for the solutions of (2.8) to achieve the objective (2.1).

Assumption 1. (A1) Time delay šœ(š‘”) is a differential function with 0ā‰¤šœ(š‘”)ā‰¤ā„Ž, Ģ‡šœ(š‘”)ā‰¤šœ‡<1, where ā„Ž and šœ‡ are positive constants. Obviously, this assumption holds for constant šœ(š‘”).

Assumption 2. (A2) Suppose that š‘“š‘–(ā‹…) is bounded and there exists a nonnegative constant š›¼ such that ā€–ā€–š‘“š‘–ī€·š‘„š‘–ī€ø(š‘”)āˆ’š‘“š‘–ī€·Ī¦ī€·š‘¦š‘–ā€–ā€–ā€–ā€–š‘’(š‘”)ī€øī€øā‰¤š›¼š‘–ā€–ā€–(š‘”),š‘–=1,2,ā€¦,š‘.(2.11)

Assumption 3. (A3) Suppose that ā„Ž(ā‹…) is bounded and there exists a nonnegative constant š›½ such that ā€–ā€–ā„Žī€·š‘„š‘–ī€øī€·Ī¦ī€·š‘¦(š‘”)āˆ’ā„Žš‘–ā€–ā€–ā€–ā€–š‘’(š‘”)ī€øī€øā‰¤š›½š‘–ā€–ā€–(š‘”),š‘–=1,2,ā€¦,š‘.(2.12)

Remark 2.3. The condition (2.12) is reasonable due to [21, 22]. For example, the Hopfield neural network [23] is described by dš‘„š‘–(š‘”)š‘„dš‘”=āˆ’š‘–(š‘”)š‘…š‘–+š‘ī“š‘—=1š‘¤š‘–š‘—š‘“š‘—ī€·š‘„š‘—ī€·š‘”āˆ’šœš‘–š‘—(š‘”)ī€øī€ø+š¼š‘–,š‘–=1,2,ā€¦,š‘.(2.13) Take š‘“š‘—(š‘„š‘—)=(šœ‹/2)arctan((šœ‹/2)šœ†š‘„š‘—),where šœ† is positive constant. It is obvious that š‘“š‘—(ā‹…) satisfies Assumption 3.

Lemma 2.4. For any vectors š‘‹,š‘Œāˆˆš‘…š‘›,the following inequality holds 2š‘‹š‘‡š‘Œā‰¤š‘‹š‘‡š‘‹+š‘Œš‘‡š‘Œ.(2.14)

Next section, we will give some sufficient conditions of complex dynamical networks (2.2) and (2.7) obtaining GS.

2.2. Main Results

Theorem 2.5. Suppose that (A1)ā€“(A3) hold. Using the following controller: š‘¢š‘–ī€·š‘¦=š·Ī¦š‘–ī€ø(š‘”)āˆ’1š‘‘š‘–š‘’š‘–(š‘”)(2.15) and the update laws Ģ‡š‘‘š‘–=š‘˜š‘–š‘’š‘‡š‘–(š‘”)š‘’š‘–(š‘”),(2.16)Ģ‡Ģ‚š‘š‘–š‘—=š›æš‘–š‘—š‘’š‘‡š‘–ī€·Ī¦ī€·š‘¦(š‘”)ā„Žš‘—,(š‘”āˆ’šœ(š‘”))ī€øī€ø(2.17) where š‘–,š‘—=1,2,ā€¦,š‘,š‘‘š‘– is feedback strength, and š›æš‘–š‘—>0,š‘˜š‘–>0 are arbitrary constants, then the complex dynamical networks (2.2) and (2.7) will achieve GS with respect to Ī¦.

Proof. Select a Lyapunov-Krasovskii functional candidate as š‘‰(š‘”,š‘’(š‘”))=š‘ī“š‘–=1š‘’š‘‡š‘–(š‘”)š‘’š‘–(š‘”)+š‘ī“š‘š‘–=1ī“š‘—=11š›æš‘–š‘—Ģƒš‘2š‘–š‘—+š‘ī“š‘–=11š‘˜š‘–ī€·š‘‘š‘–ī€øāˆ’š‘‘2+š‘š›½2ī€œ1āˆ’šœ‡š‘”š‘š‘”āˆ’šœ(š‘”)ī“š‘–=1š‘’š‘‡š‘–(šœ‰)š‘’š‘–(šœ‰)dšœ‰,(2.18) where š‘’(š‘”)=(š‘’š‘‡1(š‘”),š‘’š‘‡2(š‘”),ā€¦,š‘’š‘‡š‘(š‘”))š‘‡ and š‘‘ is a positive constant to be determined.
The time derivative of š‘‰ along the solution of the error system (2.8) isdš‘‰=dš‘”š‘ī“š‘–=12š‘’š‘‡š‘–(š‘”)Ģ‡š‘’š‘–(š‘”)+š‘ī“š‘š‘–=1ī“š‘—=12š›æš‘–š‘—Ģƒš‘š‘–š‘—Ģ‡Ģ‚š‘š‘–š‘—+š‘ī“š‘–=12š‘˜š‘–ī€·š‘‘š‘–ī€øĢ‡š‘‘āˆ’š‘‘š‘–+š‘š›½21āˆ’šœ‡š‘ī“š‘–=1š‘’š‘‡š‘–(š‘”)š‘’š‘–āˆ’(š‘”)1āˆ’Ģ‡šœ(š‘”)1āˆ’šœ‡š‘š›½2š‘ī“š‘–=1š‘’š‘‡š‘–(š‘”āˆ’šœ(š‘”))š‘’š‘–=(š‘”āˆ’šœ(š‘”))š‘ī“š‘–=12š‘’š‘‡š‘–īƒ¬š‘“(š‘”)š‘–ī€·š‘„š‘–ī€ø(š‘”)āˆ’š‘“š‘–ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1š‘š‘–š‘—š»ī€·š‘’š‘—ī€øāˆ’(š‘”āˆ’šœ(š‘”))š‘ī“š‘—=1Ģƒš‘š‘–š‘—ā„Žī€·Ī¦ī€·š‘¦š‘—ī€·š‘¦(š‘”āˆ’šœ(š‘”))ī€øī€øāˆ’š·Ī¦š‘–ī€øš‘¢(š‘”)š‘–īƒ­+(š‘”)š‘ī“š‘š‘–=1ī“š‘—=12š›æš‘–š‘—Ģƒš‘š‘–š‘—Ģ‡Ģ‚š‘š‘–š‘—+š‘ī“š‘–=12š‘˜š‘–ī€·š‘‘š‘–ī€øĢ‡š‘‘āˆ’š‘‘š‘–+š‘š›½21āˆ’šœ‡š‘ī“š‘–=1š‘’š‘‡š‘–(š‘”)š‘’š‘–(š‘”)āˆ’1āˆ’Ģ‡šœ(š‘”)1āˆ’šœ‡š‘š›½2š‘ī“š‘–=1š‘’š‘‡š‘–(š‘”āˆ’šœ(š‘”))š‘’š‘–(š‘”āˆ’šœ(š‘”)).(2.19) Substituting the controller (2.15) and update laws (2.16)-(2.17) into (2.19) and considering Assumption 2, we obtain dš‘‰ā‰¤dš‘”š‘ī“š‘–=1ī‚µ2š›¼āˆ’2š‘‘+š‘š›½2ī‚¶š‘’1āˆ’šœ‡š‘‡š‘–(š‘”)š‘’š‘–(š‘”)+2š‘ī“š‘š‘–=1ī“š‘—=1š‘’š‘‡š‘–(š‘”)š‘š‘–š‘—š»ī€·š‘’š‘—ī€øāˆ’(š‘”āˆ’šœ(š‘”))1āˆ’Ģ‡šœ(š‘”)1āˆ’šœ‡š‘š›½2š‘ī“š‘–=1š‘’š‘‡š‘–(š‘”āˆ’šœ(š‘”))š‘’š‘–(š‘”āˆ’šœ(š‘”)).(2.20) By Lemma 2.4 and considering Assumptions 1 and 3, we have 1āˆ’Ģ‡šœ(š‘”)21āˆ’šœ‡ā‰„1,š‘ī“š‘š‘–=1ī“š‘—=1š‘’š‘‡š‘–(š‘”)š‘š‘–š‘—š»ī€·š‘’š‘—ī€øā‰¤(š‘”āˆ’šœ(š‘”))š‘ī“š‘š‘–=1ī“š‘—=1š‘2š‘–š‘—š‘’š‘‡š‘–(š‘”)š‘’š‘–(š‘”)+š‘ī“š‘š‘–=1ī“š‘—=1š»š‘‡ī€·š‘’š‘—ī€øš»ī€·š‘’(š‘”āˆ’šœ(š‘”))š‘—ī€ø(š‘”āˆ’šœ(š‘”))ā‰¤š‘max1ā‰¤š‘–,š‘—ā‰¤š‘ī€½š‘2š‘–š‘—ī€¾š‘ī“š‘–=1š‘’š‘‡š‘–(š‘”)š‘’š‘–(š‘”)+š‘š›½2š‘ī“š‘—=1š‘’š‘‡š‘—(š‘”āˆ’šœ(š‘”))š‘’š‘—(š‘”āˆ’šœ(š‘”)),(2.21) then dš‘‰ā‰¤dš‘”š‘ī“š‘–=1ī‚µ2š›¼āˆ’2š‘‘+š‘š›½21āˆ’šœ‡+š‘max1ā‰¤š‘–,š‘—ā‰¤š‘ī€½š‘2š‘–š‘—ī€¾ī‚¶š‘’š‘‡š‘–(š‘”)š‘’š‘–ī‚µ(š‘”)=āˆ’2š‘‘āˆ’2š›¼āˆ’š‘š›½21āˆ’šœ‡āˆ’š‘max1ā‰¤š‘–,š‘—ā‰¤š‘ī€½š‘2š‘–š‘—ī€¾ī‚¶š‘’š‘‡(š‘”)š‘’(š‘”).(2.22)
Note that we can choose constant d to make dš‘‰/dš‘”ā‰¤āˆ’š‘’š‘‡(š‘”)š‘’(š‘”)ā‰¤0, thus š‘‰ is nonincreasing in š‘”ā‰„0. One has š‘‰ is bounded since 0ā‰¤š‘‰(š‘”,š‘’(š‘”))ā‰¤š‘‰(0,š‘’(0)), so limš‘”ā†’+āˆžš‘‰(š‘”,š‘’(š‘”)) exists andlimš‘”ā†’+āˆžī€œš‘”0š‘’š‘‡(š‘ )š‘’(š‘ )dš‘ ā‰¤āˆ’limš‘”ā†’+āˆžī€œš‘”0dš‘‰dš‘ dš‘ =š‘‰(0,š‘’(0))āˆ’limš‘”ā†’+āˆžš‘‰(š‘”,š‘’(š‘”)).(2.23) From (2.18), we have 0ā‰¤š‘’š‘‡(š‘”)š‘’(š‘”)ā‰¤š‘‰(š‘”,š‘’(š‘”)), so š‘’š‘‡(š‘”)š‘’(š‘”) is bounded. According to error system (2.8), (d/dš‘”)š‘’š‘‡(š‘”)š‘’(š‘”)=2š‘’š‘‡(š‘”)Ģ‡š‘’(š‘”) is bounded for š‘”ā‰„0 due to the boundedness of š‘“š‘–(ā‹…) and ā„Ž(ā‹…). From the above, we can see that š‘’(š‘”)āˆˆšæ2āˆ©šæāˆž and Ģ‡š‘’(š‘”)āˆˆšæāˆž. By using another form of BarbĒŽlat lemma [24], one has limš‘”ā†’+āˆžš‘’š‘‡(š‘”)š‘’(š‘”)=0, so limš‘”ā†’+āˆžš‘’(š‘”)=0 and the complex dynamical networks (2.2) and (2.7) can obtain generalized synchronization under the controller (2.15) and update laws (2.16)-(2.17). This completes the proof.

Remark 2.6. If limš‘”ā†’+āˆžĢ‡š‘’(š‘”) exists,then we can obtain limš‘”ā†’+āˆžĢ‡š‘’(š‘”)=0 for limš‘”ā†’+āˆžš‘’(š‘”)=0. According to error system (2.8), we have limš‘”ā†’+āˆžāˆ‘š‘š‘—=1Ģƒš‘š‘–š‘—ā„Ž(Ī¦(š‘¦š‘—(š‘”āˆ’šœ(š‘”))))=0. When {ā„Ž(Ī¦(š‘¦š‘—(š‘”āˆ’šœ(š‘”)))}š‘š‘—=1 are linearly independent on the orbit {š‘¦š‘—(š‘”āˆ’šœ(š‘”))}š‘š‘—=1 of synchronization manifold, limš‘”ā†’+āˆžĢƒš‘š‘–š‘—=0. We can get limš‘”ā†’+āˆžĢ‚š‘š‘–š‘—=š‘š‘–š‘—,š‘–,š‘—=1,2,ā€¦,š‘; that is, the uncertain coupling matrix š¶ can be successfully estimated using the update laws (2.17).
In a special case Ī¦(š‘¦š‘–)=šœ†š‘¦š‘–(šœ†is nonzero constant), based on Theorem 2.5, we can construct the following response networkĢ‡š‘¦š‘–1(š‘”)=šœ†īƒ¬š‘“š‘–ī€·šœ†š‘¦š‘–ī€ø+(š‘”)š‘ī“š‘—=1Ģ‚š‘š‘–š‘—ā„Žī€·šœ†š‘¦š‘—ī€øīƒ­(š‘”āˆ’šœ(š‘”))+š‘¢š‘–,š‘–=1,2,ā€¦,š‘.(2.24)

Corollary 2.7. Suppose that (A1)ā€“(A3) hold. Using the controller š‘¢š‘–=1šœ†š‘‘š‘–š‘’š‘–(š‘”)(2.25) and the update laws Ģ‡š‘‘š‘–=š‘˜š‘–š‘’š‘‡š‘–(š‘”)š‘’š‘–(š‘”),Ģ‡Ģ‚š‘š‘–š‘—=š›æš‘–š‘—š‘’š‘‡š‘–ī€·(š‘”)ā„Žšœ†š‘¦š‘—ī€ø,(š‘”āˆ’šœ(š‘”))(2.26) where š‘–,š‘—=1,2,ā€¦,š‘,š‘‘š‘– is feedback strength, and š›æš‘–š‘—>0,š‘˜š‘–>0 are arbitrary constants, then the complex dynamical networks (2.2) and (2.24) will obtain PS.

To networks (2.4), according to Theorem 2.5, one can construct the following response network:Ģ‡š‘¦š‘–ī€·š‘¦(š‘”)=š·Ī¦š‘–ī€ø(š‘”)āˆ’1īƒ¬š‘“ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1Ģ‚š‘š‘–š‘—ā„Žī€·Ī¦ī€·š‘¦š‘—īƒ­(š‘”āˆ’šœ(š‘”))ī€øī€ø+š‘¢š‘–,š‘–=1,ā€¦,š‘™,Ģ‡š‘¦š‘–ī€·š‘¦(š‘”)=š·Ī¦š‘–ī€ø(š‘”)āˆ’1īƒ¬š‘”ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1Ģ‚š‘š‘–š‘—ā„Žī€·Ī¦ī€·š‘¦š‘—īƒ­(š‘”āˆ’šœ(š‘”))ī€øī€ø+š‘¢š‘–,š‘–=š‘™+1,ā€¦,š‘(2.27) and get the following corollary:

Corollary 2.8. Suppose that (A1)ā€“(A3) hold. Using the controller š‘¢š‘–ī€·š‘¦=š·Ī¦š‘–ī€ø(š‘”)āˆ’1š‘‘š‘–š‘’š‘–(š‘”)(2.28) and the update laws Ģ‡š‘‘š‘–=š‘˜š‘–š‘’š‘‡š‘–(š‘”)š‘’š‘–(š‘”),Ģ‡Ģ‚š‘š‘–š‘—=š›æš‘–š‘—š‘’š‘‡š‘–ī€·Ī¦ī€·š‘¦(š‘”)ā„Žš‘—,(š‘”āˆ’šœ(š‘”))ī€øī€ø(2.29) where š‘–,š‘—=1,2,ā€¦,š‘,š‘‘š‘– is feedback strength, š›æš‘–š‘—>0,š‘˜š‘–>0 are arbitrary constants, then the complex dynamical network networks (2.4) and (2.27) will achieve GS with respect to Ī¦.

If coupling function ā„Ž(š‘„š‘–)=š“š‘„š‘–; that is, the network is linearly coupled, then the complex network (2.2) degenerates into (2.5). Note that ā€–š“š‘’š‘–(š‘”)ā€–ā‰¤ā€–š“ā€–ā‹…ā€–š‘’š‘–(š‘”)ā€–,š‘–=1,2,ā€¦,š‘ hold. We construct the following response network:Ģ‡š‘¦š‘–ī€·š‘¦(š‘”)=š·Ī¦š‘–ī€ø(š‘”)āˆ’1īƒ¬š‘“š‘–ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1Ģ‚š‘š‘–š‘—ī€·š‘¦š“Ī¦š‘—ī€øīƒ­(š‘”āˆ’šœ(š‘”))+š‘¢š‘–,š‘–=1,2,ā€¦,š‘.(2.30)

Corollary 2.9. Suppose that (A1) and (A2) hold. Using the controller š‘¢š‘–ī€·š‘¦=š·Ī¦š‘–ī€ø(š‘”)āˆ’1š‘‘š‘–š‘’š‘–(š‘”)(2.31) and the update laws Ģ‡š‘‘š‘–=š‘˜š‘–š‘’š‘‡š‘–(š‘”)š‘’š‘–(š‘”),Ģ‡Ģ‚š‘š‘–š‘—=š›æš‘–š‘—š‘’š‘‡š‘–ī€·š‘¦(š‘”)š“Ī¦š‘—ī€ø,(š‘”āˆ’šœ(š‘”))(2.32) where š‘–,š‘—=1,2,ā€¦,š‘,š‘‘š‘– is feedback strength, and š›æš‘–š‘—>0,š‘˜š‘–>0 are arbitrary constants, then the complex dynamical networks (2.5) and (2.30) will obtain GS.

Using different control, we can obtain the following theorem.

Theorem 2.10. Suppose that (A1) and (A3) hold. Using the following controller: š‘¢š‘–ī€·š‘¦=š·Ī¦š‘–ī€ø(š‘”)āˆ’1ī€ŗš‘‘š‘–š‘’š‘–(š‘”)+š‘“š‘–ī€·š‘„š‘–ī€ø(š‘”)āˆ’š‘“š‘–ī€·Ī¦ī€·š‘¦š‘–(š‘”)ī€øī€øī€»,(2.33) and the update laws Ģ‡š‘‘š‘–=š‘˜š‘–š‘’š‘‡š‘–(š‘”)š‘’š‘–(š‘”),(2.34)Ģ‡Ģ‚š‘š‘–š‘—=š›æš‘–š‘—š‘’š‘‡š‘–ī€·Ī¦ī€·š‘¦(š‘”)ā„Žš‘—,(š‘”āˆ’šœ(š‘”))ī€øī€ø(2.35) where š‘–,š‘—=1,2,ā€¦,š‘,š‘‘š‘– is feedback strength, and š›æš‘–š‘—>0,š‘˜š‘–>0 are arbitrary constants, then the complex dynamical networks (2.2) and (2.7) will achieve GS with respect to Ī¦.

Proof. Select the same Lyapunov-Krasovskii function as Theorem 2.5, then dš‘‰dš‘”=2š‘ī“š‘–=1š‘’š‘‡š‘–īƒ¬š‘“(š‘”)š‘–ī€·š‘„š‘–ī€ø(š‘”)āˆ’š‘“š‘–ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1š‘š‘–š‘—š»ī€·š‘’š‘—ī€øāˆ’(š‘”āˆ’šœ(š‘”))š‘ī“š‘—=1Ģƒš‘š‘–š‘—ā„Žī€·Ī¦ī€·š‘¦š‘—ī€·š‘¦(š‘”āˆ’šœ(š‘”))ī€øī€øāˆ’š·Ī¦š‘–ī€øš‘¢(š‘”)š‘–īƒ­+(š‘”)š‘ī“š‘š‘–=1ī“š‘—=12š›æš‘–š‘—Ģƒš‘š‘–š‘—Ģ‡Ģ‚š‘š‘–š‘—+š‘ī“š‘–=12š‘˜š‘–ī€·š‘‘š‘–ī€øĢ‡š‘‘āˆ’š‘‘š‘–+š‘š›½21āˆ’šœ‡š‘ī“š‘–=1š‘’š‘‡š‘–(š‘”)š‘’š‘–(š‘”)āˆ’1āˆ’Ģ‡šœ(š‘”)1āˆ’šœ‡š‘š›½2š‘ī“š‘–=1š‘’š‘‡š‘–(š‘”āˆ’šœ(š‘”))š‘’š‘–ā‰¤(š‘”āˆ’šœ(š‘”))š‘ī“š‘–=1ī‚µāˆ’2š‘‘+š‘š›½2ī‚¶š‘’1āˆ’šœ‡š‘‡š‘–(š‘”)š‘’š‘–+(š‘”)š‘ī“š‘š‘–=1ī“š‘—=1š‘’š‘‡š‘–(š‘”)š‘š‘–š‘—š»ī€·š‘’š‘—ī€øāˆ’(š‘”āˆ’šœ(š‘”))1āˆ’Ģ‡šœ(š‘”)1āˆ’šœ‡š‘š›½2š‘ī“š‘–=1š‘’š‘‡š‘–(š‘”āˆ’šœ(š‘”))š‘’š‘–(š‘”āˆ’šœ(š‘”)).(2.36) The rest of the proof is similar to Theorem 2.5 and omitted here. This completes the proof.

Remark 2.11. According to Remark 2.6, when {ā„Ž(Ī¦(š‘¦š‘—(š‘”āˆ’šœ(š‘”))))}š‘š‘—=1 are linearly independent on the orbit {š‘¦š‘—(š‘”āˆ’šœ(š‘”))}š‘š‘—=1 of synchronization manifold, we can get limš‘”ā†’+āˆžĢ‚š‘š‘–š‘—=š‘š‘–š‘—,š‘–,š‘—=1,2,ā€¦,š‘; that is, the uncertain coupling matrix š¶ can be successfully estimated using the updating laws (2.35).

Remark 2.12. Based on Theorem 2.10, we can get corollaries corresponding to Corollaries 2.7ā€“2.9.

3. Illustrative Numerical Examples

In this section, two groups of drive and response networks are concretely presented to demonstrate the effectiveness of the proposed method in the previous section.

It is well known that the unified chaotic system [25] is described byāŽ›āŽœāŽœāŽœāŽĢ‡š‘„1Ģ‡š‘„2Ģ‡š‘„3āŽžāŽŸāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽœāŽāˆ’ī€·š‘„(25š›½+10)1āˆ’š‘„2ī€øāˆ’š‘„1š‘„3+(28āˆ’35š›½)š‘„1+(29š›½āˆ’1)š‘„2š‘„1š‘„2āˆ’(š›½+8)3š‘„3āŽžāŽŸāŽŸāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽœāŽī€·š‘„(3.1)āˆ’101āˆ’š‘„2ī€øāˆ’š‘„1š‘„3+28š‘„1āˆ’š‘„2š‘„1š‘„2āˆ’83š‘„3āŽžāŽŸāŽŸāŽŸāŽŸāŽ +āŽ›āŽœāŽœāŽœāŽœāŽī€·š‘„āˆ’251āˆ’š‘„2ī€øāˆ’35š‘„1+29š‘„2āˆ’13š‘„3āŽžāŽŸāŽŸāŽŸāŽŸāŽ š›½=š¹(š‘„)+šŗ(š‘„)š›½,(3.2) which is chaotic if š›½āˆˆ[0,1]. Obviously, system (3.2) is the original Lorenz system for š›½=0 while system (3.2) belongs to the original Chen system for š›½=1. In fact, system (3.2) bridges the gap between the Lorenz system and Chen system.

The unified new chaotic system [26] can be described asāŽ›āŽœāŽœāŽœāŽĢ‡š‘„1Ģ‡š‘„2Ģ‡š‘„3āŽžāŽŸāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽœāŽš‘Žš‘„1āˆ’š‘„2š‘„3š‘š‘„2+š‘„1š‘„3š‘š‘„3+13š‘„1š‘„2āŽžāŽŸāŽŸāŽŸāŽŸāŽ =š‘”(š‘„).(3.3) It is chaotic when š‘Ž=5.0,š‘=āˆ’10.0, and š‘=āˆ’3.8.

In the following, we will take these two chaotic systems as node dynamics to validate the effectiveness of Theorems 2.5 and 2.10. To do that, we first verify that function š‘“(š‘„)=š¹(š‘„)+šŗ(š‘„)š›½(š›½āˆˆ[0,1]) satisfies Assumption 2.

Since the attractor is confined to a bounded region, there exists a constant š‘€>0, satisfying for allš‘¦=(š‘¦1,š‘¦2,š‘¦3),š‘§=(š‘§1,š‘§2,š‘§3)āˆˆš‘…3,||š‘¦||ā‰¤š‘€,||š‘§||ā‰¤š‘€; therefore,ā€–š‘“(š‘¦)āˆ’š‘“(š‘§)ā€–2=(25š›½+10)2š‘¦ī€ŗī€·2āˆ’š‘¦1ī€øāˆ’ī€·š‘§2āˆ’š‘§1ī€øī€»2+ī€ŗš‘§1š‘§3āˆ’š‘¦1š‘¦3ī€·š‘¦+(28āˆ’35š›½)1āˆ’š‘§1ī€øī€·š‘¦+(29š›½āˆ’1)2āˆ’š‘§2ī€øī€»2+ī‚øš‘¦1š‘¦2āˆ’š‘§1š‘§2āˆ’(š›½+8)3ī€·š‘¦3āˆ’š‘§3ī€øī‚¹2=(25š›½+10)2š‘¦ī€ŗī€·2āˆ’š‘§2ī€øāˆ’ī€·š‘¦1āˆ’š‘§1ī€øī€»2+ī€ŗš‘§1ī€·š‘§3āˆ’š‘¦3ī€ø+ī€·āˆ’š‘¦3š‘¦+28āˆ’35š›½ī€øī€·1āˆ’š‘§1ī€øī€·š‘¦+(29š›½āˆ’1)2āˆ’š‘§2ī€øī€»2+ī‚øš‘¦1ī€·š‘¦2āˆ’š‘§2ī€ø+š‘§2ī€·š‘¦1āˆ’š‘§1ī€øāˆ’(š›½+8)3ī€·š‘¦3āˆ’š‘§3ī€øī‚¹2ā‰¤352ī‚ƒ2ī€·š‘¦2āˆ’š‘§2ī€ø2ī€·š‘¦+21āˆ’š‘§1ī€ø2ī‚„+3š‘€2ī€·š‘¦3āˆ’š‘§3ī€ø2ī€·+6282+š‘€2š‘¦ī€øī€·1āˆ’š‘§1ī€ø2+3Ɨ282ī€·š‘¦2āˆ’š‘§2ī€ø2+3š‘€2ī€·š‘¦2āˆ’š‘§2ī€ø2+3š‘€2ī€·š‘¦1āˆ’š‘§1ī€ø2ī€·š‘¦+93āˆ’š‘§3ī€ø2ā‰¤ī€·2Ɨ352+6Ɨ282+9š‘€2ī€øā€–š‘¦āˆ’š‘§ā€–2.(3.4) Thus, function š‘“(š‘„)=š¹(š‘„)+šŗ(š‘„)š›½(š›½āˆˆ[0,1]) satisfies Assumption 2. By the same process, we can obtain that function š‘”(š‘„) satisfies Assumption 2, too.

Example 3.1. In this subsection, we consider a weighted complex dynamical network with coupling delay consisting of 3 Lorenz systems and 2 new chaotic systems (3.3). The entire networked system is given as Ģ‡š‘„š‘–ī€·š‘„(š‘”)=š¹š‘–ī€ø+(š‘”)5ī“š‘—=1š‘š‘–š‘—ā„Žī€·š‘„š‘—ī€ø(š‘”āˆ’šœ(š‘”)),š‘–=1,2,3,Ģ‡š‘„š‘–ī€·š‘„(š‘”)=š‘”š‘–ī€ø+(š‘”)5ī“š‘—=1š‘š‘–š‘—ā„Žī€·š‘„š‘—ī€ø(š‘”āˆ’šœ(š‘”)),š‘–=4,5,(3.5) where š‘„š‘–(š‘”)=(š‘„š‘–1(š‘”),š‘„š‘–2(š‘”),š‘„š‘–3(š‘”))š‘‡,š‘–=1,2,ā€¦,5.šœ(š‘”)=0.1, the weight configuration matrix ī€·š‘š¶=š‘–š‘—ī€ø5Ɨ5=āŽ›āŽœāŽœāŽœāŽœāŽœāŽœāŽœāŽāŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽŸāŽ āˆ’513101āˆ’210031āˆ’400100āˆ’210001āˆ’1.(3.6) The coupling functions are ā„Ž(š‘„š‘—(š‘”))=(sin(š‘„š‘—1(š‘”)),arctan(š‘„š‘—2(š‘”)),arctan(š‘„š‘—3(š‘”)))š‘‡,š‘—=1,2,ā€¦,5.
Let Ī¦(š‘¦š‘–)=(š‘¦š‘–1+š‘¦š‘–2,2š‘¦š‘–2,2š‘¦š‘–3)š‘‡,then š·Ī¦(š‘¦š‘–ī‚€)=110020002ī‚,š‘–=1,2,ā€¦,5.
Since (A1)ā€“(A3) hold, therefore, according to Theorem 2.5, we can use the following response network:Ģ‡š‘¦š‘–ī€·š‘¦(š‘”)=š·Ī¦š‘–ī€ø(š‘”)āˆ’1īƒ¬š¹ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1Ģ‚š‘š‘–š‘—ā„Žī€·Ī¦ī€·š‘¦š‘—īƒ­(š‘”āˆ’šœ(š‘”))ī€øī€ø+š‘¢š‘–,š‘–=1,2,3,Ģ‡š‘¦š‘–ī€·š‘¦(š‘”)=š·Ī¦š‘–ī€ø(š‘”)āˆ’1īƒ¬š‘”ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1Ģ‚š‘š‘–š‘—ā„Žī€·Ī¦ī€·š‘¦š‘—īƒ­(š‘”āˆ’šœ(š‘”))ī€øī€ø+š‘¢š‘–,š‘–=4,5.(3.7)
The controller and update laws are given by (2.15)ā€“(2.17). The initial values are given as follows: Ģ‚š‘š‘–š‘—(0)=3,š›æš‘–š‘—=1,š‘‘š‘–(0)=1,š‘˜š‘–=1,š‘„š‘–(0)=(12+š‘–Ć—0.1,15+š‘–Ć—0.1,30+š‘–Ć—0.15)š‘‡,Ģ‚š‘„š‘–(0)=(5+š‘–Ć—0.1,7.5+š‘–Ć—0.1,15+š‘–Ć—0.1),š‘–,š‘—=1,2,ā€¦,5. Figure 1 shows GS errors ā€–š‘’š‘–(š‘”)ā€–, š‘–=1,2,ā€¦,5. One can see that all nodesā€™ errors converge to zero. Some elements of matrix īš¶ are displayed in Figure 2. The numerical results show that this adaptive scheme is effective and we can get limš‘”ā†’āˆžĢ‚š‘š‘–š‘—=š‘š‘–š‘—,š‘–,š‘—=1,2,ā€¦,5.

Example 3.2. In the following simulation, we choose a weighted complex dynamical network with coupling delay consisting of 5 unified chaotic systems. The entire networked system is given as Ģ‡š‘„š‘–(š‘”)=š‘“š‘–ī€·š‘„š‘–ī€ø+(š‘”)5ī“š‘—=1š‘š‘–š‘—ā„Žī€·š‘„š‘—ī€ø(š‘”āˆ’šœ(š‘”)),š‘–=1,2,ā€¦,5,(3.8) where š‘„š‘–(š‘”)=(š‘„š‘–1(š‘”),š‘„š‘–2(š‘”),š‘„š‘–3(š‘”))š‘‡,š‘“š‘–(š‘„)=š¹(š‘„)š›½š‘–+šŗ(š‘„),š›½š‘–=0.1Ɨ(š‘–āˆ’1),š‘–=1,2,ā€¦,5. We assume šœ(š‘”)=0.3,ā„Ž(š‘„š‘—(š‘”))=(arctan(š‘„š‘—1(š‘”)),arctan(š‘„š‘—2(š‘”)),arctan(š‘„š‘—3(š‘”)))š‘‡,š‘—=1,2,ā€¦,5. C is the same as that in model (3.5).
Let Ī¦(š‘¦š‘–)=(š‘¦š‘–1+š‘¦š‘–2,2š‘¦š‘–2,š‘¦3š‘–3+š‘¦š‘–3)š‘‡,š·Ī¦(š‘¦š‘–ī‚µ)=110020003š‘¦2š‘–3+1ī‚¶,š‘–=1,2,ā€¦,5.
According to Theorem 2.10, the response network is given byĢ‡š‘¦š‘–ī€·š‘¦(š‘”)=š·Ī¦š‘–ī€ø(š‘”)āˆ’1īƒ¬š‘“š‘–ī€·Ī¦ī€·š‘¦š‘–+(š‘”)ī€øī€øš‘ī“š‘—=1Ģ‚š‘š‘–š‘—ā„Žī€·Ī¦ī€·š‘¦š‘—īƒ­(š‘”āˆ’šœ(š‘”))ī€øī€ø+š‘¢š‘–,š‘–=1,2,ā€¦,5.(3.9) The controller and update laws are given by (2.33)ā€“(2.35). The initial values are given as follows: Ģ‚š‘š‘–š‘—(0)=6,š›æš‘–š‘—=1,š‘‘š‘–(0)=1,š‘˜š‘–=1,š‘„š‘–(0)=(12,15,30)š‘‡,Ģ‚š‘„š‘–(0)=(5,7.5,3)š‘‡,š‘–,š‘—=1,2,ā€¦,5. Figures 3 and 4 show the GS errors ā€–š‘’š‘–(š‘”)ā€–, š‘–=1,2,ā€¦,5 and some elements of matrix īš¶, respectively. The results illustrate that this scheme is effective and we can get limš‘”ā†’+āˆžĢ‚š‘š‘–š‘—=š‘š‘–š‘—,š‘–,š‘—=1,2,ā€¦,5.

4. Conclusion

In this paper, we have investigated GS between two complex networks with different node dynamics and proposed some new GS schemes via nonlinear control using Lyapunov theory and BarbĒŽlat lemma. Our results generalize CS of complex dynamical network with linear coupling and without delay in [20] to GS of complex dynamical network with nonidentical nodes and time-varying delay nonlinear coupling. Numerical examples are provided to verify the effectiveness of the theoretical results. This work extends the study of GS.

Acknowledgment

This research is supported by the National Natural Science Foundation of China under Grant no. 70871056.

References

  1. D. J. Watts and S. H. Strogatz, ā€œCollective dynamics of ā€œsmall-worldā€ networks,ā€ Nature, vol. 393, no. 6684, pp. 440ā€“442, 1998. View at: Publisher Site | Google Scholar
  2. A. L. BarabĆ”si, R. Albert, and H. Jeong, ā€œMean-field theory for scale-free random network,ā€ Physica A, vol. 272, no. 1-2, pp. 173ā€“187, 1999. View at: Publisher Site | Google Scholar
  3. S. H. Strogatz, ā€œExploring complex networks,ā€ Nature, vol. 410, no. 6825, pp. 268ā€“276, 2001. View at: Publisher Site | Google Scholar
  4. L. Y. Xiang, Z. Q. Chen, Z. X. Liu, and Z. Z. Yuane, ā€œResearch on the modelling, analysis and control of complex dynamical networks,ā€ Progress in Natural Science, vol. 16, no. 13, pp. 1543ā€“1551, 2006. View at: Google Scholar
  5. T. Zhou, Z. Q. Fu, and B. H. Wang, ā€œEpidemic dynamics on complex networks,ā€ Progress in Natural Science, vol. 16, no. 5, pp. 452ā€“457, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. M. C. Ho, Y. C. Hung, and C. H. Chou, ā€œPhase and anti-phase synchronization of two chaotic systems by using active control,ā€ Physics Letters A, vol. 296, no. 1, pp. 43ā€“48, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. H. H. Chen, G. J. Sheu, Y. L. Lin, and C. S. Chen, ā€œChaos synchronization between two different chaotic systems via nonlinear feedback control,ā€ Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4393ā€“4401, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. J. Zhou, J. A. Lu, and J. H. LĆ¼, ā€œAdaptive synchronization of an uncertain complex dynamical network,ā€ IEEE Transactions on Automatic Control, vol. 51, no. 4, pp. 652ā€“656, 2006. View at: Publisher Site | Google Scholar
  9. S. M. Cai, J. Zhou, L. Xiang, and Z. R. Liu, ā€œRobust impulsive synchronization of complex delayed dynamical networks,ā€ Physics Letters A, vol. 372, no. 30, pp. 4990ā€“4995, 2008. View at: Publisher Site | Google Scholar
  10. X. R. Chen and C. X. Liu, ā€œPassive control on a unified chaotic system,ā€ Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 683ā€“687, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. L. M. Pecora and T. L. Carroll, ā€œMaster stability functions for synchronized coupled systems,ā€ Physical Review Letters, vol. 80, no. 10, pp. 2109ā€“2112, 1998. View at: Publisher Site | Google Scholar
  12. A. Arenas, A. DĆ­az-Guilera, and C. J. PĆ©rez-Vicente, ā€œSynchronization reveals topological scales in complex networks,ā€ Physical Review Letters, vol. 96, no. 11, Article ID 114102, 4 pages, 2006. View at: Publisher Site | Google Scholar
  13. M. Sun, C. Y. Zeng, and L. X. Tian, ā€œProjective synchronization in drive-response dynamical networks of partially linear systems with time-varying coupling delay,ā€ Physics Letters A, vol. 372, no. 46, pp. 6904ā€“6908, 2008. View at: Publisher Site | Google Scholar
  14. S. Zheng, Q. S. Bi, and G. L. Cai, ā€œAdaptive projective synchronization in complex networks with time-varying coupling delay,ā€ Physics Letters A, vol. 373, no. 17, pp. 1553ā€“1559, 2009. View at: Publisher Site | Google Scholar
  15. X. Q. Yu, Q. S. Ren, J. L. Hou, and J. Y. Zhao, ā€œThe chaotic phase synchronization in adaptively coupled-delayed complex networks,ā€ Physics Letters A, vol. 373, no. 14, pp. 1276ā€“1282, 2009. View at: Publisher Site | Google Scholar
  16. X. Li, ā€œPhase synchronization in complex networks with decayed long-range interactions,ā€ Physica D, vol. 223, no. 2, pp. 242ā€“247, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  17. X. L. Xu, Z. Q. Chen, G. Y. Si, X. F. Hu, and P. Luo, ā€œA novel definition of generalized synchronization on networks and a numerical simulation example,ā€ Computers & Mathematics with Applications, vol. 56, no. 11, pp. 2789ā€“2794, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  18. Y. C. Hung, Y. T. Huang, M. C. Ho, and C. K. Hu, ā€œPaths to globally generalized synchronization in scale-free networks,ā€ Physical Review E, vol. 77, no. 1, Article ID 016202, 8 pages, 2008. View at: Publisher Site | Google Scholar
  19. J. F. Li, N. Li, Y. P. Liu, and Y. Gan, ā€œLinear and nonlinear generalized synchronization of a class of chaotic systems by using a single driving variable,ā€ Acta Physica Sinica, vol. 58, no. 2, pp. 779ā€“784, 2009. View at: Google Scholar | Zentralblatt MATH
  20. J. Zhou and J. A. Lu, ā€œTopology identification of weighted complex dynamical networks,ā€ Physica A, vol. 386, no. 1, pp. 481ā€“491, 2007. View at: Publisher Site | Google Scholar
  21. X. W. Liu and T. P. Chen, ā€œExponential synchronization of nonlinear coupled dynamical networks with a delayed coupling,ā€ Physica A, vol. 381, no. 15, pp. 82ā€“92, 2007. View at: Publisher Site | Google Scholar
  22. G. M. He and J. Y. Yang, ā€œAdaptive synchronization in nonlinearly coupled dynamical networks,ā€ Chaos, Solitons and Fractals, vol. 38, no. 5, pp. 1254ā€“1259, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. J. G. Peng, H. Qiao, and Z. B. Xu, ā€œA new approach to stability of neural networks with time-varying delays,ā€ Neural Networks, vol. 15, no. 1, pp. 95ā€“103, 2002. View at: Publisher Site | Google Scholar
  24. G. Tao, ā€œA simple alternative to the BarbĒŽlat lemma,ā€ IEEE Transactions on Automatic Control, vol. 42, no. 5, p. 698, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  25. J. H. LĆ¼, G. R. Chen, D. Z. Cheng, and S. Celikovsky, ā€œBridge the gap between the Lorenz system and the Chen system,ā€ International Journal of Bifurcation and Chaos, vol. 12, no. 12, pp. 2917ā€“2926, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  26. H. K. Chen and C. I. Lee, ā€œAnti-control of chaos in rigid body motion,ā€ Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 957ā€“965, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright Ā© 2011 Qiuxiang Bian and Hongxing Yao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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