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Mathematical Problems in Engineering
Volume 2011, Article ID 978612, 15 pages
http://dx.doi.org/10.1155/2011/978612
Research Article

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

1Faculty of Science, Jiangsu University, Jiangsu, Zhenjiang, 212013, China
2Faculty of Mathematics and Physics, Jiangsu University of Science and Technology, Jiangsu, Zhenjiang, 212003, China

Received 30 August 2010; Revised 26 March 2011; Accepted 17 May 2011

Academic Editor: E. E. N. Macau

Copyright © 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.

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 [15]. 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 Barbǎlat 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 𝑥𝑖=(𝑥𝑖1,𝑥𝑖2,,𝑥𝑖𝑛)𝑇𝑅𝑛,𝑦𝑖=(𝑦𝑖1,𝑦𝑖2,,𝑦𝑖𝑛)𝑇𝑅𝑛,𝑖=1,2,,𝑁 are the state variables of the drive network and the response network, respectively. Given the smooth vector function Φ𝑅𝑛𝑅𝑛, the drive network and response network are said to achieve GS with respect to Φ. If lim𝑡𝑒𝑖(𝑡)=0,𝑖=1,2,,𝑁,(2.1) where 𝑒𝑖(𝑡)=𝑥𝑖(𝑡)Φ(𝑦𝑖(𝑡)),𝑖=1,2,,𝑁, the norm |||| of a vector 𝑥 is defined as ||𝑥||=(𝑥𝑇𝑥)1/2.

Remark 2.2. If Φ(𝑦𝑖)=𝑦𝑖, then GS is CS in [20]. If Φ(𝑦𝑖)=𝜆𝑦𝑖, 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:̇𝑥𝑖(𝑡)=𝑓𝑖𝑥𝑖+(𝑡)𝑁𝑗=1𝑐𝑖𝑗𝑥𝑗(𝑡𝜏(𝑡)),𝑖=1,2,,𝑁,(2.2) where 𝑥𝑖=(𝑥𝑖1,𝑥𝑖2,,𝑥𝑖𝑛)𝑇𝑅𝑛,𝑖=1,2,,𝑁 are the state variables of the drive network, 𝑓𝑖𝑅𝑛𝑅𝑛, 𝑅𝑛𝑅𝑛are smooth nonlinear vector functions, and 𝜏(𝑡) is time-varying delay. 𝐶=(𝑐𝑖𝑗)𝑁×𝑁 is unknown or uncertain coupling matrix; if there is a connection between node 𝑖 and node 𝑗(𝑗𝑖), then 𝑐𝑖𝑗0, otherwise𝑐𝑖𝑗=0(𝑖𝑗),and the diagonal elements of 𝐶 are defined by𝑐𝑖𝑖=𝑁𝑗=1𝑗𝑖𝑐𝑖𝑗.(2.3)

It should be noted that the complex dynamical network model (2.2) is quite general. If 𝑓𝑖=𝑓, 𝑖=1,2,,𝑙; 𝑔𝑖=𝑔,𝑖=𝑙+1,𝑙+2,,𝑁, then we can get the following complex dynamical network:̇𝑥𝑖𝑥(𝑡)=𝑓𝑖+(𝑡)𝑁𝑗=1𝑐𝑖𝑗𝑥𝑗(𝑡𝜏(𝑡)),𝑖=1,,𝑙,̇𝑥𝑖𝑥(𝑡)=𝑔𝑖+(𝑡)𝑁𝑗=1𝑐𝑖𝑗𝑥𝑗(𝑡𝜏(𝑡)),𝑖=𝑙+1,,𝑁.(2.4) On the other hand, if (𝑥𝑖)=𝐴𝑥𝑖, with 𝐴=(𝑎𝑖𝑗)𝑁×𝑁 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:̇𝑥𝑖(𝑡)=𝑓𝑖𝑥𝑖+(𝑡)𝑁𝑗=1𝑐𝑖𝑗𝐴𝑥𝑗(𝑡𝜏(𝑡)),𝑖=1,2,,𝑁.(2.5) Let𝑦𝐷Φ𝑖=𝜕𝜙1𝑦𝑖𝜕𝑦𝑖1𝜕𝜙1𝑦𝑖𝜕𝑦𝑖2𝜕𝜙1𝑦𝑖𝜕𝑦𝑖𝑛𝜕𝜙2𝑦𝑖𝜕𝑦𝑖1𝜕𝜙2𝑦𝑖𝜕𝑦𝑖2𝜕𝜙2𝑦𝑖𝜕𝑦𝑖𝑛𝜕𝜙𝑛𝑦𝑖𝜕𝑦𝑖1𝜕𝜙𝑛𝑦𝑖𝜕𝑦𝑖2𝜕𝜙𝑛𝑦𝑖𝜕𝑦𝑖𝑛(2.6) be the Jacobian matrix of the mapping Φ(𝑦𝑖)=(𝜙1(𝑦𝑖),𝜙2(𝑦𝑖),,𝜙𝑛(𝑦𝑖))𝑇,𝜙𝑗(𝑦𝑖)𝑅, 𝑖=1,2,,𝑁, 𝑗=1,2,,𝑛. When matrix 𝐷Φ(𝑦𝑖(𝑡)) is reversible, we can give the following controller response network:̇𝑦𝑖𝑦(𝑡)=𝐷Φ𝑖(𝑡)1𝑓𝑖Φ𝑦𝑖+(𝑡)𝑁𝑗=1̂𝑐𝑖𝑗Φ𝑦𝑗(𝑡𝜏(𝑡))+𝑢𝑖,𝑖=1,2,,𝑁,(2.7) where 𝑦𝑖=(𝑦𝑖1,𝑦𝑖2,,𝑦𝑖𝑛)𝑇𝑅𝑛,𝑖=1,2,,𝑁 are the state variables of the response network, 𝑢𝑖𝑅𝑛,𝑖=1,2,,𝑁 are nonlinear controllers to be designed, and 𝐶=(̂𝑐𝑖𝑗)𝑁×𝑁 is the estimate of the unknown coupling matrix 𝐶=(𝑐𝑖𝑗)𝑁×𝑁.

Let 𝑒𝑖(𝑡)=𝑥𝑖(𝑡)Φ(𝑦𝑖(𝑡)), with the aid of (2.2) and (2.7), the following error network can be obtained:̇𝑒𝑖(𝑡)=̇𝑥𝑖𝑦(𝑡)𝐷Φ𝑖(𝑡)̇𝑦𝑖(𝑡)=𝑓𝑖𝑥𝑖(𝑡)𝑓𝑖Φ𝑦𝑖+(𝑡)𝑁𝑗=1𝑐𝑖𝑗𝑥𝑗(𝑡𝜏(𝑡))𝑁𝑗=1̂𝑐𝑖𝑗Φ𝑦𝑗𝑦(𝑡𝜏(𝑡))𝐷Φ𝑖𝑢(𝑡)𝑖=𝑓𝑖𝑥𝑖(𝑡)𝑓𝑖Φ𝑦𝑖+(𝑡)𝑁𝑗=1𝑐𝑖𝑗𝐻𝑒𝑗(𝑡𝜏(𝑡))𝑁𝑗=1̃𝑐𝑖𝑗Φ𝑦𝑗𝑦(𝑡𝜏(𝑡))𝐷Φ𝑖𝑢(𝑡)𝑖,(2.8) where 𝐻𝑒𝑗𝑥(𝑡)=𝑗Φ𝑦(𝑡)𝑗(𝑡),(2.9)̃𝑐𝑖𝑗=̂𝑐𝑖𝑗𝑐𝑖𝑗.(2.10)

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

Assumption 1. (A1) Time delay 𝜏(𝑡) is a differential function with 0𝜏(𝑡), ̇𝜏(𝑡)𝜇<1, where and 𝜇 are positive constants. Obviously, this assumption holds for constant 𝜏(𝑡).

Assumption 2. (A2) Suppose that 𝑓𝑖() is bounded and there exists a nonnegative constant 𝛼 such that 𝑓𝑖𝑥𝑖(𝑡)𝑓𝑖Φ𝑦𝑖𝑒(𝑡)𝛼𝑖(𝑡),𝑖=1,2,,𝑁.(2.11)

Assumption 3. (A3) Suppose that () is bounded and there exists a nonnegative constant 𝛽 such that 𝑥𝑖Φ𝑦(𝑡)𝑖𝑒(𝑡)𝛽𝑖(𝑡),𝑖=1,2,,𝑁.(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 d𝑥𝑖(𝑡)𝑥d𝑡=𝑖(𝑡)𝑅𝑖+𝑁𝑗=1𝑤𝑖𝑗𝑓𝑗𝑥𝑗𝑡𝜏𝑖𝑗(𝑡)+𝐼𝑖,𝑖=1,2,,𝑁.(2.13) Take 𝑓𝑗(𝑥𝑗)=(𝜋/2)arctan((𝜋/2)𝜆𝑥𝑗),where 𝜆 is positive constant. It is obvious that 𝑓𝑗() satisfies Assumption 3.

Lemma 2.4. For any vectors 𝑋,𝑌𝑅𝑛,the following inequality holds 2𝑋𝑇𝑌𝑋𝑇𝑋+𝑌𝑇𝑌.(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)–(A3) hold. Using the following controller: 𝑢𝑖𝑦=𝐷Φ𝑖(𝑡)1𝑑𝑖𝑒𝑖(𝑡)(2.15) and the update laws ̇𝑑𝑖=𝑘𝑖𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡),(2.16)̇̂𝑐𝑖𝑗=𝛿𝑖𝑗𝑒𝑇𝑖Φ𝑦(𝑡)𝑗,(𝑡𝜏(𝑡))(2.17) where 𝑖,𝑗=1,2,,𝑁,𝑑𝑖 is feedback strength, and 𝛿𝑖𝑗>0,𝑘𝑖>0 are arbitrary constants, then the complex dynamical networks (2.2) and (2.7) will achieve GS with respect to Φ.

Proof. Select a Lyapunov-Krasovskii functional candidate as 𝑉(𝑡,𝑒(𝑡))=𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)+𝑁𝑁𝑖=1𝑗=11𝛿𝑖𝑗̃𝑐2𝑖𝑗+𝑁𝑖=11𝑘𝑖𝑑𝑖𝑑2+𝑁𝛽21𝜇𝑡𝑁𝑡𝜏(𝑡)𝑖=1𝑒𝑇𝑖(𝜉)𝑒𝑖(𝜉)d𝜉,(2.18) where 𝑒(𝑡)=(𝑒𝑇1(𝑡),𝑒𝑇2(𝑡),,𝑒𝑇𝑁(𝑡))𝑇 and 𝑑 is a positive constant to be determined.
The time derivative of 𝑉 along the solution of the error system (2.8) isd𝑉=d𝑡𝑁𝑖=12𝑒𝑇𝑖(𝑡)̇𝑒𝑖(𝑡)+𝑁𝑁𝑖=1𝑗=12𝛿𝑖𝑗̃𝑐𝑖𝑗̇̂𝑐𝑖𝑗+𝑁𝑖=12𝑘𝑖𝑑𝑖̇𝑑𝑑𝑖+𝑁𝛽21𝜇𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)1̇𝜏(𝑡)1𝜇𝑁𝛽2𝑁𝑖=1𝑒𝑇𝑖(𝑡𝜏(𝑡))𝑒𝑖=(𝑡𝜏(𝑡))𝑁𝑖=12𝑒𝑇𝑖𝑓(𝑡)𝑖𝑥𝑖(𝑡)𝑓𝑖Φ𝑦𝑖+(𝑡)𝑁𝑗=1𝑐𝑖𝑗𝐻𝑒𝑗(𝑡𝜏(𝑡))𝑁𝑗=1̃𝑐𝑖𝑗Φ𝑦𝑗𝑦(𝑡𝜏(𝑡))𝐷Φ𝑖𝑢(𝑡)𝑖+(𝑡)𝑁𝑁𝑖=1𝑗=12𝛿𝑖𝑗̃𝑐𝑖𝑗̇̂𝑐𝑖𝑗+𝑁𝑖=12𝑘𝑖𝑑𝑖̇𝑑𝑑𝑖+𝑁𝛽21𝜇𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)1̇𝜏(𝑡)1𝜇𝑁𝛽2𝑁𝑖=1𝑒𝑇𝑖(𝑡𝜏(𝑡))𝑒𝑖(𝑡𝜏(𝑡)).(2.19) Substituting the controller (2.15) and update laws (2.16)-(2.17) into (2.19) and considering Assumption 2, we obtain d𝑉d𝑡𝑁𝑖=12𝛼2𝑑+𝑁𝛽2𝑒1𝜇𝑇𝑖(𝑡)𝑒𝑖(𝑡)+2𝑁𝑁𝑖=1𝑗=1𝑒𝑇𝑖(𝑡)𝑐𝑖𝑗𝐻𝑒𝑗(𝑡𝜏(𝑡))1̇𝜏(𝑡)1𝜇𝑁𝛽2𝑁𝑖=1𝑒𝑇𝑖(𝑡𝜏(𝑡))𝑒𝑖(𝑡𝜏(𝑡)).(2.20) By Lemma 2.4 and considering Assumptions 1 and 3, we have 1̇𝜏(𝑡)21𝜇1,𝑁𝑁𝑖=1𝑗=1𝑒𝑇𝑖(𝑡)𝑐𝑖𝑗𝐻𝑒𝑗(𝑡𝜏(𝑡))𝑁𝑁𝑖=1𝑗=1𝑐2𝑖𝑗𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)+𝑁𝑁𝑖=1𝑗=1𝐻𝑇𝑒𝑗𝐻𝑒(𝑡𝜏(𝑡))𝑗(𝑡𝜏(𝑡))𝑁max1𝑖,𝑗𝑁𝑐2𝑖𝑗𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)+𝑁𝛽2𝑁𝑗=1𝑒𝑇𝑗(𝑡𝜏(𝑡))𝑒𝑗(𝑡𝜏(𝑡)),(2.21) then d𝑉d𝑡𝑁𝑖=12𝛼2𝑑+𝑁𝛽21𝜇+𝑁max1𝑖,𝑗𝑁𝑐2𝑖𝑗𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)=2𝑑2𝛼𝑁𝛽21𝜇𝑁max1𝑖,𝑗𝑁𝑐2𝑖𝑗𝑒𝑇(𝑡)𝑒(𝑡).(2.22)
Note that we can choose constant d to make d𝑉/d𝑡𝑒𝑇(𝑡)𝑒(𝑡)0, thus 𝑉 is nonincreasing in 𝑡0. One has 𝑉 is bounded since 0𝑉(𝑡,𝑒(𝑡))𝑉(0,𝑒(0)), so lim𝑡+𝑉(𝑡,𝑒(𝑡)) exists andlim𝑡+𝑡0𝑒𝑇(𝑠)𝑒(𝑠)d𝑠lim𝑡+𝑡0d𝑉d𝑠d𝑠=𝑉(0,𝑒(0))lim𝑡+𝑉(𝑡,𝑒(𝑡)).(2.23) From (2.18), we have 0𝑒𝑇(𝑡)𝑒(𝑡)𝑉(𝑡,𝑒(𝑡)), so 𝑒𝑇(𝑡)𝑒(𝑡) is bounded. According to error system (2.8), (d/d𝑡)𝑒𝑇(𝑡)𝑒(𝑡)=2𝑒𝑇(𝑡)̇𝑒(𝑡) is bounded for 𝑡0 due to the boundedness of 𝑓𝑖() and (). From the above, we can see that 𝑒(𝑡)𝐿2𝐿 and ̇𝑒(𝑡)𝐿. By using another form of Barbǎlat lemma [24], one has lim𝑡+𝑒𝑇(𝑡)𝑒(𝑡)=0, so lim𝑡+𝑒(𝑡)=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 lim𝑡+̇𝑒(𝑡) exists,then we can obtain lim𝑡+̇𝑒(𝑡)=0 for lim𝑡+𝑒(𝑡)=0. According to error system (2.8), we have lim𝑡+𝑁𝑗=1̃𝑐𝑖𝑗(Φ(𝑦𝑗(𝑡𝜏(𝑡))))=0. When {(Φ(𝑦𝑗(𝑡𝜏(𝑡)))}𝑁𝑗=1 are linearly independent on the orbit {𝑦𝑗(𝑡𝜏(𝑡))}𝑁𝑗=1 of synchronization manifold, lim𝑡+̃𝑐𝑖𝑗=0. We can get lim𝑡+̂𝑐𝑖𝑗=𝑐𝑖𝑗,𝑖,𝑗=1,2,,𝑁; that is, the uncertain coupling matrix 𝐶 can be successfully estimated using the update laws (2.17).
In a special case Φ(𝑦𝑖)=𝜆𝑦𝑖(𝜆is nonzero constant), based on Theorem 2.5, we can construct the following response network̇𝑦𝑖1(𝑡)=𝜆𝑓𝑖𝜆𝑦𝑖+(𝑡)𝑁𝑗=1̂𝑐𝑖𝑗𝜆𝑦𝑗(𝑡𝜏(𝑡))+𝑢𝑖,𝑖=1,2,,𝑁.(2.24)

Corollary 2.7. Suppose that (A1)–(A3) hold. Using the controller 𝑢𝑖=1𝜆𝑑𝑖𝑒𝑖(𝑡)(2.25) and the update laws ̇𝑑𝑖=𝑘𝑖𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡),̇̂𝑐𝑖𝑗=𝛿𝑖𝑗𝑒𝑇𝑖(𝑡)𝜆𝑦𝑗,(𝑡𝜏(𝑡))(2.26) where 𝑖,𝑗=1,2,,𝑁,𝑑𝑖 is feedback strength, and 𝛿𝑖𝑗>0,𝑘𝑖>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:̇𝑦𝑖𝑦(𝑡)=𝐷Φ𝑖(𝑡)1𝑓Φ𝑦𝑖+(𝑡)𝑁𝑗=1̂𝑐𝑖𝑗Φ𝑦𝑗(𝑡𝜏(𝑡))+𝑢𝑖,𝑖=1,,𝑙,̇𝑦𝑖𝑦(𝑡)=𝐷Φ𝑖(𝑡)1𝑔Φ𝑦𝑖+(𝑡)𝑁𝑗=1̂𝑐𝑖𝑗Φ𝑦𝑗(𝑡𝜏(𝑡))+𝑢𝑖,𝑖=𝑙+1,,𝑁(2.27) and get the following corollary:

Corollary 2.8. Suppose that (A1)–(A3) hold. Using the controller 𝑢𝑖𝑦=𝐷Φ𝑖(𝑡)1𝑑𝑖𝑒𝑖(𝑡)(2.28) and the update laws ̇𝑑𝑖=𝑘𝑖𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡),̇̂𝑐𝑖𝑗=𝛿𝑖𝑗𝑒𝑇𝑖Φ𝑦(𝑡)𝑗,(𝑡𝜏(𝑡))(2.29) where 𝑖,𝑗=1,2,,𝑁,𝑑𝑖 is feedback strength, 𝛿𝑖𝑗>0,𝑘𝑖>0 are arbitrary constants, then the complex dynamical network networks (2.4) and (2.27) will achieve GS with respect to Φ.

If coupling function (𝑥𝑖)=𝐴𝑥𝑖; that is, the network is linearly coupled, then the complex network (2.2) degenerates into (2.5). Note that 𝐴𝑒𝑖(𝑡)𝐴𝑒𝑖(𝑡),𝑖=1,2,,𝑁 hold. We construct the following response network:̇𝑦𝑖𝑦(𝑡)=𝐷Φ𝑖(𝑡)1𝑓𝑖Φ𝑦𝑖+(𝑡)𝑁𝑗=1̂𝑐𝑖𝑗𝑦𝐴Φ𝑗(𝑡𝜏(𝑡))+𝑢𝑖,𝑖=1,2,,𝑁.(2.30)

Corollary 2.9. Suppose that (A1) and (A2) hold. Using the controller 𝑢𝑖𝑦=𝐷Φ𝑖(𝑡)1𝑑𝑖𝑒𝑖(𝑡)(2.31) and the update laws ̇𝑑𝑖=𝑘𝑖𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡),̇̂𝑐𝑖𝑗=𝛿𝑖𝑗𝑒𝑇𝑖𝑦(𝑡)𝐴Φ𝑗,(𝑡𝜏(𝑡))(2.32) where 𝑖,𝑗=1,2,,𝑁,𝑑𝑖 is feedback strength, and 𝛿𝑖𝑗>0,𝑘𝑖>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: 𝑢𝑖𝑦=𝐷Φ𝑖(𝑡)1𝑑𝑖𝑒𝑖(𝑡)+𝑓𝑖𝑥𝑖(𝑡)𝑓𝑖Φ𝑦𝑖(𝑡),(2.33) and the update laws ̇𝑑𝑖=𝑘𝑖𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡),(2.34)̇̂𝑐𝑖𝑗=𝛿𝑖𝑗𝑒𝑇𝑖Φ𝑦(𝑡)𝑗,(𝑡𝜏(𝑡))(2.35) where 𝑖,𝑗=1,2,,𝑁,𝑑𝑖 is feedback strength, and 𝛿𝑖𝑗>0,𝑘𝑖>0 are arbitrary constants, then the complex dynamical networks (2.2) and (2.7) will achieve GS with respect to Φ.

Proof. Select the same Lyapunov-Krasovskii function as Theorem 2.5, then d𝑉d𝑡=2𝑁𝑖=1𝑒𝑇𝑖𝑓(𝑡)𝑖𝑥𝑖(𝑡)𝑓𝑖Φ𝑦𝑖+(𝑡)𝑁𝑗=1𝑐𝑖𝑗𝐻𝑒𝑗(𝑡𝜏(𝑡))𝑁𝑗=1̃𝑐𝑖𝑗Φ𝑦𝑗𝑦(𝑡𝜏(𝑡))𝐷Φ𝑖𝑢(𝑡)𝑖+(𝑡)𝑁𝑁𝑖=1𝑗=12𝛿𝑖𝑗̃𝑐𝑖𝑗̇̂𝑐𝑖𝑗+𝑁𝑖=12𝑘𝑖𝑑𝑖̇𝑑𝑑𝑖+𝑁𝛽21𝜇𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)1̇𝜏(𝑡)1𝜇𝑁𝛽2𝑁𝑖=1𝑒𝑇𝑖(𝑡𝜏(𝑡))𝑒𝑖(𝑡𝜏(𝑡))𝑁𝑖=12𝑑+𝑁𝛽2𝑒1𝜇𝑇𝑖(𝑡)𝑒𝑖+(𝑡)𝑁𝑁𝑖=1𝑗=1𝑒𝑇𝑖(𝑡)𝑐𝑖𝑗𝐻𝑒𝑗(𝑡𝜏(𝑡))1̇𝜏(𝑡)1𝜇𝑁𝛽2𝑁𝑖=1𝑒𝑇𝑖(𝑡𝜏(𝑡))𝑒𝑖(𝑡𝜏(𝑡)).(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 {(Φ(𝑦𝑗(𝑡𝜏(𝑡))))}𝑁𝑗=1 are linearly independent on the orbit {𝑦𝑗(𝑡𝜏(𝑡))}𝑁𝑗=1 of synchronization manifold, we can get lim𝑡+̂𝑐𝑖𝑗=𝑐𝑖𝑗,𝑖,𝑗=1,2,,𝑁; that is, the uncertain coupling matrix 𝐶 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 2.72.9.

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 bẏ𝑥1̇𝑥2̇𝑥3=𝑥(25𝛽+10)1𝑥2𝑥1𝑥3+(2835𝛽)𝑥1+(29𝛽1)𝑥2𝑥1𝑥2(𝛽+8)3𝑥3=𝑥(3.1)101𝑥2𝑥1𝑥3+28𝑥1𝑥2𝑥1𝑥283𝑥3+𝑥251𝑥235𝑥1+29𝑥213𝑥3𝛽=𝐹(𝑥)+𝐺(𝑥)𝛽,(3.2) which is chaotic if 𝛽[0,1]. Obviously, system (3.2) is the original Lorenz system for 𝛽=0 while system (3.2) belongs to the original Chen system for 𝛽=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 aṡ𝑥1̇𝑥2̇𝑥3=𝑎𝑥1𝑥2𝑥3𝑏𝑥2+𝑥1𝑥3𝑐𝑥3+13𝑥1𝑥2=𝑔(𝑥).(3.3) It is chaotic when 𝑎=5.0,𝑏=10.0, and 𝑐=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 𝑓(𝑥)=𝐹(𝑥)+𝐺(𝑥)𝛽(𝛽[0,1]) satisfies Assumption 2.

Since the attractor is confined to a bounded region, there exists a constant 𝑀>0, satisfying for all𝑦=(𝑦1,𝑦2,𝑦3),𝑧=(𝑧1,𝑧2,𝑧3)𝑅3,||𝑦||𝑀,||𝑧||𝑀; therefore,𝑓(𝑦)𝑓(𝑧)2=(25𝛽+10)2𝑦2𝑦1𝑧2𝑧12+𝑧1𝑧3𝑦1𝑦3𝑦+(2835𝛽)1𝑧1𝑦+(29𝛽1)2𝑧22+𝑦1𝑦2𝑧1𝑧2(𝛽+8)3𝑦3𝑧32=(25𝛽+10)2𝑦2𝑧2𝑦1𝑧12+𝑧1𝑧3𝑦3+𝑦3𝑦+2835𝛽1𝑧1𝑦+(29𝛽1)2𝑧22+𝑦1𝑦2𝑧2+𝑧2𝑦1𝑧1(𝛽+8)3𝑦3𝑧323522𝑦2𝑧22𝑦+21𝑧12+3𝑀2𝑦3𝑧32+6282+𝑀2𝑦1𝑧12+3×282𝑦2𝑧22+3𝑀2𝑦2𝑧22+3𝑀2𝑦1𝑧12𝑦+93𝑧322×352+6×282+9𝑀2𝑦𝑧2.(3.4) Thus, function 𝑓(𝑥)=𝐹(𝑥)+𝐺(𝑥)𝛽(𝛽[0,1]) satisfies Assumption 2. By the same process, we can obtain that function 𝑔(𝑥) 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 ̇𝑥𝑖𝑥(𝑡)=𝐹𝑖+(𝑡)5𝑗=1𝑐𝑖𝑗𝑥𝑗(𝑡𝜏(𝑡)),𝑖=1,2,3,̇𝑥𝑖𝑥(𝑡)=𝑔𝑖+(𝑡)5𝑗=1𝑐𝑖𝑗𝑥𝑗(𝑡𝜏(𝑡)),𝑖=4,5,(3.5) where 𝑥𝑖(𝑡)=(𝑥𝑖1(𝑡),𝑥𝑖2(𝑡),𝑥𝑖3(𝑡))𝑇,𝑖=1,2,,5.𝜏(𝑡)=0.1, the weight configuration matrix 𝑐𝐶=𝑖𝑗5×5=5131012100314001002100011.(3.6) The coupling functions are (𝑥𝑗(𝑡))=(sin(𝑥𝑗1(𝑡)),arctan(𝑥𝑗2(𝑡)),arctan(𝑥𝑗3(𝑡)))𝑇,𝑗=1,2,,5.
Let Φ(𝑦𝑖)=(𝑦𝑖1+𝑦𝑖2,2𝑦𝑖2,2𝑦𝑖3)𝑇,then 𝐷Φ(𝑦𝑖)=110020002,𝑖=1,2,,5.
Since (A1)–(A3) hold, therefore, according to Theorem 2.5, we can use the following response network:̇𝑦𝑖𝑦(𝑡)=𝐷Φ𝑖(𝑡)1𝐹Φ𝑦𝑖+(𝑡)𝑁𝑗=1̂𝑐𝑖𝑗Φ𝑦𝑗(𝑡𝜏(𝑡))+𝑢𝑖,𝑖=1,2,3,̇𝑦𝑖𝑦(𝑡)=𝐷Φ𝑖(𝑡)1𝑔Φ𝑦𝑖+(𝑡)𝑁𝑗=1̂𝑐𝑖𝑗Φ𝑦𝑗(𝑡𝜏(𝑡))+𝑢𝑖,𝑖=4,5.(3.7)
The controller and update laws are given by (2.15)–(2.17). The initial values are given as follows: ̂𝑐𝑖𝑗(0)=3,𝛿𝑖𝑗=1,𝑑𝑖(0)=1,𝑘𝑖=1,𝑥𝑖(0)=(12+𝑖×0.1,15+𝑖×0.1,30+𝑖×0.15)𝑇,̂𝑥𝑖(0)=(5+𝑖×0.1,7.5+𝑖×0.1,15+𝑖×0.1),𝑖,𝑗=1,2,,5. Figure 1 shows GS errors 𝑒𝑖(𝑡), 𝑖=1,2,,5. One can see that all nodes’ errors converge to zero. Some elements of matrix 𝐶 are displayed in Figure 2. The numerical results show that this adaptive scheme is effective and we can get lim𝑡̂𝑐𝑖𝑗=𝑐𝑖𝑗,𝑖,𝑗=1,2,,5.

fig1
Figure 1: GS errors of model (3.5) and (3.7) with respect to Φ(𝑦)=(𝑦1+𝑦2,2𝑦2,2𝑦3)𝑇.
fig2
Figure 2: Estimation of topology of model (3.5).

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 ̇𝑥𝑖(𝑡)=𝑓𝑖𝑥𝑖+(𝑡)5𝑗=1𝑐𝑖𝑗𝑥𝑗(𝑡𝜏(𝑡)),𝑖=1,2,,5,(3.8) where 𝑥𝑖(𝑡)=(𝑥𝑖1(𝑡),𝑥𝑖2(𝑡),𝑥𝑖3(𝑡))𝑇,𝑓𝑖(𝑥)=𝐹(𝑥)𝛽𝑖+𝐺(𝑥),𝛽𝑖=0.1×(𝑖1),𝑖=1,2,,5. We assume 𝜏(𝑡)=0.3,(𝑥𝑗(𝑡))=(arctan(𝑥𝑗1(𝑡)),arctan(𝑥𝑗2(𝑡)),arctan(𝑥𝑗3(𝑡)))𝑇,𝑗=1,2,,5. C is the same as that in model (3.5).
Let Φ(𝑦𝑖)=(𝑦𝑖1+𝑦𝑖2,2𝑦𝑖2,𝑦3𝑖3+𝑦𝑖3)𝑇,𝐷Φ(𝑦𝑖)=110020003𝑦2𝑖3+1,𝑖=1,2,,5.
According to Theorem 2.10, the response network is given bẏ𝑦𝑖𝑦(𝑡)=𝐷Φ𝑖(𝑡)1𝑓𝑖Φ𝑦𝑖+(𝑡)𝑁𝑗=1̂𝑐𝑖𝑗Φ𝑦𝑗(𝑡𝜏(𝑡))+𝑢𝑖,𝑖=1,2,,5.(3.9) The controller and update laws are given by (2.33)–(2.35). The initial values are given as follows: ̂𝑐𝑖𝑗(0)=6,𝛿𝑖𝑗=1,𝑑𝑖(0)=1,𝑘𝑖=1,𝑥𝑖(0)=(12,15,30)𝑇,̂𝑥𝑖(0)=(5,7.5,3)𝑇,𝑖,𝑗=1,2,,5. Figures 3 and 4 show the GS errors 𝑒𝑖(𝑡), 𝑖=1,2,,5 and some elements of matrix 𝐶, respectively. The results illustrate that this scheme is effective and we can get lim𝑡+̂𝑐𝑖𝑗=𝑐𝑖𝑗,𝑖,𝑗=1,2,,5.

fig3
Figure 3: GS errors of model (3.8) and (3.9) with respect to Φ(𝑦)=(𝑦1+𝑦2,2𝑦2,𝑦33+𝑦3)𝑇.
fig4
Figure 4: Estimation of topology of model (3.8).

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 Barbǎlat 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.

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