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 [1–5]. 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 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 π‘₯𝑖=(π‘₯𝑖1,π‘₯𝑖2,…,π‘₯𝑖𝑛)π‘‡βˆˆπ‘…π‘›,𝑦𝑖=(𝑦𝑖1,𝑦𝑖2,…,𝑦𝑖𝑛)π‘‡βˆˆπ‘…π‘›,𝑖=1,2,…,𝑁 are the state variables of the drive network and the response network, respectively. Given the smooth vector function Ξ¦βˆΆπ‘…π‘›β†’π‘…π‘›, the drive network and response network are said to achieve GS with respect to Ξ¦. If limπ‘‘β†’βˆžβ€–β€–π‘’π‘–β€–β€–(𝑑)=0,𝑖=1,2,…,𝑁,(2.1) where 𝑒𝑖(𝑑)=π‘₯𝑖(𝑑)βˆ’Ξ¦(𝑦𝑖(𝑑)),𝑖=1,2,…,𝑁, the norm ||β‹…|| of a vector π‘₯ is defined as ||π‘₯||=(π‘₯𝑇π‘₯)1/2.

Remark 2.2. If Ξ¦(𝑦𝑖)=𝑦𝑖, then GS is CS in [20]. If Ξ¦(𝑦𝑖)=πœ†π‘¦π‘–, 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:Μ‡π‘₯𝑖(𝑑)=𝑓𝑖π‘₯𝑖+(𝑑)𝑁𝑗=1π‘π‘–π‘—β„Žξ€·π‘₯𝑗(π‘‘βˆ’πœ(𝑑)),𝑖=1,2,…,𝑁,(2.2) where π‘₯𝑖=(π‘₯𝑖1,π‘₯𝑖2,…,π‘₯𝑖𝑛)π‘‡βˆˆπ‘…π‘›,𝑖=1,2,…,𝑁 are the state variables of the drive network, π‘“π‘–βˆΆπ‘…π‘›β†’π‘…π‘›, β„ŽβˆΆπ‘…π‘›β†’π‘…π‘›are smooth nonlinear vector functions, and 𝜏(𝑑) is time-varying delay. 𝐢=(𝑐𝑖𝑗)𝑁×𝑁 is unknown or uncertain coupling matrix; if there is a connection between node 𝑖 and node 𝑗(𝑗≠𝑖), then 𝑐𝑖𝑗≠0, otherwise𝑐𝑖𝑗=0(𝑖≠𝑗),and the diagonal elements of 𝐢 are defined by𝑐𝑖𝑖=βˆ’π‘ξ“π‘—=1𝑗≠𝑖𝑐𝑖𝑗.(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:Μ‡π‘₯𝑖π‘₯(𝑑)=𝑓𝑖+(𝑑)𝑁𝑗=1π‘π‘–π‘—β„Žξ€·π‘₯𝑗(π‘‘βˆ’πœ(𝑑)),𝑖=1,…,𝑙,Μ‡π‘₯𝑖π‘₯(𝑑)=𝑔𝑖+(𝑑)𝑁𝑗=1π‘π‘–π‘—β„Žξ€·π‘₯𝑗(π‘‘βˆ’πœ(𝑑)),𝑖=𝑙+1,…,𝑁.(2.4) On the other hand, if β„Ž(π‘₯𝑖)=𝐴π‘₯𝑖, with 𝐴=(π‘Žπ‘–π‘—)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:Μ‡π‘₯𝑖(𝑑)=𝑓𝑖π‘₯𝑖+(𝑑)𝑁𝑗=1𝑐𝑖𝑗𝐴π‘₯𝑗(π‘‘βˆ’πœ(𝑑)),𝑖=1,2,…,𝑁.(2.5) Let𝑦𝐷Φ𝑖=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπœ•πœ™1ξ€·π‘¦π‘–ξ€Έπœ•π‘¦π‘–1πœ•πœ™1ξ€·π‘¦π‘–ξ€Έπœ•π‘¦π‘–2β‹―πœ•πœ™1ξ€·π‘¦π‘–ξ€Έπœ•π‘¦π‘–π‘›πœ•πœ™2ξ€·π‘¦π‘–ξ€Έπœ•π‘¦π‘–1πœ•πœ™2ξ€·π‘¦π‘–ξ€Έπœ•π‘¦π‘–2β‹―πœ•πœ™2ξ€·π‘¦π‘–ξ€Έπœ•π‘¦π‘–π‘›β‹―β‹―β‹―β‹―πœ•πœ™π‘›ξ€·π‘¦π‘–ξ€Έπœ•π‘¦π‘–1πœ•πœ™π‘›ξ€·π‘¦π‘–ξ€Έπœ•π‘¦π‘–2β‹―πœ•πœ™π‘›ξ€·π‘¦π‘–ξ€Έπœ•π‘¦π‘–π‘›βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ (2.6) be the Jacobian matrix of the mapping Ξ¦(𝑦𝑖)=(πœ™1(𝑦𝑖),πœ™2(𝑦𝑖),…,πœ™π‘›(𝑦𝑖))𝑇,πœ™π‘—(𝑦𝑖)βˆˆπ‘…, 𝑖=1,2,…,𝑁, 𝑗=1,2,…,𝑛. When matrix 𝐷Φ(𝑦𝑖(𝑑)) is reversible, we can give the following controller response network:̇𝑦𝑖𝑦(𝑑)=𝐷Φ𝑖(𝑑)βˆ’1𝑓𝑖Φ𝑦𝑖+(𝑑)𝑁𝑗=1Μ‚π‘π‘–π‘—β„Žξ€·Ξ¦ξ€·π‘¦π‘—ξƒ­(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ+𝑒𝑖,𝑖=1,2,…,𝑁,(2.7) where 𝑦𝑖=(𝑦𝑖1,𝑦𝑖2,…,𝑦𝑖𝑛)π‘‡βˆˆπ‘…π‘›,𝑖=1,2,…,𝑁 are the state variables of the response network, π‘’π‘–βˆˆπ‘…π‘›,𝑖=1,2,…,𝑁 are nonlinear controllers to be designed, and 𝐢=(̂𝑐𝑖𝑗)𝑁×𝑁 is the estimate of the unknown coupling matrix 𝐢=(𝑐𝑖𝑗)𝑁×𝑁.

Let 𝑒𝑖(𝑑)=π‘₯𝑖(𝑑)βˆ’Ξ¦(𝑦𝑖(𝑑)), with the aid of (2.2) and (2.7), the following error network can be obtained:̇𝑒𝑖(𝑑)=Μ‡π‘₯𝑖𝑦(𝑑)βˆ’π·Ξ¦π‘–(𝑑)̇𝑦𝑖(𝑑)=𝑓𝑖π‘₯𝑖(𝑑)βˆ’π‘“π‘–ξ€·Ξ¦ξ€·π‘¦π‘–+(𝑑)𝑁𝑗=1π‘π‘–π‘—β„Žξ€·π‘₯π‘—ξ€Έβˆ’(π‘‘βˆ’πœ(𝑑))𝑁𝑗=1Μ‚π‘π‘–π‘—β„Žξ€·Ξ¦ξ€·π‘¦π‘—ξ€·π‘¦(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έβˆ’π·Ξ¦π‘–ξ€Έπ‘’(𝑑)𝑖=𝑓𝑖π‘₯𝑖(𝑑)βˆ’π‘“π‘–ξ€·Ξ¦ξ€·π‘¦π‘–+(𝑑)𝑁𝑗=1π‘π‘–π‘—π»ξ€·π‘’π‘—ξ€Έβˆ’(π‘‘βˆ’πœ(𝑑))𝑁𝑗=1Μƒπ‘π‘–π‘—β„Žξ€·Ξ¦ξ€·π‘¦π‘—ξ€·π‘¦(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έβˆ’π·Ξ¦π‘–ξ€Έπ‘’(𝑑)𝑖,(2.8) where 𝐻𝑒𝑗π‘₯(𝑑)=β„Žπ‘—ξ€Έξ€·Ξ¦ξ€·π‘¦(𝑑)βˆ’β„Žπ‘—(𝑑)ξ€Έξ€Έ,(2.9)̃𝑐𝑖𝑗=Μ‚π‘π‘–π‘—βˆ’π‘π‘–π‘—.(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(𝑑) is a differential function with 0β‰€πœ(𝑑)β‰€β„Ž, Μ‡πœ(𝑑)β‰€πœ‡<1, where β„Ž and πœ‡ are positive constants. Obviously, this assumption holds for constant 𝜏(𝑑).

Assumption 2. (A2) Suppose that 𝑓𝑖(β‹…) is bounded and there exists a nonnegative constant 𝛼 such that ‖‖𝑓𝑖π‘₯𝑖(𝑑)βˆ’π‘“π‘–ξ€·Ξ¦ξ€·π‘¦π‘–β€–β€–β€–β€–π‘’(𝑑)≀𝛼𝑖‖‖(𝑑),𝑖=1,2,…,𝑁.(2.11)

Assumption 3. (A3) Suppose that β„Ž(β‹…) is bounded and there exists a nonnegative constant 𝛽 such that β€–β€–β„Žξ€·π‘₯𝑖Φ𝑦(𝑑)βˆ’β„Žπ‘–β€–β€–β€–β€–π‘’(𝑑)≀𝛽𝑖‖‖(𝑑),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 dπ‘₯𝑖(𝑑)π‘₯d𝑑=βˆ’π‘–(𝑑)𝑅𝑖+𝑁𝑗=1𝑀𝑖𝑗𝑓𝑗π‘₯π‘—ξ€·π‘‘βˆ’πœπ‘–π‘—(𝑑)ξ€Έξ€Έ+𝐼𝑖,𝑖=1,2,…,𝑁.(2.13) Take 𝑓𝑗(π‘₯𝑗)=(πœ‹/2)arctan((πœ‹/2)πœ†π‘₯𝑗),where πœ† is positive constant. It is obvious that 𝑓𝑗(β‹…) satisfies Assumption 3.

Lemma 2.4. For any vectors 𝑋,π‘Œβˆˆπ‘…π‘›,the following inequality holds 2π‘‹π‘‡π‘Œβ‰€π‘‹π‘‡π‘‹+π‘Œπ‘‡π‘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)–(A3) hold. Using the following controller: 𝑒𝑖𝑦=𝐷Φ𝑖(𝑑)βˆ’1𝑑𝑖𝑒𝑖(𝑑)(2.15) and the update laws ̇𝑑𝑖=π‘˜π‘–π‘’π‘‡π‘–(𝑑)𝑒𝑖(𝑑),(2.16)̇̂𝑐𝑖𝑗=𝛿𝑖𝑗𝑒𝑇𝑖Φ𝑦(𝑑)β„Žπ‘—,(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ(2.17) where 𝑖,𝑗=1,2,…,𝑁,𝑑𝑖 is feedback strength, and 𝛿𝑖𝑗>0,π‘˜π‘–>0 are arbitrary constants, then the complex dynamical networks (2.2) and (2.7) will achieve GS with respect to Ξ¦.

Proof. Select a Lyapunov-Krasovskii functional candidate as 𝑉(𝑑,𝑒(𝑑))=𝑁𝑖=1𝑒𝑇𝑖(𝑑)𝑒𝑖(𝑑)+𝑁𝑁𝑖=1𝑗=11𝛿𝑖𝑗̃𝑐2𝑖𝑗+𝑁𝑖=11π‘˜π‘–ξ€·π‘‘π‘–ξ€Έβˆ’π‘‘2+𝑁𝛽2ξ€œ1βˆ’πœ‡π‘‘π‘π‘‘βˆ’πœ(𝑑)𝑖=1𝑒𝑇𝑖(πœ‰)𝑒𝑖(πœ‰)dπœ‰,(2.18) where 𝑒(𝑑)=(𝑒𝑇1(𝑑),𝑒𝑇2(𝑑),…,𝑒𝑇𝑁(𝑑))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=d𝑑𝑁𝑖=12𝑒𝑇𝑖(𝑑)̇𝑒𝑖(𝑑)+𝑁𝑁𝑖=1𝑗=12𝛿𝑖𝑗̃𝑐𝑖𝑗̇̂𝑐𝑖𝑗+𝑁𝑖=12π‘˜π‘–ξ€·π‘‘π‘–ξ€ΈΜ‡π‘‘βˆ’π‘‘π‘–+𝑁𝛽21βˆ’πœ‡π‘ξ“π‘–=1𝑒𝑇𝑖(𝑑)π‘’π‘–βˆ’(𝑑)1βˆ’Μ‡πœ(𝑑)1βˆ’πœ‡π‘π›½2𝑁𝑖=1𝑒𝑇𝑖(π‘‘βˆ’πœ(𝑑))𝑒𝑖=(π‘‘βˆ’πœ(𝑑))𝑁𝑖=12𝑒𝑇𝑖𝑓(𝑑)𝑖π‘₯𝑖(𝑑)βˆ’π‘“π‘–ξ€·Ξ¦ξ€·π‘¦π‘–+(𝑑)𝑁𝑗=1π‘π‘–π‘—π»ξ€·π‘’π‘—ξ€Έβˆ’(π‘‘βˆ’πœ(𝑑))𝑁𝑗=1Μƒπ‘π‘–π‘—β„Žξ€·Ξ¦ξ€·π‘¦π‘—ξ€·π‘¦(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έβˆ’π·Ξ¦π‘–ξ€Έπ‘’(𝑑)𝑖+(𝑑)𝑁𝑁𝑖=1𝑗=12𝛿𝑖𝑗̃𝑐𝑖𝑗̇̂𝑐𝑖𝑗+𝑁𝑖=12π‘˜π‘–ξ€·π‘‘π‘–ξ€ΈΜ‡π‘‘βˆ’π‘‘π‘–+𝑁𝛽21βˆ’πœ‡π‘ξ“π‘–=1𝑒𝑇𝑖(𝑑)𝑒𝑖(𝑑)βˆ’1βˆ’Μ‡πœ(𝑑)1βˆ’πœ‡π‘π›½2𝑁𝑖=1𝑒𝑇𝑖(π‘‘βˆ’πœ(𝑑))𝑒𝑖(π‘‘βˆ’πœ(𝑑)).(2.19) Substituting the controller (2.15) and update laws (2.16)-(2.17) into (2.19) and considering Assumption 2, we obtain d𝑉≀d𝑑𝑁𝑖=1ξ‚΅2π›Όβˆ’2𝑑+𝑁𝛽2𝑒1βˆ’πœ‡π‘‡π‘–(𝑑)𝑒𝑖(𝑑)+2𝑁𝑁𝑖=1𝑗=1𝑒𝑇𝑖(𝑑)π‘π‘–π‘—π»ξ€·π‘’π‘—ξ€Έβˆ’(π‘‘βˆ’πœ(𝑑))1βˆ’Μ‡πœ(𝑑)1βˆ’πœ‡π‘π›½2𝑁𝑖=1𝑒𝑇𝑖(π‘‘βˆ’πœ(𝑑))𝑒𝑖(π‘‘βˆ’πœ(𝑑)).(2.20) By Lemma 2.4 and considering Assumptions 1 and 3, we have 1βˆ’Μ‡πœ(𝑑)21βˆ’πœ‡β‰₯1,𝑁𝑁𝑖=1𝑗=1𝑒𝑇𝑖(𝑑)𝑐𝑖𝑗𝐻𝑒𝑗≀(π‘‘βˆ’πœ(𝑑))𝑁𝑁𝑖=1𝑗=1𝑐2𝑖𝑗𝑒𝑇𝑖(𝑑)𝑒𝑖(𝑑)+𝑁𝑁𝑖=1𝑗=1𝐻𝑇𝑒𝑗𝐻𝑒(π‘‘βˆ’πœ(𝑑))𝑗(π‘‘βˆ’πœ(𝑑))≀𝑁max1≀𝑖,𝑗≀𝑁𝑐2𝑖𝑗𝑁𝑖=1𝑒𝑇𝑖(𝑑)𝑒𝑖(𝑑)+𝑁𝛽2𝑁𝑗=1𝑒𝑇𝑗(π‘‘βˆ’πœ(𝑑))𝑒𝑗(π‘‘βˆ’πœ(𝑑)),(2.21) then d𝑉≀d𝑑𝑁𝑖=1ξ‚΅2π›Όβˆ’2𝑑+𝑁𝛽21βˆ’πœ‡+𝑁max1≀𝑖,𝑗≀𝑁𝑐2𝑖𝑗𝑒𝑇𝑖(𝑑)𝑒𝑖(𝑑)=βˆ’2π‘‘βˆ’2π›Όβˆ’π‘π›½21βˆ’πœ‡βˆ’π‘max1≀𝑖,𝑗≀𝑁𝑐2𝑖𝑗𝑒𝑇(𝑑)𝑒(𝑑).(2.22)
Note that we can choose constant d to make d𝑉/dπ‘‘β‰€βˆ’π‘’π‘‡(𝑑)𝑒(𝑑)≀0, thus 𝑉 is nonincreasing in 𝑑β‰₯0. One has 𝑉 is bounded since 0≀𝑉(𝑑,𝑒(𝑑))≀𝑉(0,𝑒(0)), so lim𝑑→+βˆžπ‘‰(𝑑,𝑒(𝑑)) exists andlim𝑑→+βˆžξ€œπ‘‘0𝑒𝑇(𝑠)𝑒(𝑠)dπ‘ β‰€βˆ’lim𝑑→+βˆžξ€œπ‘‘0d𝑉d𝑠d𝑠=𝑉(0,𝑒(0))βˆ’lim𝑑→+βˆžπ‘‰(𝑑,𝑒(𝑑)).(2.23) From (2.18), we have 0≀𝑒𝑇(𝑑)𝑒(𝑑)≀𝑉(𝑑,𝑒(𝑑)), so 𝑒𝑇(𝑑)𝑒(𝑑) is bounded. According to error system (2.8), (d/d𝑑)𝑒𝑇(𝑑)𝑒(𝑑)=2𝑒𝑇(𝑑)̇𝑒(𝑑) is bounded for 𝑑β‰₯0 due to the boundedness of 𝑓𝑖(β‹…) and β„Ž(β‹…). From the above, we can see that 𝑒(𝑑)∈𝐿2∩𝐿∞ and ̇𝑒(𝑑)∈𝐿∞. By using another form of Barbǎlat lemma [24], one has lim𝑑→+βˆžπ‘’π‘‡(𝑑)𝑒(𝑑)=0, so lim𝑑→+βˆžπ‘’(𝑑)=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 lim𝑑→+βˆžΜ‡π‘’(𝑑) exists,then we can obtain lim𝑑→+βˆžΜ‡π‘’(𝑑)=0 for lim𝑑→+βˆžπ‘’(𝑑)=0. According to error system (2.8), we have lim𝑑→+βˆžβˆ‘π‘π‘—=1Μƒπ‘π‘–π‘—β„Ž(Ξ¦(𝑦𝑗(π‘‘βˆ’πœ(𝑑))))=0. When {β„Ž(Ξ¦(𝑦𝑗(π‘‘βˆ’πœ(𝑑)))}𝑁𝑗=1 are linearly independent on the orbit {𝑦𝑗(π‘‘βˆ’πœ(𝑑))}𝑁𝑗=1 of synchronization manifold, lim𝑑→+βˆžΜƒπ‘π‘–π‘—=0. We can get lim𝑑→+βˆžΜ‚π‘π‘–π‘—=𝑐𝑖𝑗,𝑖,𝑗=1,2,…,𝑁; that is, the uncertain coupling matrix 𝐢 can be successfully estimated using the update laws (2.17).
In a special case Ξ¦(𝑦𝑖)=πœ†π‘¦π‘–(πœ†is nonzero constant), based on Theorem 2.5, we can construct the following response network̇𝑦𝑖1(𝑑)=πœ†ξƒ¬π‘“π‘–ξ€·πœ†π‘¦π‘–ξ€Έ+(𝑑)𝑁𝑗=1Μ‚π‘π‘–π‘—β„Žξ€·πœ†π‘¦π‘—ξ€Έξƒ­(π‘‘βˆ’πœ(𝑑))+𝑒𝑖,𝑖=1,2,…,𝑁.(2.24)

Corollary 2.7. Suppose that (A1)–(A3) hold. Using the controller 𝑒𝑖=1πœ†π‘‘π‘–π‘’π‘–(𝑑)(2.25) and the update laws ̇𝑑𝑖=π‘˜π‘–π‘’π‘‡π‘–(𝑑)𝑒𝑖(𝑑),̇̂𝑐𝑖𝑗=𝛿𝑖𝑗𝑒𝑇𝑖(𝑑)β„Žπœ†π‘¦π‘—ξ€Έ,(π‘‘βˆ’πœ(𝑑))(2.26) where 𝑖,𝑗=1,2,…,𝑁,𝑑𝑖 is feedback strength, and 𝛿𝑖𝑗>0,π‘˜π‘–>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:̇𝑦𝑖𝑦(𝑑)=𝐷Φ𝑖(𝑑)βˆ’1𝑓Φ𝑦𝑖+(𝑑)𝑁𝑗=1Μ‚π‘π‘–π‘—β„Žξ€·Ξ¦ξ€·π‘¦π‘—ξƒ­(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ+𝑒𝑖,𝑖=1,…,𝑙,̇𝑦𝑖𝑦(𝑑)=𝐷Φ𝑖(𝑑)βˆ’1𝑔Φ𝑦𝑖+(𝑑)𝑁𝑗=1Μ‚π‘π‘–π‘—β„Žξ€·Ξ¦ξ€·π‘¦π‘—ξƒ­(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ+𝑒𝑖,𝑖=𝑙+1,…,𝑁(2.27) and get the following corollary:

Corollary 2.8. Suppose that (A1)–(A3) hold. Using the controller 𝑒𝑖𝑦=𝐷Φ𝑖(𝑑)βˆ’1𝑑𝑖𝑒𝑖(𝑑)(2.28) and the update laws ̇𝑑𝑖=π‘˜π‘–π‘’π‘‡π‘–(𝑑)𝑒𝑖(𝑑),̇̂𝑐𝑖𝑗=𝛿𝑖𝑗𝑒𝑇𝑖Φ𝑦(𝑑)β„Žπ‘—,(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ(2.29) where 𝑖,𝑗=1,2,…,𝑁,𝑑𝑖 is feedback strength, 𝛿𝑖𝑗>0,π‘˜π‘–>0 are arbitrary constants, then the complex dynamical network networks (2.4) and (2.27) will achieve GS with respect to Ξ¦.

If coupling function β„Ž(π‘₯𝑖)=𝐴π‘₯𝑖; that is, the network is linearly coupled, then the complex network (2.2) degenerates into (2.5). Note that ‖𝐴𝑒𝑖(𝑑)‖≀‖𝐴‖⋅‖𝑒𝑖(𝑑)β€–,𝑖=1,2,…,𝑁 hold. We construct the following response network:̇𝑦𝑖𝑦(𝑑)=𝐷Φ𝑖(𝑑)βˆ’1𝑓𝑖Φ𝑦𝑖+(𝑑)𝑁𝑗=1̂𝑐𝑖𝑗𝑦𝐴Φ𝑗(π‘‘βˆ’πœ(𝑑))+𝑒𝑖,𝑖=1,2,…,𝑁.(2.30)

Corollary 2.9. Suppose that (A1) and (A2) hold. Using the controller 𝑒𝑖𝑦=𝐷Φ𝑖(𝑑)βˆ’1𝑑𝑖𝑒𝑖(𝑑)(2.31) and the update laws ̇𝑑𝑖=π‘˜π‘–π‘’π‘‡π‘–(𝑑)𝑒𝑖(𝑑),̇̂𝑐𝑖𝑗=𝛿𝑖𝑗𝑒𝑇𝑖𝑦(𝑑)𝐴Φ𝑗,(π‘‘βˆ’πœ(𝑑))(2.32) where 𝑖,𝑗=1,2,…,𝑁,𝑑𝑖 is feedback strength, and 𝛿𝑖𝑗>0,π‘˜π‘–>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: 𝑒𝑖𝑦=𝐷Φ𝑖(𝑑)βˆ’1𝑑𝑖𝑒𝑖(𝑑)+𝑓𝑖π‘₯𝑖(𝑑)βˆ’π‘“π‘–ξ€·Ξ¦ξ€·π‘¦π‘–(𝑑)ξ€Έξ€Έξ€»,(2.33) and the update laws ̇𝑑𝑖=π‘˜π‘–π‘’π‘‡π‘–(𝑑)𝑒𝑖(𝑑),(2.34)̇̂𝑐𝑖𝑗=𝛿𝑖𝑗𝑒𝑇𝑖Φ𝑦(𝑑)β„Žπ‘—,(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ(2.35) where 𝑖,𝑗=1,2,…,𝑁,𝑑𝑖 is feedback strength, and 𝛿𝑖𝑗>0,π‘˜π‘–>0 are arbitrary constants, then the complex dynamical networks (2.2) and (2.7) will achieve GS with respect to Ξ¦.

Proof. Select the same Lyapunov-Krasovskii function as Theorem 2.5, then d𝑉d𝑑=2𝑁𝑖=1𝑒𝑇𝑖𝑓(𝑑)𝑖π‘₯𝑖(𝑑)βˆ’π‘“π‘–ξ€·Ξ¦ξ€·π‘¦π‘–+(𝑑)𝑁𝑗=1π‘π‘–π‘—π»ξ€·π‘’π‘—ξ€Έβˆ’(π‘‘βˆ’πœ(𝑑))𝑁𝑗=1Μƒπ‘π‘–π‘—β„Žξ€·Ξ¦ξ€·π‘¦π‘—ξ€·π‘¦(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έβˆ’π·Ξ¦π‘–ξ€Έπ‘’(𝑑)𝑖+(𝑑)𝑁𝑁𝑖=1𝑗=12𝛿𝑖𝑗̃𝑐𝑖𝑗̇̂𝑐𝑖𝑗+𝑁𝑖=12π‘˜π‘–ξ€·π‘‘π‘–ξ€ΈΜ‡π‘‘βˆ’π‘‘π‘–+𝑁𝛽21βˆ’πœ‡π‘ξ“π‘–=1𝑒𝑇𝑖(𝑑)𝑒𝑖(𝑑)βˆ’1βˆ’Μ‡πœ(𝑑)1βˆ’πœ‡π‘π›½2𝑁𝑖=1𝑒𝑇𝑖(π‘‘βˆ’πœ(𝑑))𝑒𝑖≀(π‘‘βˆ’πœ(𝑑))𝑁𝑖=1ξ‚΅βˆ’2𝑑+𝑁𝛽2𝑒1βˆ’πœ‡π‘‡π‘–(𝑑)𝑒𝑖+(𝑑)𝑁𝑁𝑖=1𝑗=1𝑒𝑇𝑖(𝑑)π‘π‘–π‘—π»ξ€·π‘’π‘—ξ€Έβˆ’(π‘‘βˆ’πœ(𝑑))1βˆ’Μ‡πœ(𝑑)1βˆ’πœ‡π‘π›½2𝑁𝑖=1𝑒𝑇𝑖(π‘‘βˆ’πœ(𝑑))𝑒𝑖(π‘‘βˆ’πœ(𝑑)).(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 {β„Ž(Ξ¦(𝑦𝑗(π‘‘βˆ’πœ(𝑑))))}𝑁𝑗=1 are linearly independent on the orbit {𝑦𝑗(π‘‘βˆ’πœ(𝑑))}𝑁𝑗=1 of synchronization manifold, we can get lim𝑑→+βˆžΜ‚π‘π‘–π‘—=𝑐𝑖𝑗,𝑖,𝑗=1,2,…,𝑁; that is, the uncertain coupling matrix 𝐢 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 2.7–2.9.

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 byβŽ›βŽœβŽœβŽœβŽΜ‡π‘₯1Μ‡π‘₯2Μ‡π‘₯3⎞⎟⎟⎟⎠=βŽ›βŽœβŽœβŽœβŽœβŽβˆ’ξ€·π‘₯(25𝛽+10)1βˆ’π‘₯2ξ€Έβˆ’π‘₯1π‘₯3+(28βˆ’35𝛽)π‘₯1+(29π›½βˆ’1)π‘₯2π‘₯1π‘₯2βˆ’(𝛽+8)3π‘₯3⎞⎟⎟⎟⎟⎠=βŽ›βŽœβŽœβŽœβŽœβŽξ€·π‘₯(3.1)βˆ’101βˆ’π‘₯2ξ€Έβˆ’π‘₯1π‘₯3+28π‘₯1βˆ’π‘₯2π‘₯1π‘₯2βˆ’83π‘₯3⎞⎟⎟⎟⎟⎠+βŽ›βŽœβŽœβŽœβŽœβŽξ€·π‘₯βˆ’251βˆ’π‘₯2ξ€Έβˆ’35π‘₯1+29π‘₯2βˆ’13π‘₯3βŽžβŽŸβŽŸβŽŸβŽŸβŽ π›½=𝐹(π‘₯)+𝐺(π‘₯)𝛽,(3.2) which is chaotic if π›½βˆˆ[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 asβŽ›βŽœβŽœβŽœβŽΜ‡π‘₯1Μ‡π‘₯2Μ‡π‘₯3⎞⎟⎟⎟⎠=βŽ›βŽœβŽœβŽœβŽœβŽπ‘Žπ‘₯1βˆ’π‘₯2π‘₯3𝑏π‘₯2+π‘₯1π‘₯3𝑐π‘₯3+13π‘₯1π‘₯2⎞⎟⎟⎟⎟⎠=𝑔(π‘₯).(3.3) It is chaotic when π‘Ž=5.0,𝑏=βˆ’10.0, and 𝑐=βˆ’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(π‘₯)=𝐹(π‘₯)+𝐺(π‘₯)𝛽(π›½βˆˆ[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)βˆˆπ‘…3,||𝑦||≀𝑀,||𝑧||≀𝑀; therefore,‖𝑓(𝑦)βˆ’π‘“(𝑧)β€–2=(25𝛽+10)2𝑦2βˆ’π‘¦1ξ€Έβˆ’ξ€·π‘§2βˆ’π‘§1ξ€Έξ€»2+𝑧1𝑧3βˆ’π‘¦1𝑦3𝑦+(28βˆ’35𝛽)1βˆ’π‘§1𝑦+(29π›½βˆ’1)2βˆ’π‘§2ξ€Έξ€»2+𝑦1𝑦2βˆ’π‘§1𝑧2βˆ’(𝛽+8)3𝑦3βˆ’π‘§3ξ€Έξ‚Ή2=(25𝛽+10)2𝑦2βˆ’π‘§2ξ€Έβˆ’ξ€·π‘¦1βˆ’π‘§1ξ€Έξ€»2+𝑧1𝑧3βˆ’π‘¦3ξ€Έ+ξ€·βˆ’π‘¦3𝑦+28βˆ’35𝛽1βˆ’π‘§1𝑦+(29π›½βˆ’1)2βˆ’π‘§2ξ€Έξ€»2+𝑦1𝑦2βˆ’π‘§2ξ€Έ+𝑧2𝑦1βˆ’π‘§1ξ€Έβˆ’(𝛽+8)3𝑦3βˆ’π‘§3ξ€Έξ‚Ή2≀3522𝑦2βˆ’π‘§2ξ€Έ2𝑦+21βˆ’π‘§1ξ€Έ2ξ‚„+3𝑀2𝑦3βˆ’π‘§3ξ€Έ2ξ€·+6282+𝑀2𝑦1βˆ’π‘§1ξ€Έ2+3Γ—282𝑦2βˆ’π‘§2ξ€Έ2+3𝑀2𝑦2βˆ’π‘§2ξ€Έ2+3𝑀2𝑦1βˆ’π‘§1ξ€Έ2𝑦+93βˆ’π‘§3ξ€Έ2≀2Γ—352+6Γ—282+9𝑀2ξ€Έβ€–π‘¦βˆ’π‘§β€–2.(3.4) Thus, function 𝑓(π‘₯)=𝐹(π‘₯)+𝐺(π‘₯)𝛽(π›½βˆˆ[0,1]) satisfies Assumption 2. By the same process, we can obtain that function 𝑔(π‘₯) 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 Μ‡π‘₯𝑖π‘₯(𝑑)=𝐹𝑖+(𝑑)5𝑗=1π‘π‘–π‘—β„Žξ€·π‘₯𝑗(π‘‘βˆ’πœ(𝑑)),𝑖=1,2,3,Μ‡π‘₯𝑖π‘₯(𝑑)=𝑔𝑖+(𝑑)5𝑗=1π‘π‘–π‘—β„Žξ€·π‘₯𝑗(π‘‘βˆ’πœ(𝑑)),𝑖=4,5,(3.5) where π‘₯𝑖(𝑑)=(π‘₯𝑖1(𝑑),π‘₯𝑖2(𝑑),π‘₯𝑖3(𝑑))𝑇,𝑖=1,2,…,5.𝜏(𝑑)=0.1, the weight configuration matrix 𝑐𝐢=𝑖𝑗5Γ—5=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βˆ’513101βˆ’210031βˆ’400100βˆ’210001βˆ’1.(3.6) The coupling functions are β„Ž(π‘₯𝑗(𝑑))=(sin(π‘₯𝑗1(𝑑)),arctan(π‘₯𝑗2(𝑑)),arctan(π‘₯𝑗3(𝑑)))𝑇,𝑗=1,2,…,5.
Let Ξ¦(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:̇𝑦𝑖𝑦(𝑑)=𝐷Φ𝑖(𝑑)βˆ’1𝐹Φ𝑦𝑖+(𝑑)𝑁𝑗=1Μ‚π‘π‘–π‘—β„Žξ€·Ξ¦ξ€·π‘¦π‘—ξƒ­(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ+𝑒𝑖,𝑖=1,2,3,̇𝑦𝑖𝑦(𝑑)=𝐷Φ𝑖(𝑑)βˆ’1𝑔Φ𝑦𝑖+(𝑑)𝑁𝑗=1Μ‚π‘π‘–π‘—β„Žξ€·Ξ¦ξ€·π‘¦π‘—ξƒ­(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ+𝑒𝑖,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,π‘˜π‘–=1,π‘₯𝑖(0)=(12+𝑖×0.1,15+𝑖×0.1,30+𝑖×0.15)𝑇,Μ‚π‘₯𝑖(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(𝑑)β€–, 𝑖=1,2,…,5. One can see that all nodes’ errors converge to zero. Some elements of matrix 𝐢 are displayed in Figure 2. The numerical results show that this adaptive scheme is effective and we can get limπ‘‘β†’βˆžΜ‚π‘π‘–π‘—=𝑐𝑖𝑗,𝑖,𝑗=1,2,…,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 Μ‡π‘₯𝑖(𝑑)=𝑓𝑖π‘₯𝑖+(𝑑)5𝑗=1π‘π‘–π‘—β„Žξ€·π‘₯𝑗(π‘‘βˆ’πœ(𝑑)),𝑖=1,2,…,5,(3.8) where π‘₯𝑖(𝑑)=(π‘₯𝑖1(𝑑),π‘₯𝑖2(𝑑),π‘₯𝑖3(𝑑))𝑇,𝑓𝑖(π‘₯)=𝐹(π‘₯)𝛽𝑖+𝐺(π‘₯),𝛽𝑖=0.1Γ—(π‘–βˆ’1),𝑖=1,2,…,5. We assume 𝜏(𝑑)=0.3,β„Ž(π‘₯𝑗(𝑑))=(arctan(π‘₯𝑗1(𝑑)),arctan(π‘₯𝑗2(𝑑)),arctan(π‘₯𝑗3(𝑑)))𝑇,𝑗=1,2,…,5. C is the same as that in model (3.5).
Let Ξ¦(𝑦𝑖)=(𝑦𝑖1+𝑦𝑖2,2𝑦𝑖2,𝑦3𝑖3+𝑦𝑖3)𝑇,𝐷Φ(𝑦𝑖)=110020003𝑦2𝑖3+1ξ‚Ά,𝑖=1,2,…,5.
According to Theorem 2.10, the response network is given bẏ𝑦𝑖𝑦(𝑑)=𝐷Φ𝑖(𝑑)βˆ’1𝑓𝑖Φ𝑦𝑖+(𝑑)𝑁𝑗=1Μ‚π‘π‘–π‘—β„Žξ€·Ξ¦ξ€·π‘¦π‘—ξƒ­(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ+𝑒𝑖,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,π‘˜π‘–=1,π‘₯𝑖(0)=(12,15,30)𝑇,Μ‚π‘₯𝑖(0)=(5,7.5,3)𝑇,𝑖,𝑗=1,2,…,5. Figures 3 and 4 show the GS errors ‖𝑒𝑖(𝑑)β€–, 𝑖=1,2,…,5 and some elements of matrix 𝐢, respectively. The results illustrate that this scheme is effective and we can get lim𝑑→+βˆžΜ‚π‘π‘–π‘—=𝑐𝑖𝑗,𝑖,𝑗=1,2,…,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 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.