Adaptive projective synchronization in complex networks with time-varying coupling delay

Physics Letters A 373 (2009) 1553–1559

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Physics Letters A www.elsevier.com/locate/pla

Adaptive projective synchronization in complex networks with time-varying coupling delay Song Zheng a,∗ , Qinsheng Bi b , Guoliang Cai b a b

School of Mechanical Engineering, Jiangsu University, Zhenjiang Jiangsu 212013, PR China Faculty of Science, Jiangsu University, Zhenjiang Jiangsu 212013, PR China

a r t i c l e

i n f o

Article history: Received 11 December 2008 Received in revised form 18 February 2009 Accepted 2 March 2009 Available online 5 March 2009 Communicated by A.R. Bishop

a b s t r a c t In this Letter, adaptive projective synchronization (PS) between two complex networks with time-varying coupling delay is investigated by the adaptive control method, and this method has been applied to identify the exact topology of a weighted general complex network. To validate the proposed method, the Lü and Qi systems as the nodes of the networks are detailed analysis, and some numerical results show the effectiveness of the present method. © 2009 Elsevier B.V. All rights reserved.

PACS: 05.45.Xt 05.45.Gg 89.75.Hc Keywords: Time-varying coupling delay Complex networks Adaptive control PS

1. Introduction Complex networks exist in various fields of real world such as in the Internet, the World Wide Web, food webs, electricity distribution networks, etc., and thus become more and more important in our daily life. A complex network consists of a large number of nodes and the connections between them. These nodes can have different meanings in different situations, such as microprocessors, computers, companies, etc. The nature of complex networks is the complexity, including topological structure, dynamical evolution, node diversity, and so on. Among various complex dynamical behaviors, synchronization is a significant and interesting phenomenon. Historically, the synchronization of complex dynamical networks has been studied in various fields of science and engineering [1–12]. Most of them focused on synchronization in a network that was called ‘inner synchronization’ [11] as it was a collective behavior within a network. On the other hand, one may also consider ‘outer synchronization’ [12] which illustrated the synchronization between two or more complex networks, but the synchronization of the inner network was ignored in their work. In the literatures, the projective synchronization (PS) is widely studied. However, most of them have mainly focused on two coupled chaotic systems about the PS [13,14]. In Ref. [13], the authors introduced the PS in two coupled partial linear systems. In Ref. [14], Jia discussed the PS of a new hyperchaotic Lorenz system. Later, the PS between two complex networks has obtained much more attention [15–17]. In Ref. [15], Guo et al. studied the PS in drive-response networks via impulsive control, while Hu et al. [16] discussed the PS of a drive-response dynamical networks model without the time delay. Actually, the characteristic of time-delayed coupling commonly exists in nature, and the effects of time delay on the dynamics of various coupled dynamical systems are very significant in science. Recently, Sun et al. [17] proposed a linear controller and an updated law to realize the PS in drive-response dynamical networks of partially linear systems with time-varying coupling delay. In Ref. [18], the global synchronization problem of a class of complex network systems with time delay and nonlinear coupling via adaptive method is investigated by Hua et al. While in Ref. [19], Wu and Jiao studied synchronization

*

Corresponding author. Tel.: +86 511 88972662. E-mail address: [email protected] (S. Zheng).

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S. Zheng et al. / Physics Letters A 373 (2009) 1553–1559

based on a new general delayed complex dynamical network model with non-symmetric coupling. Based on the LaSalle invariant principle of functional differential equations and the adaptive feedback control technique, Wang et al. investigated the adaptive synchronization of neural networks with time-varying coupling delay and distributed delay [20]. However, for the actual drive networks and response networks, the coupling configuration matrix is generally unknown or uncertain. More recently, Ref. [21] shown an adaptive linear feedback technique to identify the exact topology of a weighted general complex dynamical network model with time-varying coupling delay. Based on the above discussions, in this Letter, the real network noted as a drive network, and we construct another response network receiving the evolution of each node. Using the adaptive control method, we design two different nonlinear controllers to achieve PS between drive and response complex networks with time-varying coupling delay, which can identify the exact topology of the real network. The rest of this Letter is organized as follows. A complex dynamical network model with coupling time-varying delay and some preliminaries are presented in Section 2. In Section 3, some sufficient conditions for the PS are derived by the adaptive method. In Section 4, two illustrative examples are given for supporting the theory results. Finally, conclusions are given in Section 5. 2. Model description and preliminaries In this Letter, a complex dynamical network with time-varying coupling delay consisting of N identical nodes with linear couplings is considered, which is characterized by





x˙ i (t ) = Bxi (t ) + f t , xi (t ) +

N 





c i j Ax j t − τ (t ) ,

i = 1, 2, . . . , N ,

(1)

j =1

where xi (t ) = (xi1 (t ), xi2 (t ), . . . , xin (t )) T ∈ R n is the state vector of the ith node, B ∈ R n×n is a constant matrix, f : R × R n → R n is a smooth nonlinear function, τ (t )  0 is the time-varying coupling delay. A ∈ R n×n is inner-coupling matrix and C = (c i j ) N × N ∈ R N × N is the unknown or uncertain weight configuration matrix. If there is a connection from node i to node j ( j = i ), then the coupling c i j = 0; otherwise, c i j = 0 ( j = i ), and the diagonal elements of matrix C are defined as N 

c ii = −

i = 1, 2, . . . , N .

ci j ,

j =1, j =i

Assumption 1. Time delay τ (t ) is a constant.

τ (t ) is a differential function with 0  τ˙ (t )  ε < 1. Clearly, this assumption is ensured if the coupling delay

Assumption 2. Suppose there exists a constant L > 0, such that  f (t , x(t )) − f (t , y (t ))  L x(t ) − y (t ) holds for any time-varying vectors x(t ), y (t ), and the norm  ·  of a vector x is defined as x = (x T x)1/2 . Next, we will introduce a lemma, which is needed in the proof of the main theorems. Lemma. (See [22].) For any vectors x, y ∈ R n and positive definite matrix Q ∈ R n×n , the following matrix inequality holds: 2x T y  x T Q x + y T Q −1 y. 3. Adaptive controlling method In this section, we will make drive-response complex dynamical networks with time-varying coupling delay achieve adaptive projective synchronization by using adaptive controlling method. For simplicity, we refer to model (1) as the drive complex dynamical network, and consider a response network described as following:





y˙ i (t ) = B y i (t ) + f t , y i (t ) +

N 





cˆ i j Ay j t − τ (t ) + u i ,

i = 1, 2, . . . , N ,

(2)

j =1

where y i (t ) = ( y i1 (t ), y i2 (t ), . . . , y in (t )) T ∈ R n is the response state vector of the ith node, u i (i = 1, 2, . . . , N) are nonlinear controllers to be designed, and Cˆ = (ˆc i j ) N × N is estimation of the weight matrix C . Let e i (t ) = xi (t ) − λ y i (t ) (λ(= 0) is a scaling factor) and c˜ i j = cˆ i j − c i j , with the aid of Eqs. (1) and (2), the following error dynamical network can be obtained:









e˙ i (t ) = Be i (t ) + f t , xi (t ) − λ f t , y i (t ) − λ

N 





c˜ i j Ay j t − τ (t ) +

j =1

N 





c i j Ae j t − τ (t ) − λu i ,

i = 1, 2, . . . , N .

(3)

j =1

Theorem 1. Suppose Assumption 1 holds. Using the following adaptive controllers and updated laws:

    1 di e i (t ) + f t , xi (t ) − λ f t , y i (t ) , i = 1, 2, . . . , N , λ   c˙ˆ i j = λδi j e iT (t ) Ay j t − τ (t ) , i , j = 1, 2, . . . , N ,

(5)

d˙ i = ki e T (t )e i (t ),

(6)

ui =

i

i = 1, 2, . . . , N ,

(4)

where d = (d1 , d2 , . . . , d N ) T ∈ R N is the adaptive feedback gain vector to be designed, δi j > 0, ki > 0 (i , j = 1, 2, . . . , N) are arbitrary constants, then the response network (2) can synchronize with the drive network (1), and the weight configuration matrix C of network (1) can be identified by Cˆ , i.e., limt →∞ e i (t ) = limt →∞ (ˆc i j − c i j ) = 0, i = 1, 2, . . . , N.

S. Zheng et al. / Physics Letters A 373 (2009) 1553–1559

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Proof. Choose the following Lyapunov function: V =

N 1

2

e iT (t )e i (t ) +

i =1

N N 1  1

2

i =1 j =1

c˜ 2i j +

δi j

t  N N 2 1 1 1 di − d∗i + e iT (θ)e i (θ) dθ 2 ki 2(1 − ε ) i =1

(7)

t −τ (t ) i =1

where d∗i is a positive constant to be determined. Calculating the derivative of (7) along the trajectories of (3), and with adaptive controllers (4) and updated laws (5) and (6). Thus, we obtain V˙ =

N 



e iT



N N              Be i (t ) + f t , xi (t ) − λ f t , y i (t ) − λ c˜ i j Ay j t − τ (t ) + c i j Ae j t − τ (t ) − di e i (t ) − f t , xi (t ) + λ f t , y i (t )



i =1

+

j =1

N N   i =1 j =1

=

N 

1

δi j

N 

c˜ i j c˙ˆ i j +

i =1

e iT (t ) Be i (t ) +

i =1

N N  

j =1

 1 1 di − d∗i d˙ i + ki 2(1 − ε ) 



c i j e iT (t ) Ae j t − τ (t ) −

i =1 j =1

N 

e iT (t )e i (t ) −

i =1

N 

N 1 − τ˙ (t ) 

2(1 − ε )

N 

2(1 − ε )

i =1



i =1

1

d∗i e iT (t )e i (t ) +

 



e iT t − τ (t ) e i t − τ (t )

e iT (t )e i (t ) −

i =1

N 1 − τ˙ (t ) 

2(1 − ε )

 





e iT t − τ (t ) e i t − τ (t ) .

i =1

Let e (t ) = (e 1T (t ), e 2T (t ), . . . , e TN (t )) T ∈ R nN , P = (C ⊗ A ), where ⊗ represents the Kronecker product. Then by Lemma, we have





1

V˙ = e T (t ) Be (t ) + e T (t ) P e t − τ (t ) − e T (t ) D ∗ e (t ) +

e T (t )e (t ) −

1 − τ˙ (t )

 





e T t − τ (t ) e t − τ (t )

2(1 − ε ) 2(1 − ε )       1 1 − τ˙ (t ) T  T ∗ e T (t )e (t ) − e t − τ (t ) e t − τ (t ) .  e (t ) Be (t ) + e (t ) P P e (t ) + e t − τ (t ) e t − τ (t ) − e (t ) D e (t ) + 2 2 2(1 − ε ) 2(1 − ε ) 1

T

T

1

T

T



From Assumption 1, we get 1 2



1 − τ˙ (t ) 2(1 − ε )

thus we have

,



V˙  e T (t ) B +

1 2

P PT +

1 2(1 − ε )



I − D ∗ e (t )

(8)

where I is the identity maximal, D ∗ = diag(d∗1 , d∗2 , . . . , d∗N ). The constants d∗i (i = 1, 2, . . . , N) can be properly chosen to make V˙  0. Therefore, based on the Lyapunov stability theory, the errors

vector e (t ) → 0 and Cˆ → C as t → ∞. This implies the unknown weights c i j can be successfully estimated using adaptive controllers (4) and update laws (5) and (6). 2 Theorem 2. Suppose Assumptions 1 and 2 hold. Using the following nonlinear controller and updated laws:

    1 di e i (t ) + f t , λ y i (t ) − λ f t , y i (t ) , i = 1, 2, . . . , N , λ   c˙ˆ i j = λδi j e iT (t ) Ay j t − τ (t ) , i , j = 1, 2, . . . , N ,

(10)

d˙ i = ki e iT (t )e i (t ),

(11)

ui =

(9)

i = 1, 2, . . . , N ,

where δi j > 0, ki > 0 (i , j = 1, 2, . . . , N) are arbitrary constants, then the response network (2) can synchronize with the drive network (1), and the

weight configuration matrix C of network (1) can be identified by Cˆ , i.e., limt →∞ e i (t ) = limt →∞ (ˆc i j − c i j ) = 0, i = 1, 2, . . . , N. Proof. Choose the same Lyapunov function as Theorem 1, then V˙ =

N 











e iT Be i (t ) + f t , xi (t ) − λ f t , y i (t ) − λ

i =1

+

i =1 j =1

=

1

δi j

c˜ i j c˙ˆ i j +

e iT (t ) Be i (t ) +

i =1

−

i =1 N 

d∗i e iT (t )e i (t ) +

i =1



i =1

N 

 1 1 di − d∗i d˙ i + ki 2(1 − ε )

 







e iT f t , xi (t ) − f t , λ y i (t )

i =1

N 

N 





c˜ i j Ay j t − τ (t ) +

j =1

N N  

N 

N 

e iT (t ) Be i (t ) + L

N 





j =1 N 

e iT (t )e i (t ) −

i =1

+

N N  

N 1 − τ˙ (t ) 

2(1 − ε )





1 2(1 − ε )

N  i =1

e iT (t )e i (t ) −

i =1

e iT (t )e i (t ) +

N N   i =1 j =1

N 1 − τ˙ (t ) 

2(1 − ε )

i =1



c i j e iT (t ) Ae j t − τ (t )



 





e iT t − τ (t ) e i t − τ (t )

i =1



c i j e iT (t ) Ae j t − τ (t ) −

N  i =1

 



e iT t − τ (t ) e i t − τ (t )

i =1 j =1 N 









c i j Ae j t − τ (t ) − di e i (t ) − f t , λ y i (t ) + λ f t , y i (t )

e iT (t )d∗i e i (t )



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S. Zheng et al. / Physics Letters A 373 (2009) 1553–1559

+

N 

1 2(1 − ε )

N 1 − τ˙ (t ) 

e iT (t )e i (t ) −

2(1 − ε )

i =1

 





e iT t − τ (t ) e i t − τ (t ) .

i =1

The rest of proof is similar to Theorem 1 and omitted here, therefore, this theorem has been proofed.

2

4. Numerical simulations In this section, to verify and demonstrate the effectiveness of the proposed methods, we consider two numerical examples, that is, the Qi chaotic system [23] and the Lü chaotic system [24]. It is well known that the Qi chaotic system is described by

⎛ x˙ ⎞

⎛ a(x − x ) + x x ⎞ ⎛x ⎞ 2 1 2 3 1 ⎝ x˙ 2 ⎠ = ⎝ cx1 − x2 − x1 x3 ⎠ = B ⎝ x2 ⎠ + f (t , x) x˙ 3 −bx3 + x1 x2 x3 1

(12)

which has a chaotic attractor when a = 35, b = 8/3, c = 80, where B =

⎛ x˙ ⎞

⎛ a(x − x ) ⎞ ⎛x ⎞ 1 2 1 1 ⎝ x˙ 2 ⎠ = ⎝ cx2 − x1 x3 ⎠ = B ⎝ x2 ⎠ + f (t , x) x˙ 3 −bx3 + x1 x2 x3

−a a 0 

0  where a = 36, b = 3, c = 20, B = 0 c 0 , f (t , x) = −x1 x3 . 0 0 −b

−a

a 0 c −1 0 0 0 −b



, f (t , x) =

x2 x3



−x1 x3 , while the Lü chaotic system is x1 x2

(13)

x1 x2

Then we will investigate these two chaotic systems in detail to validate the effectiveness of Theorems 1 and 2. Example 1. Choose the Qi chaotic system to verify the effectiveness of Theorem 1. Now, we consider a weighted linearly coupled complex dynamical network (1) with coupling delay consisting of 5 identical Qi chaotic systems. Taking the weight configuration coupling matrix

⎛ −6 ⎜ 3 ⎜ ⎜ ⎜ ⎝ 1

3

⎞

2

0

1

0 ⎟ ⎟

−4

1

0

C = (c i j )5×5 = ⎜ 2

1

−3

0

0

0

−2

0

0

0

5

⎟ ⎟ 1 ⎠

0 ⎟.

−5

Then the drive network system is defined as





x˙ i (t ) = Bxi (t ) + f t , xi (t ) +

N 





c i j Ax j t − τ (t ) ,

i = 1, 2, . . . , 5.

(14)

j =1 et 1+et

t

, then τ˙ (t ) = (1+eet )2 ∈ [0, 12 ). Let scaling factor λ = 2, the initial values are givens as follows: ki = 3, δi j = 1, cˆ i j (0) = 3, xi (0) = (0.1 + 0.1i , 0.2 + 0.1i , 0.4 + 0.1i ) T , y i (0) = (1.6 + 0.1i , 0.6 + 0.1i , 0.7 + 0.1i ) T , where 1  i , j  5. Therefore, according to Theorem 1, the PS is achieved by using the following response network, the controller and updated laws given by For simplicity, we assume that A = I 3 ,





y˙ i (t ) = B y i (t ) + f t , y i (t ) +

N 

τ (t ) =





cˆ i j Ay j t − τ (t ) + u i ,

i = 1, 2, . . . , 5,

j =1

    1 di e i (t ) + f t , xi (t ) − λ f t , y i (t ) , i = 1, 2, . . . , 5, λ   c˙ˆ i j = λδi j e iT (t ) Ay j t − τ (t ) , i , j = 1, 2, . . . , 5, ui =

d˙ i = ki e iT (t )e i (t ),

i = 1, 2, . . . , 5.

(15)

Then, some elements of matrix Cˆ and the synchronization errors e i (t ) (1  i  5) are shown respectively in Figs. 1 and 2. The numerical results show that adaptive scheme for the drive-response complex network is effective in Theorem 1. Example 2. We consider the Lü system to verify the effectiveness of Theorem 2. Considering a weighted linearly coupled complex dynamical network (1) with coupling delay consisting of 5 identical Lü chaotic systems, the coupling configuration matrix

⎛ −6

⎜ 3 ⎜ ⎜ C = (c i j )5×5 = ⎜ 2 ⎜ ⎝ 1 0

⎞

3

2

0

1

−4

1

0

1

−3

0 ⎟ ⎟

0

0

0

−5

0

0

4

⎟ ⎟ 4 ⎠

0 ⎟,

−4

S. Zheng et al. / Physics Letters A 373 (2009) 1553–1559

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Fig. 1. Estimation of the weight matrix C with time t.

Fig. 2. Projective synchronization errors of the drive and response networks, where λ = 2.

and assume that A = I 3 ,

τ (t ) =

et 1+et

. Let scaling factor λ = −0.5, the initial values are givens as follows: ki = 1, δi j = 1, cˆ i j (0) = 3,

xi (0) = (0.1 + 0.1i , 0.2 + 0.1i , 0.4 + 0.1i ) T , y i (0) = (1.6 + 0.1i , 0.6 + 0.1i , 0.7 + 0.1i ) T , where 1  i , j  5. Based on Theorem 2, the response network, the controller and updated laws are given as:





y˙ i (t ) = B y i (t ) + f t , y i (t ) +

N 





cˆ i j Ay j t − τ (t ) + u i ,

i = 1, 2, . . . , 5,

j =1

    1  f t , λ y i (t ) − λ f t , xi (t ) + di e i (t ) , i = 1, 2, . . . , 5, λ   c˙ˆ i j = λδi j e iT (t ) Ay j t − τ (t ) , i , j = 1, 2, . . . , 5, ui =

d˙ i = ki e iT (t )e i (t ),

i = 1, 2, . . . , 5.

(16)

Then, some elements of matrix Cˆ and the synchronization errors e i (t ) (1  i  6) are shown respectively in Figs. 3 and 4. The numerical results show that adaptive scheme for the drive-response complex network is correct in Theorem 2. 5. Conclusion In this Letter, the nonlinear controllers and adaptive updated laws have been proposed to study the PS between two complex networks with time-varying coupling delay. With the Lyapunov stability theory and the adaptive control method, two PS theorems have been proposed, and the weight matrix C can be also identified. Numerical results demonstrate that the proposed approach is effective and feasible.

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S. Zheng et al. / Physics Letters A 373 (2009) 1553–1559

Fig. 3. Estimation of the weight matrix C with time t.

Fig. 4. Projective synchronization errors of the drive and response networks, where λ = −0.5.

Acknowledgements The authors thank the referees and the editor for their valuable comments and suggestions on improvement of this Letter. This work was supported by the National Natural Science Foundation of China (Nos. 70571030, 90610031) and the Advanced Talents’ Foundation of Jiangsu University (No. 07JDG054). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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