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Synchronization analysis of networks with both delayed and non-delayed couplings via adaptive pinning control method
Commun Nonlinear Sci Numer Simulat 15 (2010) 4202–4208
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Synchronization analysis of networks with both delayed and non-delayed couplings via adaptive pinning control method q Zengyun Wang a, Lihong Huang a,*, Yaonan Wang b, Yi Zuo c,d a
College of Mathematics and Economtrics, Hunan University, Changsha 410082, China College of Electric and Information Technology, Hunan University, Changsha 410082, China c The School of Energy and Power Engineering, Changsha University of Science and Technology, Changsha, Hunan, 410004, PR China d Department of Applied Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 b
a r t i c l e
i n f o
Article history: Received 23 June 2009 Received in revised form 1 February 2010 Accepted 1 February 2010 Available online 11 March 2010 Keywords: Complex networks Synchronization Time delay Pinning adaptive control method
a b s t r a c t In this paper, simple controllers are designed to realize the synchronization of complex networks with time delays, in which the coupling configuration matrix and inner coupling matrix are not restricted to be symmetric matrix. Several adaptive synchronization criteria are obtained based on Lyapunov stability theory. These criteria relay on the coupling strength and the number of nodes pinning to the networks. For a given complex dynamical network with both delayed and non-delayed couplings, we give the minimum number of controllers under which synchronization can be achieved. One example shows the effectiveness of the proposed pinning adaptive controller. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Since complex network behaviors have become a hot topic in many fields, such as biology, social systems, linguistic networks and technological systems [1–3], the analysis and controllability of complex networks have attracted lots of attention in the past few years [4–6]. A large number of nodes and links exist in the complex networks and each node represents a fundamental cell with specific activity. In addition, time delay is inevitable because the information spreading through a complex network is characterized by the finite speed of signal transmission over distance. Therefore, complex networks with time delay have become more and more increasing in various areas of science and engineering. As an interesting example of dynamics, synchronization has received a great deal of attention [7–11]. Recently, Wang et al. [12] give some criteria for both delay-dependent and delay-independent stabilities of the synchronization state. The stability condition of the synchronization for complex networks with time-varying couplings was discussed in [13]. However, all these criteria are depended on various parameters of the networks including the coupling strength, weight distribution and time delay. For the complexity of the dynamical network, it is difficult to realize the synchronization by adding controllers to all nodes, such as [14]. To reduce the number of the controllers, a natural way is using pinning control method. Since Grigoriev et al. [15] studied the pinning control of spatiotemporal chaos, many papers have discussed this method. For example, Sinha et al. [16] discussed the global and local control of spatiotemporal chaos in coupled map lattices. Sorrentino et al. [6] explored the controllability of complex networks via pinning. Furthermore, Zhou et al. [17] solved two fundamental questions in pinning control of complex networks: (i) How many nodes should a network with fixed network structure and coupling strength be pinned to q This work was supported by National Natural Science Foundation of China (10771055 and 60835004), Key Project of Hunan Province Programs for Applied Fundamental Research (2008FJ2008) and Graduate Innovation Foundation of Hunan Province (cx2009B065). * Corresponding author. E-mail addresses: [email protected] (L. Huang), [email protected] (Y. Wang), [email protected] (Y. Zuo).
1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.02.001
Z. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 4202–4208
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reach network synchronization? (ii) How much coupling strength should a network with fixed network structure and pinning nodes be applied to realize network synchronization? However, there is few result for the delayed complex networks by using pinning control method. Although authors in Refs. [18,19,21,22] discussed the pinning controllers for the complex networks with time delays, the pinning controllers need the accurate time delay and their results did not solve the fundamental questions proposed in [17]. In this paper, we analyze the synchronization of complex networks with both delayed and non-delayed coupling using pinning adaptive control method and solve the fundamental questions. These synchronization criteria derive only depend on non-delayed terns, and are of practical importance since the exact information about time delayed coupling and time delay is really difficult in real applications (see, e.g., Remark 3 of [20]). The structure of this paper is outlined as follows. In Section 2, we introduce a model of delayed dynamical networks and give some preliminaries. Section 3 proposes the adaptive pinning controllers and suggests the main criteria for the synchronization of the delayed complex networks. An numerical examples is given in Section 4 to show the effectiveness of our proposed results. Finally, concluding remarks are made in Section 5. 2. Problem statement and preliminaries We consider a complex delayed dynamical network consisting of N identically coupled nodes with each node being an n-dimensional dynamical system. The state equations of the entire network are
x_ i ðtÞ ¼ f ðxi ðtÞÞ þ
N X
cij Axj þ
N X
j¼1
csij As xj ðt sÞ þ ui ;
i ¼ 1; 2; . . . ; N;
ð2:1Þ
j¼1
where xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ xin ðtÞÞT 2 Rn is a state vector. f ¼ f ðxi ðtÞÞ : Rn ! Rn is a vector-value function standing for the activity of an individual subsystem, s P 0 represents time delay, which may be unknown. A 2 Rnn and As 2 Rnn describe the individual coupling between node i and node j at time t and t s. The coupling matrices C ¼ ðcij ÞNN and C s ¼ ðcsij ÞNN represent the coupling strength and the underlying topology for non-delayed configuration and delayed one, respectively. If there is a link from node i to node j ði–jÞ, then cij > 0 and otherwise cij ¼ 0. Similarly, if there is a link from node i to node j ði–jÞ underlying topology for delayed configuration, then csij > 0 and otherwise csij ¼ 0. Assume C and C s are diffusive matrices satisfying:
cii ¼
N X
cij ;
N X
csii ¼
j¼1;j–i
csij ;
ð2:2Þ
j¼1;j–i
which ensure the existence of synchronization of system (2.1). When the delayed system (2.1) achieves synchronization, namely, the states x1 ðtÞ ! x2 ðtÞ ! ! xN ðtÞ ! sðtÞ, as t ! 1, where sðtÞ 2 Rn is a solution of an isolate node, namely
s_ ðtÞ ¼ f ðsðtÞÞ:
ð2:3Þ
The rigorous mathematical definition of synchronization for the delayed dynamical networks (2.1) is introduced as follows. Definition 2.1 [14]. Let xi ðt; t 0 ; /Þ; i ¼ 1; 2; . . . ; N be a solution of the delayed network(2.1), where / ¼ ð/T1 ; /T2 /TN ÞT ; /i ¼ /i ðhÞ 2 Cð½s; 0; Rn Þ are initial conditions, f : R X ! Rn is continuously differentiable, X # Rn . If there is a nonempty subset K X, such that /i take values in K and xi ðt; t 0 ; /Þ 2 X for all t P t 0 ; i ¼ 1; 2; . . . ; N, and
lim kxi ðt; t0 ; /Þ sðt; t 0 ; s0 Þk ¼ 0;
t!1
i ¼ 1; 2; . . . ; N;
ð2:4Þ
where kk is the Euclidean norm, sðt; t0 ; s0 Þ is a solution of the system (2.3) with s0 2 X, then the delayed network (2.1) is said to realize synchronization, and K K K is called the region of synchrony for the delayed network (2.1). We assume that s(t) can be an equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit. The object is designing controllers ui to synchronize the dynamical network (2.1). That is, the trajectories of the closed-loop systems satisfy:
lim kxi ðtÞ sðtÞk ¼ 0;
t!1
i ¼ 1; 2; . . . ; N:
ð2:5Þ
Define the error vector by:
ei ¼ xi ðtÞ sðtÞ;
i ¼ 1; 2; . . . ; N:
ð2:6Þ
One gets the following error dynamical systems:
e_ i ðtÞ ¼ f ðxi ðtÞÞ f ðsðtÞÞ þ
N X j¼1
cij Aej þ
N X
csij As ej ðt sÞ þ ui :
ð2:7Þ
j¼1
Then the synchronization problem of the dynamical network (2.1) is equivalent to the problem of stabilization of the error dynamical system (2.7).
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To achieve the objective (2.5), we need the following assumptions and lemmas. A(1) There exists a nonnegative constant a, such that kf ðxi Þ f ðsÞk 6 akxi sk; i ¼ 1; 2; . . . ; N. A(2) The individual coupling matrix kAk ¼ c > 0 and kAs k 6 k. Denote qmin as the minimum eigenvalue of the matrix ðA þ AT Þ=2. bþC b T =2 by removing A(3) Assume ki ð1 6 i 6 NÞ be the maximum eigenvalues of C i , where C i is the minor matrix of C b is a modify matrix of C via replacing the diagonal elements cii by qmin cii . the first i 1 rows and columns, in which C c b bT 0 A (3) Assume ki ð1 6 i 6 NÞ be the maximum eigenvalues of C i , where C i is the minor matrix of C þ C =2 by removing b the first i 1 rows and columns, in which C is a modify matrix of C via replacing the diagonal elements cii by
qmin c c ii .
A(4) The diagonal strength for delayed configuration matrix, namely, the diagonal elements of the matrix C s satisfies jcsii j 6 g. Lemma 2.2. For any vectors x; y 2 Rm and positive definite matrix Q 2 Rmm , the following matrix inequality holds: 2xT y 6 xT Qx þ yT Q 1 y. If not specified otherwise, inequality Q > 0 ðQ < 0; Q P 0; Q 6 0Þ means Q is a positive (or negative, or semi-positive, or semi-negative) definite matrix. Lemma 2.3. Assume that A ¼
A3 B1 ; B¼ A2 0
A1 AT3
0 , where A; B 2 RNN ; A1 ; B1 2 Rrr ð1 6 r 6 NÞ; B1 ¼ diagfb1 ; . . . ; br g is a 0
diagonal positive definite matrix, AT1 ¼ A1 and AT2 ¼ A2 . Then A B < 0 is equivalent to A2 < 0 for large enough bi ð1 6 i 6 rÞ. 3. The pinning adaptive controller for the synchronization of the delayed complex network Without loss of the generality, assume that the first l nodes 1 6 i 6 l are selected and pinned with the adaptive controllers, which are described by
(
ui ¼ pi ei ;
p_i ¼ qi kei k22
ui ¼ 0;
1 6 i 6 l;
ð3:1Þ
otherwise;
where qi ; 1 6 i 6 l are any positive constants. Thus the controlled network (2.7) can be written as follows:
8 N N P P > > > e_ i ðtÞ ¼ f ðxi ðtÞÞ f ðsðtÞÞ þ cij Aej þ csij As ej ðt sÞ pi ei > > j¼1 j¼1 > < p_i ¼ qi kei k22 ; > > > N N > P P > > : e_ i ðtÞ ¼ f ðxi ðtÞÞ f ðsðtÞÞ þ cij Aej þ csij As ej ðt sÞ; j¼1
1 6 i 6 l; 1 6 i 6 l;
ð3:2Þ
l þ 1 6 i 6 N:
j¼1
Theorem 3.1. Suppose Að1Þ—Að4Þ hold. If there exist a natural number 1 6 l 6 N and a positive constant b such klþ1 < ac holds, 2 2 2 g k . Then the delayed complex dynamical network (2.1) is globally synchronized under the pinning adaptive where a ¼ a þ b þ N 4b controllers
(
ui ¼ pi ei ;
p_i ¼ qi kei k22
ui ¼ 0;
1 6 i 6 l; otherwise;
where qi are positive constants for 1 6 i 6 l. Proof. We consider the Lyapunov function candidate
VðxÞ ¼
Z t N l N 1X 1X ðpi pÞ2 X eTi ei þ þ b eTi ðhÞei ðhÞ dh; 2 i¼1 2 i¼1 qi ts i¼1
where p is a positive constant and b is any positive constant. Thus the differential coefficient of V is described by
V_ ¼
N X
eTi e_ i þ
i¼1
¼
N X
l N N X X ðpi pÞp_ i X þ beTi ei beTi ðt sÞei ðt sÞ qi i¼1 i¼1 i¼1
eTi ðf ðxi ðtÞÞ f ðsðtÞÞÞ þ
i¼1
l X i¼1
N X N X i¼1
pi eTi ei þ
l X i¼1
pi eTi ei
l X i¼1
cij eTi Aej þ
j¼1
peTi ei þ
N X N X i¼1
N X i¼1
eTi csij As ej ðt sÞ
j¼1
beTi ei
N X i¼1
beTi ðt sÞei ðt sÞ:
ð3:3Þ
Z. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 4202–4208
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From(2.2) and csij P 0, it is obvious that jcsij j < jcsii j, according to A(2), A(4) and Lemma 2.2, we have N X N X i¼1
eTi csij As ej ðt sÞ 6
j¼1
N X N N X N 1X 1X ðcsij Þ2 ei ATs QAs ei þ eT ðt sÞQ 1 ej ðt sÞ 2 i¼1 j¼1 2 i¼1 j¼1 j
6
N N Ng 2 X NX eTi ATs QAs ei þ eT ðt sÞQ 1 ej ðt sÞ 2 j¼1 j 2 i¼1
¼
2 N N N N X X N2 g 2 X N2 g 2 k X eTi ATs As ei þ b eTj ðt sÞej ðt sÞ ¼ eTi ei þ b eTj ðt sÞej ðt sÞ; 4b i¼1 4b i¼1 j¼1 j¼1
ð3:4Þ
N where Q ¼ 2b I > 0. Using Að1Þ and substituting (3.4) into (3.3), it follows that
2 N N2 g 2 k X eT ei 4b i¼1 i i¼1 i¼1 j¼1 i¼1 i¼1 ! ! 2 T N N N N l X X X X X N2 g 2 k T T T A þA ei ei ei þ ¼ aþbþ cij ei Aej þ cii ei peTi ei 2 4b i¼1 i¼1 j¼1;j–i i¼1 i¼1 ! ! 2 N N N N l b bT þ C X X X X X N2 g 2 k C T T T T ei ei þ 6 aþbþ ccij kei k2 kej k2 þ cii qmin ei ei pei ei ¼ e aIN þ c D e; 2 4b i¼1 i¼1 j¼1;j–i i¼1 i¼1
V_ 6
N X
aeTi ei þ
N X N X
cij eTi Aej
l X
peTi ei þ
N X
beTi ei þ
ð3:5Þ
where D ¼ diag p; pl ; 0; 0 ; e ¼ ðke1 k2 ; ke2 k2 ; . . . ; keN k2 ÞT . |fflfflffl{zfflfflffl} bT b According to Lemma 2.3, aIN þ c C 2þ C D < 0 is equivalent to aINl þ cC lþ1 < 0 for large enough p. From the assumption of Theorem 3.1, if there exist a natural number 1 6 l 6 N satisfying a þ cklþ1 < 0, one has aINl þ cC lþ1 < 0. Therefore, bT b bT b aIN þ c C 2þ C D < 0. Because aIN þ c C 2þ C D is a real symmetric matrix, there exists an orthogonal matrix P satisfying bT b aIN þ c C 2þ C D ¼ P T diag ^ k1 ; ^ k2 ; . . . ; ^ kN P, where ^ k1 ; ^ k2 ; . . . ; ^ kN ðPeÞ. It follows that ki < 0. Therefore, one gets V_ 6 ðPeÞT diag ^
T Pe ! 0 as t ! 0. Since P is an orthogonal matrix,the error vector g ¼ eT1 ; eT1 ; . . . ; eTN ! 0 as t ! 0. That is, the synchronous solution S(t) of the controlled network (2.1) is globally asymptotically stable under the pinning adaptive controllers (3.1). h Remark 1. It is not easy to get the entire information on the coupling matrix C s , so we use the partial information that the diagonal elements of the coupling matrix strength for delayed configuration csii has the upper bound g. In this paper, the coupling matrix C does not need to be symmetric. Remark 2. Compared with the results in paper [18,19], In this paper, the controllers are independent on the information of the time delay, so it is much easy to implement. Also, we use the adaptive technique, this will lead to convergence more fast. Remark 3. The controllers design method proposed in this paper can be applied to some uncertain delayed network. If there exist uncertain terms DC s ; DAs , which are bounded in the network, then the bound of k, g should be changed in the theorem. Corollary 3.2. Suppose Að1Þ; Að2Þ; A0 ð3Þ and Að4Þ hold and C ¼ cC, where C is a diffusive coupling matrix with cij ¼ 0 or 1 ði–jÞ. For the given c, if there exist a natural number 1 6 l 6 N and a positive constant b such that klþ1 < cac holds, where 2 2 2 g k . Then the delayed complex dynamical network (2.1) is globally synchronized under the pinning adaptive controla ¼ a þ b þ N 4b lers (3.1). Remark 4. Compared with the results in paper [21,22], we can estimate the minimum number of the pinning nodes which can achieve network synchronization. Specially assume C is symmetric and qmin ¼ c, if c > ka2 , then the network (2.1) can achieve synchronization under the adaptive controller (3.1) with l ¼ 1. From here we know that fewer controllers are needed to realize synchronization as the size of network N gets smaller. If the network does not have the time delay term, that is As ¼ 0. Obviously k ¼ 0 in Að2Þ. We can choose 2 P P as the Lyapunov function candidate, then we will have the following corollary, which is VðxÞ ¼ 12 Ni¼1 eTi ei þ 12 li¼1 ðpi pÞ qi the theorem in [17]. Corollary 3.3. Suppose Að1Þ—Að3Þ hold. If there exists a natural number 1 6 l 6 N such klþ1 < ac holds. Then the complex dynamical network (2.1) is globally synchronized under the pinning adaptive controller
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Z. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 4202–4208
(
ui ¼ pi ei ;
p_i ¼ qi kei k22
ui ¼ 0;
1 6 i 6 l; otherwise;
where qi are positive constants for 1 6 i 6 l. 4. Numerical example In this section, we shall present an example to demonstrate the applicability of the proposed controllers. Here we consider an array of chaotic systems with time delay, in which each subsystem is a Lorenz systems. We investigate the network with 50 nodes. The node of the complex networks is described by
50 40
X3
30 20 10 0 40 20 0 −20 X2
−40
−20
10
0
−10
20
X
1
Fig. 1. Lorenz chaotic attractor.
50 40
ei1(1≤ i≤50)
30 20 10 0 −10 −20
0
2
4
time (s)
6
8
Fig. 2. Synchronization errors ei1 ð1 6 i 6 50Þ of the controlled network (4.2).
10
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Z. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 4202–4208
0
1 0 r 1 x_i1 B _ C B @ xi2 A ¼ @ r3 x_i3
0
r1
0
1 0
10
xi1
1
0
0
1
CB C B C 0 A@ xi2 A þ @ xi1 xi3 A; xi3 xi1 xi2 r2
ð4:1Þ
where the parameters are often taken r 1 ¼ 10; r2 ¼ 8=3 and r3 ¼ 28 for chaotic behavior Fig. 1. The controlled network consist of 50 identical Lorenz systems, which is described by
x_ i ðtÞ ¼ f ðxi ðtÞÞ þ
N X
cij Axj þ
j¼1
N X
csij As xj ðt sÞ þ ui ;
ð4:2Þ
j¼1
where 1 6 i 6 50; f ðxi Þ ¼ ð10xi1 þ 10xi2 ; 28xi1 xi2 xi1 xi3 ; 83 xi3 þ xi1 xi2 ÞT ; ui is expressed in (3.1) and C ¼ ðcij Þ5050 is a symmetrically diffusive coupling matrix with cij ¼ 0 or 1ði–jÞ and the diagonal elements of C s have the upper bound g ¼ 2. Here the coupling coefficient c ¼ 100 and the inner coupling matrix is given as follows: 50 40
ei2(1≤ i≤50)
30 20 10 0 −10 −20 −30
0
2
4
time (s)
6
8
10
Fig. 3. Synchronization errors ei2 ð1 6 i 6 50Þ of the controlled network (4.2).
60 50
ei3(1≤ i≤50)
40 30 20 10 0 −10
0
2
4
time (s)
6
8
Fig. 4. Synchronization errors ei3 ð1 6 i 6 50Þ of the controlled network (4.2).
10
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Z. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 4202–4208
2
1
0
1
0
0
1
3
6 7 A ¼ 4 0 1:2 1 5;
2
1 0 1
3
6 7 As ¼ 4 0 2 0 5 ; 1 0 1
Obviously we have c ¼ kAk2 ¼ 1:2; kAs k2 ¼ 2. It is well known that Lorenz attractor is bounded. There exist constants M1 ¼ 20; M2 ¼ 25 and M3 ¼ 45 satisfying kxij k; ksj k 6 Mj for 1 6 i 6 50 and 1 6 j 6 3. Thus one has kf ðxi Þ f ðsÞk2 6 88:8kei k2 ; i ¼ 1; 2; . . . ; N. Choose the positive number b ¼ Ng in the theorem, it is obvious to get a ¼ 188:8. Then one has cac ¼ 1:57. For a given random matrix C in this example one gets k14þ1 < 1:57. This means that one needs 14 nodes of 50 nodes to realize the synchronization. From the theorem and Corollary 3.2, the controlled network is globally synchronized with the solution of chaotic system (4.1) under the pinning adaptive controllers. In the simulation, we choose the initial values, xi ð0Þ ¼ ð4 þ i; 5 þ i; 6 þ iÞT ; sð0Þ ¼ ð4; 5; 6ÞT for 1 6 i 6 50 and qi ¼ 2 and pi ð0Þ ¼ 1 for 1 6 i 6 14. The errors eij ; 1 6 i 6 50; 1 6 j 6 3 are shown in Figs. 2–4. The controlled network is obviously globally asymptotically stable. 5. Conclusion In this paper, we designed simple controllers to realize the asymptotical synchronization for the complex dynamical networks with time delays. Compared with the model in [17], the proposed model additionally provides a mathematical description including the term of past states in the complex system equation. Compared with the results in [18], the simple controllers are not dependent on the accurate time delay and our results do not need to know the exact information of the matrix C s and As . Under some assumptions, the controllers are given to realize the synchronization for the delayed complex network. The theoretical result is shown by a numerical example. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Synchronization analysis of networks with both delayed and non-delayed couplings via adaptive pinning control method q Zengyun Wang a, Lihong Huang a,*, Yaonan Wang b, Yi Zuo c,d a
College of Mathematics and Economtrics, Hunan University, Changsha 410082, China College of Electric and Information Technology, Hunan University, Changsha 410082, China c The School of Energy and Power Engineering, Changsha University of Science and Technology, Changsha, Hunan, 410004, PR China d Department of Applied Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 b
a r t i c l e
i n f o
Article history: Received 23 June 2009 Received in revised form 1 February 2010 Accepted 1 February 2010 Available online 11 March 2010 Keywords: Complex networks Synchronization Time delay Pinning adaptive control method
a b s t r a c t In this paper, simple controllers are designed to realize the synchronization of complex networks with time delays, in which the coupling configuration matrix and inner coupling matrix are not restricted to be symmetric matrix. Several adaptive synchronization criteria are obtained based on Lyapunov stability theory. These criteria relay on the coupling strength and the number of nodes pinning to the networks. For a given complex dynamical network with both delayed and non-delayed couplings, we give the minimum number of controllers under which synchronization can be achieved. One example shows the effectiveness of the proposed pinning adaptive controller. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Since complex network behaviors have become a hot topic in many fields, such as biology, social systems, linguistic networks and technological systems [1–3], the analysis and controllability of complex networks have attracted lots of attention in the past few years [4–6]. A large number of nodes and links exist in the complex networks and each node represents a fundamental cell with specific activity. In addition, time delay is inevitable because the information spreading through a complex network is characterized by the finite speed of signal transmission over distance. Therefore, complex networks with time delay have become more and more increasing in various areas of science and engineering. As an interesting example of dynamics, synchronization has received a great deal of attention [7–11]. Recently, Wang et al. [12] give some criteria for both delay-dependent and delay-independent stabilities of the synchronization state. The stability condition of the synchronization for complex networks with time-varying couplings was discussed in [13]. However, all these criteria are depended on various parameters of the networks including the coupling strength, weight distribution and time delay. For the complexity of the dynamical network, it is difficult to realize the synchronization by adding controllers to all nodes, such as [14]. To reduce the number of the controllers, a natural way is using pinning control method. Since Grigoriev et al. [15] studied the pinning control of spatiotemporal chaos, many papers have discussed this method. For example, Sinha et al. [16] discussed the global and local control of spatiotemporal chaos in coupled map lattices. Sorrentino et al. [6] explored the controllability of complex networks via pinning. Furthermore, Zhou et al. [17] solved two fundamental questions in pinning control of complex networks: (i) How many nodes should a network with fixed network structure and coupling strength be pinned to q This work was supported by National Natural Science Foundation of China (10771055 and 60835004), Key Project of Hunan Province Programs for Applied Fundamental Research (2008FJ2008) and Graduate Innovation Foundation of Hunan Province (cx2009B065). * Corresponding author. E-mail addresses: [email protected] (L. Huang), [email protected] (Y. Wang), [email protected] (Y. Zuo).
1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.02.001
Z. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 4202–4208
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reach network synchronization? (ii) How much coupling strength should a network with fixed network structure and pinning nodes be applied to realize network synchronization? However, there is few result for the delayed complex networks by using pinning control method. Although authors in Refs. [18,19,21,22] discussed the pinning controllers for the complex networks with time delays, the pinning controllers need the accurate time delay and their results did not solve the fundamental questions proposed in [17]. In this paper, we analyze the synchronization of complex networks with both delayed and non-delayed coupling using pinning adaptive control method and solve the fundamental questions. These synchronization criteria derive only depend on non-delayed terns, and are of practical importance since the exact information about time delayed coupling and time delay is really difficult in real applications (see, e.g., Remark 3 of [20]). The structure of this paper is outlined as follows. In Section 2, we introduce a model of delayed dynamical networks and give some preliminaries. Section 3 proposes the adaptive pinning controllers and suggests the main criteria for the synchronization of the delayed complex networks. An numerical examples is given in Section 4 to show the effectiveness of our proposed results. Finally, concluding remarks are made in Section 5. 2. Problem statement and preliminaries We consider a complex delayed dynamical network consisting of N identically coupled nodes with each node being an n-dimensional dynamical system. The state equations of the entire network are
x_ i ðtÞ ¼ f ðxi ðtÞÞ þ
N X
cij Axj þ
N X
j¼1
csij As xj ðt sÞ þ ui ;
i ¼ 1; 2; . . . ; N;
ð2:1Þ
j¼1
where xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ xin ðtÞÞT 2 Rn is a state vector. f ¼ f ðxi ðtÞÞ : Rn ! Rn is a vector-value function standing for the activity of an individual subsystem, s P 0 represents time delay, which may be unknown. A 2 Rnn and As 2 Rnn describe the individual coupling between node i and node j at time t and t s. The coupling matrices C ¼ ðcij ÞNN and C s ¼ ðcsij ÞNN represent the coupling strength and the underlying topology for non-delayed configuration and delayed one, respectively. If there is a link from node i to node j ði–jÞ, then cij > 0 and otherwise cij ¼ 0. Similarly, if there is a link from node i to node j ði–jÞ underlying topology for delayed configuration, then csij > 0 and otherwise csij ¼ 0. Assume C and C s are diffusive matrices satisfying:
cii ¼
N X
cij ;
N X
csii ¼
j¼1;j–i
csij ;
ð2:2Þ
j¼1;j–i
which ensure the existence of synchronization of system (2.1). When the delayed system (2.1) achieves synchronization, namely, the states x1 ðtÞ ! x2 ðtÞ ! ! xN ðtÞ ! sðtÞ, as t ! 1, where sðtÞ 2 Rn is a solution of an isolate node, namely
s_ ðtÞ ¼ f ðsðtÞÞ:
ð2:3Þ
The rigorous mathematical definition of synchronization for the delayed dynamical networks (2.1) is introduced as follows. Definition 2.1 [14]. Let xi ðt; t 0 ; /Þ; i ¼ 1; 2; . . . ; N be a solution of the delayed network(2.1), where / ¼ ð/T1 ; /T2 /TN ÞT ; /i ¼ /i ðhÞ 2 Cð½s; 0; Rn Þ are initial conditions, f : R X ! Rn is continuously differentiable, X # Rn . If there is a nonempty subset K X, such that /i take values in K and xi ðt; t 0 ; /Þ 2 X for all t P t 0 ; i ¼ 1; 2; . . . ; N, and
lim kxi ðt; t0 ; /Þ sðt; t 0 ; s0 Þk ¼ 0;
t!1
i ¼ 1; 2; . . . ; N;
ð2:4Þ
where kk is the Euclidean norm, sðt; t0 ; s0 Þ is a solution of the system (2.3) with s0 2 X, then the delayed network (2.1) is said to realize synchronization, and K K K is called the region of synchrony for the delayed network (2.1). We assume that s(t) can be an equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit. The object is designing controllers ui to synchronize the dynamical network (2.1). That is, the trajectories of the closed-loop systems satisfy:
lim kxi ðtÞ sðtÞk ¼ 0;
t!1
i ¼ 1; 2; . . . ; N:
ð2:5Þ
Define the error vector by:
ei ¼ xi ðtÞ sðtÞ;
i ¼ 1; 2; . . . ; N:
ð2:6Þ
One gets the following error dynamical systems:
e_ i ðtÞ ¼ f ðxi ðtÞÞ f ðsðtÞÞ þ
N X j¼1
cij Aej þ
N X
csij As ej ðt sÞ þ ui :
ð2:7Þ
j¼1
Then the synchronization problem of the dynamical network (2.1) is equivalent to the problem of stabilization of the error dynamical system (2.7).
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To achieve the objective (2.5), we need the following assumptions and lemmas. A(1) There exists a nonnegative constant a, such that kf ðxi Þ f ðsÞk 6 akxi sk; i ¼ 1; 2; . . . ; N. A(2) The individual coupling matrix kAk ¼ c > 0 and kAs k 6 k. Denote qmin as the minimum eigenvalue of the matrix ðA þ AT Þ=2. bþC b T =2 by removing A(3) Assume ki ð1 6 i 6 NÞ be the maximum eigenvalues of C i , where C i is the minor matrix of C b is a modify matrix of C via replacing the diagonal elements cii by qmin cii . the first i 1 rows and columns, in which C c b bT 0 A (3) Assume ki ð1 6 i 6 NÞ be the maximum eigenvalues of C i , where C i is the minor matrix of C þ C =2 by removing b the first i 1 rows and columns, in which C is a modify matrix of C via replacing the diagonal elements cii by
qmin c c ii .
A(4) The diagonal strength for delayed configuration matrix, namely, the diagonal elements of the matrix C s satisfies jcsii j 6 g. Lemma 2.2. For any vectors x; y 2 Rm and positive definite matrix Q 2 Rmm , the following matrix inequality holds: 2xT y 6 xT Qx þ yT Q 1 y. If not specified otherwise, inequality Q > 0 ðQ < 0; Q P 0; Q 6 0Þ means Q is a positive (or negative, or semi-positive, or semi-negative) definite matrix. Lemma 2.3. Assume that A ¼
A3 B1 ; B¼ A2 0
A1 AT3
0 , where A; B 2 RNN ; A1 ; B1 2 Rrr ð1 6 r 6 NÞ; B1 ¼ diagfb1 ; . . . ; br g is a 0
diagonal positive definite matrix, AT1 ¼ A1 and AT2 ¼ A2 . Then A B < 0 is equivalent to A2 < 0 for large enough bi ð1 6 i 6 rÞ. 3. The pinning adaptive controller for the synchronization of the delayed complex network Without loss of the generality, assume that the first l nodes 1 6 i 6 l are selected and pinned with the adaptive controllers, which are described by
(
ui ¼ pi ei ;
p_i ¼ qi kei k22
ui ¼ 0;
1 6 i 6 l;
ð3:1Þ
otherwise;
where qi ; 1 6 i 6 l are any positive constants. Thus the controlled network (2.7) can be written as follows:
8 N N P P > > > e_ i ðtÞ ¼ f ðxi ðtÞÞ f ðsðtÞÞ þ cij Aej þ csij As ej ðt sÞ pi ei > > j¼1 j¼1 > < p_i ¼ qi kei k22 ; > > > N N > P P > > : e_ i ðtÞ ¼ f ðxi ðtÞÞ f ðsðtÞÞ þ cij Aej þ csij As ej ðt sÞ; j¼1
1 6 i 6 l; 1 6 i 6 l;
ð3:2Þ
l þ 1 6 i 6 N:
j¼1
Theorem 3.1. Suppose Að1Þ—Að4Þ hold. If there exist a natural number 1 6 l 6 N and a positive constant b such klþ1 < ac holds, 2 2 2 g k . Then the delayed complex dynamical network (2.1) is globally synchronized under the pinning adaptive where a ¼ a þ b þ N 4b controllers
(
ui ¼ pi ei ;
p_i ¼ qi kei k22
ui ¼ 0;
1 6 i 6 l; otherwise;
where qi are positive constants for 1 6 i 6 l. Proof. We consider the Lyapunov function candidate
VðxÞ ¼
Z t N l N 1X 1X ðpi pÞ2 X eTi ei þ þ b eTi ðhÞei ðhÞ dh; 2 i¼1 2 i¼1 qi ts i¼1
where p is a positive constant and b is any positive constant. Thus the differential coefficient of V is described by
V_ ¼
N X
eTi e_ i þ
i¼1
¼
N X
l N N X X ðpi pÞp_ i X þ beTi ei beTi ðt sÞei ðt sÞ qi i¼1 i¼1 i¼1
eTi ðf ðxi ðtÞÞ f ðsðtÞÞÞ þ
i¼1
l X i¼1
N X N X i¼1
pi eTi ei þ
l X i¼1
pi eTi ei
l X i¼1
cij eTi Aej þ
j¼1
peTi ei þ
N X N X i¼1
N X i¼1
eTi csij As ej ðt sÞ
j¼1
beTi ei
N X i¼1
beTi ðt sÞei ðt sÞ:
ð3:3Þ
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From(2.2) and csij P 0, it is obvious that jcsij j < jcsii j, according to A(2), A(4) and Lemma 2.2, we have N X N X i¼1
eTi csij As ej ðt sÞ 6
j¼1
N X N N X N 1X 1X ðcsij Þ2 ei ATs QAs ei þ eT ðt sÞQ 1 ej ðt sÞ 2 i¼1 j¼1 2 i¼1 j¼1 j
6
N N Ng 2 X NX eTi ATs QAs ei þ eT ðt sÞQ 1 ej ðt sÞ 2 j¼1 j 2 i¼1
¼
2 N N N N X X N2 g 2 X N2 g 2 k X eTi ATs As ei þ b eTj ðt sÞej ðt sÞ ¼ eTi ei þ b eTj ðt sÞej ðt sÞ; 4b i¼1 4b i¼1 j¼1 j¼1
ð3:4Þ
N where Q ¼ 2b I > 0. Using Að1Þ and substituting (3.4) into (3.3), it follows that
2 N N2 g 2 k X eT ei 4b i¼1 i i¼1 i¼1 j¼1 i¼1 i¼1 ! ! 2 T N N N N l X X X X X N2 g 2 k T T T A þA ei ei ei þ ¼ aþbþ cij ei Aej þ cii ei peTi ei 2 4b i¼1 i¼1 j¼1;j–i i¼1 i¼1 ! ! 2 N N N N l b bT þ C X X X X X N2 g 2 k C T T T T ei ei þ 6 aþbþ ccij kei k2 kej k2 þ cii qmin ei ei pei ei ¼ e aIN þ c D e; 2 4b i¼1 i¼1 j¼1;j–i i¼1 i¼1
V_ 6
N X
aeTi ei þ
N X N X
cij eTi Aej
l X
peTi ei þ
N X
beTi ei þ
ð3:5Þ
where D ¼ diag p; pl ; 0; 0 ; e ¼ ðke1 k2 ; ke2 k2 ; . . . ; keN k2 ÞT . |fflfflffl{zfflfflffl} bT b According to Lemma 2.3, aIN þ c C 2þ C D < 0 is equivalent to aINl þ cC lþ1 < 0 for large enough p. From the assumption of Theorem 3.1, if there exist a natural number 1 6 l 6 N satisfying a þ cklþ1 < 0, one has aINl þ cC lþ1 < 0. Therefore, bT b bT b aIN þ c C 2þ C D < 0. Because aIN þ c C 2þ C D is a real symmetric matrix, there exists an orthogonal matrix P satisfying bT b aIN þ c C 2þ C D ¼ P T diag ^ k1 ; ^ k2 ; . . . ; ^ kN P, where ^ k1 ; ^ k2 ; . . . ; ^ kN ðPeÞ. It follows that ki < 0. Therefore, one gets V_ 6 ðPeÞT diag ^
T Pe ! 0 as t ! 0. Since P is an orthogonal matrix,the error vector g ¼ eT1 ; eT1 ; . . . ; eTN ! 0 as t ! 0. That is, the synchronous solution S(t) of the controlled network (2.1) is globally asymptotically stable under the pinning adaptive controllers (3.1). h Remark 1. It is not easy to get the entire information on the coupling matrix C s , so we use the partial information that the diagonal elements of the coupling matrix strength for delayed configuration csii has the upper bound g. In this paper, the coupling matrix C does not need to be symmetric. Remark 2. Compared with the results in paper [18,19], In this paper, the controllers are independent on the information of the time delay, so it is much easy to implement. Also, we use the adaptive technique, this will lead to convergence more fast. Remark 3. The controllers design method proposed in this paper can be applied to some uncertain delayed network. If there exist uncertain terms DC s ; DAs , which are bounded in the network, then the bound of k, g should be changed in the theorem. Corollary 3.2. Suppose Að1Þ; Að2Þ; A0 ð3Þ and Að4Þ hold and C ¼ cC, where C is a diffusive coupling matrix with cij ¼ 0 or 1 ði–jÞ. For the given c, if there exist a natural number 1 6 l 6 N and a positive constant b such that klþ1 < cac holds, where 2 2 2 g k . Then the delayed complex dynamical network (2.1) is globally synchronized under the pinning adaptive controla ¼ a þ b þ N 4b lers (3.1). Remark 4. Compared with the results in paper [21,22], we can estimate the minimum number of the pinning nodes which can achieve network synchronization. Specially assume C is symmetric and qmin ¼ c, if c > ka2 , then the network (2.1) can achieve synchronization under the adaptive controller (3.1) with l ¼ 1. From here we know that fewer controllers are needed to realize synchronization as the size of network N gets smaller. If the network does not have the time delay term, that is As ¼ 0. Obviously k ¼ 0 in Að2Þ. We can choose 2 P P as the Lyapunov function candidate, then we will have the following corollary, which is VðxÞ ¼ 12 Ni¼1 eTi ei þ 12 li¼1 ðpi pÞ qi the theorem in [17]. Corollary 3.3. Suppose Að1Þ—Að3Þ hold. If there exists a natural number 1 6 l 6 N such klþ1 < ac holds. Then the complex dynamical network (2.1) is globally synchronized under the pinning adaptive controller
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(
ui ¼ pi ei ;
p_i ¼ qi kei k22
ui ¼ 0;
1 6 i 6 l; otherwise;
where qi are positive constants for 1 6 i 6 l. 4. Numerical example In this section, we shall present an example to demonstrate the applicability of the proposed controllers. Here we consider an array of chaotic systems with time delay, in which each subsystem is a Lorenz systems. We investigate the network with 50 nodes. The node of the complex networks is described by
50 40
X3
30 20 10 0 40 20 0 −20 X2
−40
−20
10
0
−10
20
X
1
Fig. 1. Lorenz chaotic attractor.
50 40
ei1(1≤ i≤50)
30 20 10 0 −10 −20
0
2
4
time (s)
6
8
Fig. 2. Synchronization errors ei1 ð1 6 i 6 50Þ of the controlled network (4.2).
10
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0
1 0 r 1 x_i1 B _ C B @ xi2 A ¼ @ r3 x_i3
0
r1
0
1 0
10
xi1
1
0
0
1
CB C B C 0 A@ xi2 A þ @ xi1 xi3 A; xi3 xi1 xi2 r2
ð4:1Þ
where the parameters are often taken r 1 ¼ 10; r2 ¼ 8=3 and r3 ¼ 28 for chaotic behavior Fig. 1. The controlled network consist of 50 identical Lorenz systems, which is described by
x_ i ðtÞ ¼ f ðxi ðtÞÞ þ
N X
cij Axj þ
j¼1
N X
csij As xj ðt sÞ þ ui ;
ð4:2Þ
j¼1
where 1 6 i 6 50; f ðxi Þ ¼ ð10xi1 þ 10xi2 ; 28xi1 xi2 xi1 xi3 ; 83 xi3 þ xi1 xi2 ÞT ; ui is expressed in (3.1) and C ¼ ðcij Þ5050 is a symmetrically diffusive coupling matrix with cij ¼ 0 or 1ði–jÞ and the diagonal elements of C s have the upper bound g ¼ 2. Here the coupling coefficient c ¼ 100 and the inner coupling matrix is given as follows: 50 40
ei2(1≤ i≤50)
30 20 10 0 −10 −20 −30
0
2
4
time (s)
6
8
10
Fig. 3. Synchronization errors ei2 ð1 6 i 6 50Þ of the controlled network (4.2).
60 50
ei3(1≤ i≤50)
40 30 20 10 0 −10
0
2
4
time (s)
6
8
Fig. 4. Synchronization errors ei3 ð1 6 i 6 50Þ of the controlled network (4.2).
10
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2
1
0
1
0
0
1
3
6 7 A ¼ 4 0 1:2 1 5;
2
1 0 1
3
6 7 As ¼ 4 0 2 0 5 ; 1 0 1
Obviously we have c ¼ kAk2 ¼ 1:2; kAs k2 ¼ 2. It is well known that Lorenz attractor is bounded. There exist constants M1 ¼ 20; M2 ¼ 25 and M3 ¼ 45 satisfying kxij k; ksj k 6 Mj for 1 6 i 6 50 and 1 6 j 6 3. Thus one has kf ðxi Þ f ðsÞk2 6 88:8kei k2 ; i ¼ 1; 2; . . . ; N. Choose the positive number b ¼ Ng in the theorem, it is obvious to get a ¼ 188:8. Then one has cac ¼ 1:57. For a given random matrix C in this example one gets k14þ1 < 1:57. This means that one needs 14 nodes of 50 nodes to realize the synchronization. From the theorem and Corollary 3.2, the controlled network is globally synchronized with the solution of chaotic system (4.1) under the pinning adaptive controllers. In the simulation, we choose the initial values, xi ð0Þ ¼ ð4 þ i; 5 þ i; 6 þ iÞT ; sð0Þ ¼ ð4; 5; 6ÞT for 1 6 i 6 50 and qi ¼ 2 and pi ð0Þ ¼ 1 for 1 6 i 6 14. The errors eij ; 1 6 i 6 50; 1 6 j 6 3 are shown in Figs. 2–4. The controlled network is obviously globally asymptotically stable. 5. Conclusion In this paper, we designed simple controllers to realize the asymptotical synchronization for the complex dynamical networks with time delays. Compared with the model in [17], the proposed model additionally provides a mathematical description including the term of past states in the complex system equation. Compared with the results in [18], the simple controllers are not dependent on the accurate time delay and our results do not need to know the exact information of the matrix C s and As . Under some assumptions, the controllers are given to realize the synchronization for the delayed complex network. The theoretical result is shown by a numerical example. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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