- Email: [email protected]
Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling
Physics Letters A 372 (2008) 6340–6346
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling Sun Wen ∗ , Shihua Chen, Wanli Guo College of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China
a r t i c l e
i n f o
Article history: Received 30 May 2008 Received in revised form 8 August 2008 Accepted 14 August 2008 Available online 2 September 2008 Communicated by A.R. Bishop PACS: 05.45.Xt 05.45.Gg
a b s t r a c t This Letter investigates the global synchronization of a general complex dynamical network with nondelayed and delayed coupling. Based on Lasalle’s invariance principle, adaptive global synchronization criteria is obtained. Analytical result shows that under the designed adaptive controllers, a general complex dynamical network with non-delayed and delayed coupling can globally asymptotically synchronize to a given trajectory. What is more, the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition and the coupling matrix is not assumed to be symmetric or irreducible. Finally, numerical simulations are presented to verify the effectiveness of the proposed synchronization criteria. © 2008 Published by Elsevier B.V.
Keywords: Complex network Global synchronization Non-delayed and delayed coupling Adaptive
1. Introduction Complex networks are shown to exist in various fields of real world [1,2]. In general, complex networks consist of a large number of nodes and the links among them. Since their flexibility and generality for representing virtually any natural and man-made structure, complex networks have been received much attention as an interdisciplinary subject. Among various complex dynamical behaviors, synchronization is a significant and interesting phenomena. The earlier studies of synchronization in complex networks focus on regular networks such as global coupling networks or star coupling networks or the nearest neighbor coupling networks. In order to describe the more realistic networks—the transition from regular networks to random networks, Watts and Strogatz [3] introduced the small-world model recently. The small-world properties, jointly with the characterization of power law of networks, laid a foundation of the modern complex network theory. Since then, large networks of coupled dynamical systems that exhibit collective phenomena have attracted a lot of great interests in variety of fields, ranging from biology to physics and engineering [4–12]. In many existing studies, a common approach is to design entities external to the network which are deputed to control the whole network onto a desired synchronous evolution. In [13], Wang and Chen pinned a complex dynamical network to its equilibrium by using local injections. In [14], Li et al. introduced a linear state feedback controller to synchronize a complex network to a desired orbit. Especially, Chen et al. provide sufficient conditions guaranteeing synchronizing a dynamical network to a homogenous solution with a single pinning controller [15]. On the other hand, the characteristic of time-delayed coupling commonly exists in biological and physical systems such as biological neural networks, epidemiological models, communication networks, electrical power grids, etc., [16,17]. Some time-delays are trivial so that can be ignored, while some others cannot be ignored, particularly in long-distance communication and traffic congestions. To simulate more realistic networks, time delay should be taken into account. In [18], Wang et al. design a simple adaptive controller that ensures weighted complex dynamical networks with coupling time-varying delays to reach the desired manifold. It is noticed that most of the studies on synchronization of dynamical network have been performed under some implicit assumptions that there exists the information communication of nodes via the edges only at time t [19–21] or at time t − τ [22,23]. However, in many circumstances, this simplification does not match satisfactorily the peculiarities of real networks: there exists the information commu-
*
Corresponding author. Tel.: +86 2787164850; fax: +86 2768752256. E-mail address: [email protected] (S. Wen).
0375-9601/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.08.059
S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
6341
nication of nodes not only at time t but also at time t − τ . Whereas the synchronization of both delay-coupled and non-delay coupled complex dynamical network have almost been ignored in the literatures. Even [24,25] investigating this kind of network with delay and non-delay coupling, the node dynamics is assumed to satisfy a very strong and conservative uniformly Lipschitz condition. Motivated by these, this Letter is to handle the problem of adaptive synchronization for a general complex dynamical network with delayed coupling and non-delayed coupling. We design a simple adaptive controller that ensures a general complex dynamical network with both delayed and non-delayed coupling to reach the desired orbit. It should be pointed out that the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition and the matrix is not assumed to be symmetric or irreducible. The rest of the Letter is organized as follows: in Section 2, the model of a general complex dynamical network with non-delayed and delayed coupling is presented and some preliminaries are also given; in Section 3, sufficient condition of global synchronization of this complex networks is obtained; numerical simulations for verifying the theoretical result are presented in Section 4; the Letter is concluded in Section 5. 2. Model and preliminaries Consider a complex network consisting of N identical diffusively coupled nodes, in which each node is an n-dimensional dynamical unit. This model can be described as follows: dxi (t ) dt
N N = f xi (t ), t + c ai j Γ x j (t ) + c b i j Γ x j (t − τ ), j =1
i = 1, 2, . . . , N ,
(2.1)
j =1
where xi = (x1i , . . . , xni ) T ∈ R n is the state vector of node i, f : R n × [0, ∞] → R n is continuous vector function, c is the coupling strength, τ 0 is the coupled delay, A = (ai j ), B = (bi j ) ∈ R N ×N are the coupling matrices with zero-sum rows, which represent the coupling strength and the underlying topology for non-delayed configuration and delayed one at time t respectively, ai j 0, b i j 0 for i = j, ai j , b i j are defined as follows: if there is a connection from node j to node i (i = j ), ai j > 0, b i j > 0, otherwise ai j = 0, b i j = 0 (i = j ) for i , j = 1, 2, . . . , N, and Γ ∈ R n×n is an positive definite diagonal matrix which describe the individual couplings between node i and node j. Network (2.1) is a general complex network model in which there exist both non-delayed coupling and delayed coupling. It means that each node communicates with other nodes the information at time t as well as at time t − τ . In effect, this phenomenon exists commonly in our real world, for example, in the stock market, decision-making of single trader is influenced by that of others at time t as well as at time t − τ . Moreover, when B = 0 or A = 0, Network (2.1) is translated into dxi (t ) dt
N = f xi (t ), t + c ai j Γ x j (t ),
i = 1, 2, . . . , N ,
(2.2)
j =1
or dxi (t ) dt
N = f xi (t ), t + c b i j Γ x j (t − τ ),
i = 1, 2, . . . , N ,
(2.3)
j =1
which was investigated intensively in [3–6,10–14] or [7–9,16–18], so our network model contains the previous network model with only non-delayed coupling (B = 0) or only delayed coupling ( A = 0) as a special case. The marked difference between this Letter and the existing work [24,25] is that the node dynamic in our model need not satisfy the very strong and conservative uniformly Lipschitz condition but the less conservative Assumption H1. Let C ([−τ , 0], R n ) be the Banach space of continuous functions mapping the interval [−τ , 0] into R n with the norm φ = sup−τ θ0 φ(θ). For the functional differential equation (2.1), its initial conditions are given by xi (t ) = φ i (t ) ∈ C ([−τ , 0], R n ). We always assume that (2.1) has a unique solution with respect to initial conditions. Our objective is to design the adaptive controller u i (t ) to synchronize the network (2.1) onto a given evolution s(t ), i.e.,
lim xi (t ) − s(t ) = 0,
t →∞
i = 1, 2, . . . , n,
where synchronous evolution s(t ) is a solution of an isolated node of the network (2.1) such that s˙ (t ) = f (s(t )). In the Letter, we have the following hypothesis and Lemma: Assumption H1. Let P = diag{ p 1 , . . . , pn }, p i > 0, i = 1, 2, . . . , N, f (x, t ) ∈ C ( R n × [0, ∞], R n ) satisfies: (x − y ) T P [ f (x, t ) − f ( y , t ) − Θ(x − y )] −α (x − y ) T (x − y ) for some α > 0, all x, y ∈ R n and t > 0, where Θ = diag{θ1 , . . . , θn }. Many chaotic systems satisfy Assumption H1, for example, Chua’s oscillators, Rössler systems, Lorenz systems, Chen systems, and Lü systems. Lemma 1 (The Shur complements [26]). Let Q and R be two symmetric matrices. Then
Q
S
ST
R
<0
is equivalent to R < 0,
Q − S R −1 S T < 0.
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S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
3. Adaptive global synchronization with delayed and non-delayed coupling In order to achieve the global synchronization, we introduce adaptive control strategy to nodes in the network (2.1). Then the controlled network is given by dxi (t ) dt
N N = f xi (t ), t + c ai j Γ x j (t ) + c b i j Γ x j (t − τ ) + u i (t ), j =1
i = 1, 2, . . . , N ,
(3.1)
j =1
where
u i (t ) = −di (t ) xi (t ) − s(t )
(i = 1, 2, . . . , N )
(3.2)
are the adaptive controllers, di (t ) are the time-varying gains. To guarantee negative feedback, the adaptive gains are designed as:
T
d˙ i (t ) = αi xi (t ) − s(t ) where
P xi (t ) − s(t ) ,
i = 1, 2, . . . , N
(3.3)
αi are positive constants to be determined. Let e i (t ) = xi (t ) − s(t ), then we have the following dynamical error equations: de i (t ) dt
N N = f xi (t ), t − f s(t ), t + c ai j Γ e j (t ) + c b i j Γ e j (t − τ ) + u i (t ), j =1
i = 1, 2, . . . , N .
(3.4)
j =1
Then we can derive the main result: Theorem 1. Let Assumption H1 hold. The controlled complex dynamical network (3.1) globally asymptotically synchronizes to the given trajectory s(t ) under the adaptive controllers (3.2) and the adaptive gains (3.3), where di (0) and αi are arbitrary positive constants. Proof. We choose a Lyapunov function candidate as
t N N n T 2 1 i T i 1 1 V (t ) = e (t ) P e (t ) + d (t ) − li + pj e˜ j (s) Q j e˜ j (s) ds 2 2 αi i i =1
i =1
j =1
t −τ
where li is a positive constant to be determined. Differentiating V (t ) along the trajectories of (3.4), by using the inequality 2x T y x T Q x + y T Q −1 y and the Shur complements, we have dV (t ) dt
=
N N T i j j i e (t ) P f x (t ), t − f s(t ), t + c ai j Γ e (t ) + c b i j Γ e (t − τ ) + u (t )
N
i
i =1
+
j =1
N
T
di (t ) − li e i (t )
P e i (t ) +
n
i =1
=
N
T
e i (t )
N
P f xi (t ), t − f s(t ), t − Θ e i (t ) +
T
Q j e˜ j (t − τ )
T
e i (t )
(Θ P − li P )e i (t )
N
N n n T T T i j j e (t ) P ai j Γ e (t ) + b i j Γ e (t − τ ) + p j e˜ j (t ) Q j e˜ j (t ) − p j e˜ j (t − τ ) Q j e˜ j (t − τ )
N i =1
N
j =1
T
e i (t )
e i (t ) +
i =1
+c
j =1 N
T
e i (t )
i =1
N N
T
e i (t )
N
n
T
p j e˜ j (t )
e i (t )
T
e i (t ) +
n
T
p j e˜ j (t )
T
p j e˜ j (t )
Q j e˜ j (t ) −
n
(θ j I N − L )˜e j (t ) + c
γ j B e˜ j (t − τ ) +
n j =1
T
e i (t ) +
n
i =1
j =1
i =1
j =1
T
p j e˜ j (t − τ )
Q j e˜ j (t − τ )
n
T
p j e˜ j (t )
γ j A e˜ j (t )
j =1
j =1
e i (t )
ai j P Γ e j (t )
j =1
j =1
N
T
i =1 j =1
b i j P Γ e j (t − τ ) +
i =1 n
j =1
N N
j =1
e i (t )
+c
j =1
(Θ P − li P )e i (t ) + c
i =1 j =1
= −η
Q j e˜ j (t ) − e˜ j (t − τ )
i =1
+c
= −η
T
p j e˜ j (t )
j =1
i =1
−η
j =1
T
p j e˜ j (t )
T
p j e˜ j (t )
Q j e˜ j (t ) −
n
T
p j e˜ j (t − τ )
Q j e˜ j (t − τ )
j =1
T (θ j I N − L )˜e j (t ) + c e˜ j (t ) γ j A e˜ j (t )
T T T + c e˜ j (t ) γ j B e˜ j (t − τ ) + e˜ j (t ) Q j e˜ j (t ) − e˜ j (t − τ ) Q j e˜ j (t − τ ) N n T i T i (c γ j )2 T −1 (θ j I N − L ) + c γ j A s + e (t ) e (t ) + p j e˜ j (t ) −η B Q j B + Q j e˜ j (t ),
4
S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
Fig. 1. Synchronization errors e 1i (t ), e 2i (t ), e 3i (t ), and E (t ) of network (3.1) with
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τ = 0.01 (i = 1, 2, . . . , 6).
Fig. 2. Synchronization errors e 1i (t ), e 2i (t ), e 3i (t ), and E (t ) of network (3.1) with
τ = 1 (i = 1, 2, . . . , 6).
where e˜ j (t ) = [˜e 1j , e˜ 2j , . . . , e˜ Nj ] T , L = diag{li , . . . , l N }. Taking suitable positive parameters θ j , l j , c , γ j , Q j such that
T
e˜ j (t )
(θ j I N − L ) + c γ j A s +
(c γ j )2 4
1 B + Q j e˜ j (t ) 0, BT Q − j
j = 1, . . . , n,
so we obtain dV (t ) dt
−η
N
T
e i (t )
e i (t ) 0.
i =1 dV (t )
It is obvious that the largest invariant set contained in set E = { dt = 0} = {e i (t ) = 0, i = 1, . . . , N } is M = {e i (t ) = 0, di (t ) = li , i = 1, . . . , N }. According to Lasalle’s invariance principle [27], starting with arbitrary initial values of (3.1), the trajectory asymptotically converges to the set M, i.e., as t → ∞, xi (t ) → s(t ) and di (t ) → li (i = 1, 2, . . . , N ), where li (i = 1, 2, . . . , N ) are positive constants. This indicates that globally asymptotical synchronization of network (3.2) is realized, we complete the proof. 2 Remark 1. In our model, there exists non-delayed coupling as well as delayed coupling, which is suitable to investigate and simulate more realistic complex networks.
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S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
Fig. 3. Synchronization errors e 1i (t ), e 2i (t ), e 3i (t ), and E (t ) of network (3.1) with
τ = 5 (i = 1, 2, . . . , 6).
Remark 2. In our Letter, the node dynamic is not assumed to satisfy the very strong and conservative uniformly Lipschitz condition but the less conservative Assumption H1. Remark 3. In our Letter, the coupling matrix A, B is not necessarily symmetric and irreducible. To this end, this theorem is applicable to a great many complex dynamical networks. Remark 4. The constant αi can be chosen properly to adjust the synchronization speed. The larger the adaptive gains the faster is to achieve synchronization of networks (3.1).
αi (i = 1, . . . , N ),
If there exists the information communication of nodes via the edges only at time t or at time t − τ , i.e., the coupling matrix B = 0 or A = 0, then we have the following corollaries. Corollary 1. With the adaptive controllers (3.2) and the adaptive gains (3.3), the coupled complex dynamical network dxi (t ) dt
N = f xi (t ), t + c ai j Γ x j (t ) + u i (t ),
i = 1, 2, . . . , N ,
(3.5)
j =1
will globally asymptotically synchronize to a given trajectory s(t ). Corollary 2. With the adaptive controllers (3.2) and the adaptive gains (3.3), the delay-coupled complex dynamical network dxi (t ) dt
N = f xi (t ), t + c b i j Γ x j (t − τ ) + u i (t ),
i = 1, 2, . . . , N ,
j =1
will globally asymptotically synchronize to a given trajectory s(t ). 4. Numerical results In this section, numerical simulations are presented to verify the theoretical result obtained in the previous section. We consider chaotic Lorenz oscillators as the node dynamic system. A single Lorenz oscillator is described by
⎧ ⎨ s˙ 1 = a(s2 − s1 ), s˙ = cs1 − s1 s3 − s2 , ⎩ 2 s˙ 3 = s1 s2 − bs3 , where a = 10, b = 8/3, c = 28. We show that a delayed and non-delayed network with 6 nodes described by
⎧ dxi (t ) 6 6 i j j i ⎪ ⎨ dt = f (x (t ), t ) + c j =1 ai j Γ x (t ) + c j =1 bi j Γ x (t − τ ) + u (t ), u i (t ) = −di (t )(xi (t ) − s(t )), ⎪ ⎩ d˙ i (t ) = αi (xi (t ) − s(t )) T P (xi (t ) − s(t )),
i = 1, 2, . . . , 6
(3.6)
S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
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Fig. 4. Synchronization errors e 1i (t ), e 2i (t ), e 3i (t ), and E (t ) of network (3.5) (i = 1, 2, . . . , 6).
Fig. 5. Synchronization errors e 1i (t ), e 2i (t ), e 3i (t ), and E (t ) of network (3.6) with
τ = 0.01(i = 1, 2, . . . , 6).
will reach a specified trajectory s(t ) of the corresponding chaotic Lorenz oscillator, where f (·) is chaotic Lorenz oscillator. The coupling matrix are
⎡
−1 1/5 1/5 ⎢ 1/4 −1 0 ⎢ ⎢ ⎢ 1/3 0 −1 A=⎢ ⎢ 1/4 1/4 1/4 ⎢ ⎢ ⎣ 1/4 1/4 1/4 1/2
1/2
0
1/5 1/5
1/5
⎤
1/4 1/4 1/4 ⎥ ⎥ ⎥ 1/3 1/3 0 ⎥ ⎥, −1 1/4 0 ⎥ ⎥ ⎥ 1/4 −1 0 ⎦ 0
0
⎡
−1 1/4 1/4 0 1/4 ⎢ 1/4 −1 0 1/4 1/4 ⎢ ⎢ ⎢ 1/3 0 −1 1/3 1/3 B =⎢ ⎢ 0 0 0 −1 1/2 ⎢ ⎢ ⎣ 0 0 0 1/4 −1
−1
0
0
0
3/4 1/4
1/4
⎤
1/4 ⎥ ⎥
⎥ ⎥, 1/2 ⎥ ⎥ ⎥ 3/4 ⎦ 0 ⎥
−1
respectively. We take P = diag{24, 0.002, 12}, Q j = diag{0.1273, 0.4818, 0.3364, 0.2818, 0.2818, 0.1909}, Θ = diag{3, 3, 3}, c = 30, Γ = diag{1, 1, 1}, sufficient large l j , such that
(θ j I N − L ) + c γ j A s +
(c γ j )2 4
1 B + Q j < 0, BT Q − j
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S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
then the coupled network with delayed and non-delayed coupling can globally synchronize to the desired orbit, and the simulation results N i 2 are plotted in Figs. 1–3. We introduce the quantity [21,22] E (t ) = i =1 x (t ) − s(t ) / N, which is used to measure the quality of the control process. It is obvious that when E (t ) no longer increases, the whole network achieves synchronization globally. When the coupling matrix B = 0 or A = 0, network (3.1) translates into network (3.5) or (3.6), then the simulation results are plotted in Figs. 4–5. 5. Conclusion This Letter handles the problem of the global synchronization of a general complex dynamical networks with non-delayed and delayed coupling. By Lasalle’s invariance principle, adaptive global synchronization criteria is obtained. Analytical result shows that under the designed adaptive controllers, a general complex dynamical networks with non-delayed and delayed coupling can globally asymptotically synchronize to a given trajectory. Moreover, the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition and the coupling matrix is not assumed to be symmetric or irreducible. Numerical simulations are presented to verify the effectiveness of the proposed synchronization criteria finally. Acknowledgements The authors thank the referees and the editor for their valuable comments and suggestions on improvement of this Letter. The work is supported by the National Nature Science Foundation of China (No. 70571059). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling Sun Wen ∗ , Shihua Chen, Wanli Guo College of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China
a r t i c l e
i n f o
Article history: Received 30 May 2008 Received in revised form 8 August 2008 Accepted 14 August 2008 Available online 2 September 2008 Communicated by A.R. Bishop PACS: 05.45.Xt 05.45.Gg
a b s t r a c t This Letter investigates the global synchronization of a general complex dynamical network with nondelayed and delayed coupling. Based on Lasalle’s invariance principle, adaptive global synchronization criteria is obtained. Analytical result shows that under the designed adaptive controllers, a general complex dynamical network with non-delayed and delayed coupling can globally asymptotically synchronize to a given trajectory. What is more, the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition and the coupling matrix is not assumed to be symmetric or irreducible. Finally, numerical simulations are presented to verify the effectiveness of the proposed synchronization criteria. © 2008 Published by Elsevier B.V.
Keywords: Complex network Global synchronization Non-delayed and delayed coupling Adaptive
1. Introduction Complex networks are shown to exist in various fields of real world [1,2]. In general, complex networks consist of a large number of nodes and the links among them. Since their flexibility and generality for representing virtually any natural and man-made structure, complex networks have been received much attention as an interdisciplinary subject. Among various complex dynamical behaviors, synchronization is a significant and interesting phenomena. The earlier studies of synchronization in complex networks focus on regular networks such as global coupling networks or star coupling networks or the nearest neighbor coupling networks. In order to describe the more realistic networks—the transition from regular networks to random networks, Watts and Strogatz [3] introduced the small-world model recently. The small-world properties, jointly with the characterization of power law of networks, laid a foundation of the modern complex network theory. Since then, large networks of coupled dynamical systems that exhibit collective phenomena have attracted a lot of great interests in variety of fields, ranging from biology to physics and engineering [4–12]. In many existing studies, a common approach is to design entities external to the network which are deputed to control the whole network onto a desired synchronous evolution. In [13], Wang and Chen pinned a complex dynamical network to its equilibrium by using local injections. In [14], Li et al. introduced a linear state feedback controller to synchronize a complex network to a desired orbit. Especially, Chen et al. provide sufficient conditions guaranteeing synchronizing a dynamical network to a homogenous solution with a single pinning controller [15]. On the other hand, the characteristic of time-delayed coupling commonly exists in biological and physical systems such as biological neural networks, epidemiological models, communication networks, electrical power grids, etc., [16,17]. Some time-delays are trivial so that can be ignored, while some others cannot be ignored, particularly in long-distance communication and traffic congestions. To simulate more realistic networks, time delay should be taken into account. In [18], Wang et al. design a simple adaptive controller that ensures weighted complex dynamical networks with coupling time-varying delays to reach the desired manifold. It is noticed that most of the studies on synchronization of dynamical network have been performed under some implicit assumptions that there exists the information communication of nodes via the edges only at time t [19–21] or at time t − τ [22,23]. However, in many circumstances, this simplification does not match satisfactorily the peculiarities of real networks: there exists the information commu-
*
Corresponding author. Tel.: +86 2787164850; fax: +86 2768752256. E-mail address: [email protected] (S. Wen).
0375-9601/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.08.059
S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
6341
nication of nodes not only at time t but also at time t − τ . Whereas the synchronization of both delay-coupled and non-delay coupled complex dynamical network have almost been ignored in the literatures. Even [24,25] investigating this kind of network with delay and non-delay coupling, the node dynamics is assumed to satisfy a very strong and conservative uniformly Lipschitz condition. Motivated by these, this Letter is to handle the problem of adaptive synchronization for a general complex dynamical network with delayed coupling and non-delayed coupling. We design a simple adaptive controller that ensures a general complex dynamical network with both delayed and non-delayed coupling to reach the desired orbit. It should be pointed out that the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition and the matrix is not assumed to be symmetric or irreducible. The rest of the Letter is organized as follows: in Section 2, the model of a general complex dynamical network with non-delayed and delayed coupling is presented and some preliminaries are also given; in Section 3, sufficient condition of global synchronization of this complex networks is obtained; numerical simulations for verifying the theoretical result are presented in Section 4; the Letter is concluded in Section 5. 2. Model and preliminaries Consider a complex network consisting of N identical diffusively coupled nodes, in which each node is an n-dimensional dynamical unit. This model can be described as follows: dxi (t ) dt
N N = f xi (t ), t + c ai j Γ x j (t ) + c b i j Γ x j (t − τ ), j =1
i = 1, 2, . . . , N ,
(2.1)
j =1
where xi = (x1i , . . . , xni ) T ∈ R n is the state vector of node i, f : R n × [0, ∞] → R n is continuous vector function, c is the coupling strength, τ 0 is the coupled delay, A = (ai j ), B = (bi j ) ∈ R N ×N are the coupling matrices with zero-sum rows, which represent the coupling strength and the underlying topology for non-delayed configuration and delayed one at time t respectively, ai j 0, b i j 0 for i = j, ai j , b i j are defined as follows: if there is a connection from node j to node i (i = j ), ai j > 0, b i j > 0, otherwise ai j = 0, b i j = 0 (i = j ) for i , j = 1, 2, . . . , N, and Γ ∈ R n×n is an positive definite diagonal matrix which describe the individual couplings between node i and node j. Network (2.1) is a general complex network model in which there exist both non-delayed coupling and delayed coupling. It means that each node communicates with other nodes the information at time t as well as at time t − τ . In effect, this phenomenon exists commonly in our real world, for example, in the stock market, decision-making of single trader is influenced by that of others at time t as well as at time t − τ . Moreover, when B = 0 or A = 0, Network (2.1) is translated into dxi (t ) dt
N = f xi (t ), t + c ai j Γ x j (t ),
i = 1, 2, . . . , N ,
(2.2)
j =1
or dxi (t ) dt
N = f xi (t ), t + c b i j Γ x j (t − τ ),
i = 1, 2, . . . , N ,
(2.3)
j =1
which was investigated intensively in [3–6,10–14] or [7–9,16–18], so our network model contains the previous network model with only non-delayed coupling (B = 0) or only delayed coupling ( A = 0) as a special case. The marked difference between this Letter and the existing work [24,25] is that the node dynamic in our model need not satisfy the very strong and conservative uniformly Lipschitz condition but the less conservative Assumption H1. Let C ([−τ , 0], R n ) be the Banach space of continuous functions mapping the interval [−τ , 0] into R n with the norm φ = sup−τ θ0 φ(θ). For the functional differential equation (2.1), its initial conditions are given by xi (t ) = φ i (t ) ∈ C ([−τ , 0], R n ). We always assume that (2.1) has a unique solution with respect to initial conditions. Our objective is to design the adaptive controller u i (t ) to synchronize the network (2.1) onto a given evolution s(t ), i.e.,
lim xi (t ) − s(t ) = 0,
t →∞
i = 1, 2, . . . , n,
where synchronous evolution s(t ) is a solution of an isolated node of the network (2.1) such that s˙ (t ) = f (s(t )). In the Letter, we have the following hypothesis and Lemma: Assumption H1. Let P = diag{ p 1 , . . . , pn }, p i > 0, i = 1, 2, . . . , N, f (x, t ) ∈ C ( R n × [0, ∞], R n ) satisfies: (x − y ) T P [ f (x, t ) − f ( y , t ) − Θ(x − y )] −α (x − y ) T (x − y ) for some α > 0, all x, y ∈ R n and t > 0, where Θ = diag{θ1 , . . . , θn }. Many chaotic systems satisfy Assumption H1, for example, Chua’s oscillators, Rössler systems, Lorenz systems, Chen systems, and Lü systems. Lemma 1 (The Shur complements [26]). Let Q and R be two symmetric matrices. Then
Q
S
ST
R
<0
is equivalent to R < 0,
Q − S R −1 S T < 0.
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3. Adaptive global synchronization with delayed and non-delayed coupling In order to achieve the global synchronization, we introduce adaptive control strategy to nodes in the network (2.1). Then the controlled network is given by dxi (t ) dt
N N = f xi (t ), t + c ai j Γ x j (t ) + c b i j Γ x j (t − τ ) + u i (t ), j =1
i = 1, 2, . . . , N ,
(3.1)
j =1
where
u i (t ) = −di (t ) xi (t ) − s(t )
(i = 1, 2, . . . , N )
(3.2)
are the adaptive controllers, di (t ) are the time-varying gains. To guarantee negative feedback, the adaptive gains are designed as:
T
d˙ i (t ) = αi xi (t ) − s(t ) where
P xi (t ) − s(t ) ,
i = 1, 2, . . . , N
(3.3)
αi are positive constants to be determined. Let e i (t ) = xi (t ) − s(t ), then we have the following dynamical error equations: de i (t ) dt
N N = f xi (t ), t − f s(t ), t + c ai j Γ e j (t ) + c b i j Γ e j (t − τ ) + u i (t ), j =1
i = 1, 2, . . . , N .
(3.4)
j =1
Then we can derive the main result: Theorem 1. Let Assumption H1 hold. The controlled complex dynamical network (3.1) globally asymptotically synchronizes to the given trajectory s(t ) under the adaptive controllers (3.2) and the adaptive gains (3.3), where di (0) and αi are arbitrary positive constants. Proof. We choose a Lyapunov function candidate as
t N N n T 2 1 i T i 1 1 V (t ) = e (t ) P e (t ) + d (t ) − li + pj e˜ j (s) Q j e˜ j (s) ds 2 2 αi i i =1
i =1
j =1
t −τ
where li is a positive constant to be determined. Differentiating V (t ) along the trajectories of (3.4), by using the inequality 2x T y x T Q x + y T Q −1 y and the Shur complements, we have dV (t ) dt
=
N N T i j j i e (t ) P f x (t ), t − f s(t ), t + c ai j Γ e (t ) + c b i j Γ e (t − τ ) + u (t )
N
i
i =1
+
j =1
N
T
di (t ) − li e i (t )
P e i (t ) +
n
i =1
=
N
T
e i (t )
N
P f xi (t ), t − f s(t ), t − Θ e i (t ) +
T
Q j e˜ j (t − τ )
T
e i (t )
(Θ P − li P )e i (t )
N
N n n T T T i j j e (t ) P ai j Γ e (t ) + b i j Γ e (t − τ ) + p j e˜ j (t ) Q j e˜ j (t ) − p j e˜ j (t − τ ) Q j e˜ j (t − τ )
N i =1
N
j =1
T
e i (t )
e i (t ) +
i =1
+c
j =1 N
T
e i (t )
i =1
N N
T
e i (t )
N
n
T
p j e˜ j (t )
e i (t )
T
e i (t ) +
n
T
p j e˜ j (t )
T
p j e˜ j (t )
Q j e˜ j (t ) −
n
(θ j I N − L )˜e j (t ) + c
γ j B e˜ j (t − τ ) +
n j =1
T
e i (t ) +
n
i =1
j =1
i =1
j =1
T
p j e˜ j (t − τ )
Q j e˜ j (t − τ )
n
T
p j e˜ j (t )
γ j A e˜ j (t )
j =1
j =1
e i (t )
ai j P Γ e j (t )
j =1
j =1
N
T
i =1 j =1
b i j P Γ e j (t − τ ) +
i =1 n
j =1
N N
j =1
e i (t )
+c
j =1
(Θ P − li P )e i (t ) + c
i =1 j =1
= −η
Q j e˜ j (t ) − e˜ j (t − τ )
i =1
+c
= −η
T
p j e˜ j (t )
j =1
i =1
−η
j =1
T
p j e˜ j (t )
T
p j e˜ j (t )
Q j e˜ j (t ) −
n
T
p j e˜ j (t − τ )
Q j e˜ j (t − τ )
j =1
T (θ j I N − L )˜e j (t ) + c e˜ j (t ) γ j A e˜ j (t )
T T T + c e˜ j (t ) γ j B e˜ j (t − τ ) + e˜ j (t ) Q j e˜ j (t ) − e˜ j (t − τ ) Q j e˜ j (t − τ ) N n T i T i (c γ j )2 T −1 (θ j I N − L ) + c γ j A s + e (t ) e (t ) + p j e˜ j (t ) −η B Q j B + Q j e˜ j (t ),
4
S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
Fig. 1. Synchronization errors e 1i (t ), e 2i (t ), e 3i (t ), and E (t ) of network (3.1) with
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τ = 0.01 (i = 1, 2, . . . , 6).
Fig. 2. Synchronization errors e 1i (t ), e 2i (t ), e 3i (t ), and E (t ) of network (3.1) with
τ = 1 (i = 1, 2, . . . , 6).
where e˜ j (t ) = [˜e 1j , e˜ 2j , . . . , e˜ Nj ] T , L = diag{li , . . . , l N }. Taking suitable positive parameters θ j , l j , c , γ j , Q j such that
T
e˜ j (t )
(θ j I N − L ) + c γ j A s +
(c γ j )2 4
1 B + Q j e˜ j (t ) 0, BT Q − j
j = 1, . . . , n,
so we obtain dV (t ) dt
−η
N
T
e i (t )
e i (t ) 0.
i =1 dV (t )
It is obvious that the largest invariant set contained in set E = { dt = 0} = {e i (t ) = 0, i = 1, . . . , N } is M = {e i (t ) = 0, di (t ) = li , i = 1, . . . , N }. According to Lasalle’s invariance principle [27], starting with arbitrary initial values of (3.1), the trajectory asymptotically converges to the set M, i.e., as t → ∞, xi (t ) → s(t ) and di (t ) → li (i = 1, 2, . . . , N ), where li (i = 1, 2, . . . , N ) are positive constants. This indicates that globally asymptotical synchronization of network (3.2) is realized, we complete the proof. 2 Remark 1. In our model, there exists non-delayed coupling as well as delayed coupling, which is suitable to investigate and simulate more realistic complex networks.
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S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
Fig. 3. Synchronization errors e 1i (t ), e 2i (t ), e 3i (t ), and E (t ) of network (3.1) with
τ = 5 (i = 1, 2, . . . , 6).
Remark 2. In our Letter, the node dynamic is not assumed to satisfy the very strong and conservative uniformly Lipschitz condition but the less conservative Assumption H1. Remark 3. In our Letter, the coupling matrix A, B is not necessarily symmetric and irreducible. To this end, this theorem is applicable to a great many complex dynamical networks. Remark 4. The constant αi can be chosen properly to adjust the synchronization speed. The larger the adaptive gains the faster is to achieve synchronization of networks (3.1).
αi (i = 1, . . . , N ),
If there exists the information communication of nodes via the edges only at time t or at time t − τ , i.e., the coupling matrix B = 0 or A = 0, then we have the following corollaries. Corollary 1. With the adaptive controllers (3.2) and the adaptive gains (3.3), the coupled complex dynamical network dxi (t ) dt
N = f xi (t ), t + c ai j Γ x j (t ) + u i (t ),
i = 1, 2, . . . , N ,
(3.5)
j =1
will globally asymptotically synchronize to a given trajectory s(t ). Corollary 2. With the adaptive controllers (3.2) and the adaptive gains (3.3), the delay-coupled complex dynamical network dxi (t ) dt
N = f xi (t ), t + c b i j Γ x j (t − τ ) + u i (t ),
i = 1, 2, . . . , N ,
j =1
will globally asymptotically synchronize to a given trajectory s(t ). 4. Numerical results In this section, numerical simulations are presented to verify the theoretical result obtained in the previous section. We consider chaotic Lorenz oscillators as the node dynamic system. A single Lorenz oscillator is described by
⎧ ⎨ s˙ 1 = a(s2 − s1 ), s˙ = cs1 − s1 s3 − s2 , ⎩ 2 s˙ 3 = s1 s2 − bs3 , where a = 10, b = 8/3, c = 28. We show that a delayed and non-delayed network with 6 nodes described by
⎧ dxi (t ) 6 6 i j j i ⎪ ⎨ dt = f (x (t ), t ) + c j =1 ai j Γ x (t ) + c j =1 bi j Γ x (t − τ ) + u (t ), u i (t ) = −di (t )(xi (t ) − s(t )), ⎪ ⎩ d˙ i (t ) = αi (xi (t ) − s(t )) T P (xi (t ) − s(t )),
i = 1, 2, . . . , 6
(3.6)
S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
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Fig. 4. Synchronization errors e 1i (t ), e 2i (t ), e 3i (t ), and E (t ) of network (3.5) (i = 1, 2, . . . , 6).
Fig. 5. Synchronization errors e 1i (t ), e 2i (t ), e 3i (t ), and E (t ) of network (3.6) with
τ = 0.01(i = 1, 2, . . . , 6).
will reach a specified trajectory s(t ) of the corresponding chaotic Lorenz oscillator, where f (·) is chaotic Lorenz oscillator. The coupling matrix are
⎡
−1 1/5 1/5 ⎢ 1/4 −1 0 ⎢ ⎢ ⎢ 1/3 0 −1 A=⎢ ⎢ 1/4 1/4 1/4 ⎢ ⎢ ⎣ 1/4 1/4 1/4 1/2
1/2
0
1/5 1/5
1/5
⎤
1/4 1/4 1/4 ⎥ ⎥ ⎥ 1/3 1/3 0 ⎥ ⎥, −1 1/4 0 ⎥ ⎥ ⎥ 1/4 −1 0 ⎦ 0
0
⎡
−1 1/4 1/4 0 1/4 ⎢ 1/4 −1 0 1/4 1/4 ⎢ ⎢ ⎢ 1/3 0 −1 1/3 1/3 B =⎢ ⎢ 0 0 0 −1 1/2 ⎢ ⎢ ⎣ 0 0 0 1/4 −1
−1
0
0
0
3/4 1/4
1/4
⎤
1/4 ⎥ ⎥
⎥ ⎥, 1/2 ⎥ ⎥ ⎥ 3/4 ⎦ 0 ⎥
−1
respectively. We take P = diag{24, 0.002, 12}, Q j = diag{0.1273, 0.4818, 0.3364, 0.2818, 0.2818, 0.1909}, Θ = diag{3, 3, 3}, c = 30, Γ = diag{1, 1, 1}, sufficient large l j , such that
(θ j I N − L ) + c γ j A s +
(c γ j )2 4
1 B + Q j < 0, BT Q − j
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S. Wen et al. / Physics Letters A 372 (2008) 6340–6346
then the coupled network with delayed and non-delayed coupling can globally synchronize to the desired orbit, and the simulation results N i 2 are plotted in Figs. 1–3. We introduce the quantity [21,22] E (t ) = i =1 x (t ) − s(t ) / N, which is used to measure the quality of the control process. It is obvious that when E (t ) no longer increases, the whole network achieves synchronization globally. When the coupling matrix B = 0 or A = 0, network (3.1) translates into network (3.5) or (3.6), then the simulation results are plotted in Figs. 4–5. 5. Conclusion This Letter handles the problem of the global synchronization of a general complex dynamical networks with non-delayed and delayed coupling. By Lasalle’s invariance principle, adaptive global synchronization criteria is obtained. Analytical result shows that under the designed adaptive controllers, a general complex dynamical networks with non-delayed and delayed coupling can globally asymptotically synchronize to a given trajectory. Moreover, the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition and the coupling matrix is not assumed to be symmetric or irreducible. Numerical simulations are presented to verify the effectiveness of the proposed synchronization criteria finally. Acknowledgements The authors thank the referees and the editor for their valuable comments and suggestions on improvement of this Letter. The work is supported by the National Nature Science Foundation of China (No. 70571059). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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