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Outer synchronization between two nonidentical networks with circumstance noise
Physica A 389 (2010) 1480–1488
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Physica A journal homepage: www.elsevier.com/locate/physa
Outer synchronization between two nonidentical networks with circumstance noise Guanjun Wang ∗ , Jinde Cao, Jianquan Lu Department of Mathematics, Southeast University, Nanjing 210096, China
article
info
Article history: Received 17 July 2009 Received in revised form 29 November 2009 Available online 22 December 2009 Keywords: Complex network Outer synchronization Adaptive controller Stochastic LaSalle theorem Lyapunov function
abstract This paper regards the outer synchronization between two delay-coupled complex dynamical networks with nonidentical topological structures and a noise perturbation. Considering one network as the drive network and the other one as the response network, the drive–response system achieves synchronous states through a suitably designed adaptive controller. The stochastic LaSalle invariance principle is employed to theoretically prove the almost sure synchronization between two networks. Finally, two numerical examples are examined in order to illustrate the proposed synchronization scheme. © 2009 Elsevier B.V. All rights reserved.
1. Introduction A complex network is a large set of interconnected nodes, in which a node is a fundamental unit with specific contents. Many real world systems can be described by complex networks, such as the World Wide Web, telephone call graphs, neural networks, scientific citation web, etc. Random graph theory is the traditional mathematical method in studying complex networks which was proposed by Erdös and Rényi [1] about fifty years ago. Recently, the development of modern computer provided a powerful computational tool to deal with large scale data from networks, which led to the discovery of the ‘‘small-world’’ and ‘‘scale-free’’ properties of complex networks [2,3]. Consequently, complex networks become a focus point of research which has attracted increasing attention from various fields of science and engineering. In particular, special attention has been focused on the synchronization problem for dynamical complex networks, in which each node is regarded as a dynamical element. Much work has been done for the synchronization of complex networks in the literature, see e.g. Refs. [4–11]. Wang and Chen [4,5] studied the synchronization problem for two specific complex networks with small-world and scale-free properties, respectively. In Ref. [6], sufficient conditions are provided to synchronize a class of delay-coupled complex dynamical networks by using the Lyapunov method. In Ref. [7], Zhou, Lu and Lü proposed an adaptive synchronization scheme for uncertain dynamical networks. Lu and Ho investigated the local and global synchronization for complex dynamical networks without assuming the symmetry and irreducibility of the coupling configuration matrices [8]. A review on synchronization of complex networks can be found in Ref. [11]. The synchronization within one network mentioned in the above references is named ‘‘inner synchronization’’ [12], which is concerned with the synchronization among the nodes within a network. Different from the ‘‘inner synchronization’’, the synchronization between two or more complex networks is called ‘‘outer synchronization’’, which was firstly studied by Li et al. [12]. And then outer synchronization was investigated by several other researchers in Refs. [13–16], etc. In Ref. [12], Li, Sun and Kurths studied the synchronization problem for two complex networks with identical topological
∗
Corresponding author. E-mail addresses: [email protected] (G. Wang), [email protected] (J. Cao), [email protected] (J. Lu).
0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.12.014
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structures by using an open-plus-closed-loop controller. Later on, the outer synchronization for two coupled complex networks was extended to the discrete time case [13]. In Ref. [14], adaptive controllers were designed to synchronize two complex networks with identical or nonidentical topological structures. In Ref. [16], Wu, Zheng and Zhou synchronized two different complex dynamical networks by constructing a nonlinear controller. For large scale networks, time delays is unavoidable and should be taken into account due to the finite information transmission and processing speeds among the network nodes. Time-delayed coupling extensively exists in many biological and physical systems such as gene regulatory networks, communication networks, and electrical power grids etc. It has also been discovered that time delays frequently have great influence on the behavior of dynamical systems [17,18]. Hence, it is very important to study the synchronization of complex networks with delay coupling. Noise is another factor which may affect the behavior of dynamical systems [19]. During the evolving process of complex dynamical networks, noises from circumstance are often inevitable, which may destroy the synchronization of complex network. Goldobina and Pikovsky [20, 21] even proved that self-sustained oscillators can be synchronized by common noise. Motivated by above discussions, we will study the outer synchronization problem for two delay-coupled complex dynamical networks with the consideration of a circumstance noise. Adaptive controllers are designed for the global outer synchronization of the drive–response network system, in which the networks have identical and nonidentical topological structures. Numerical simulations are provided to show the effectiveness of the proposed synchronization scheme. 2. Network modeling and preliminaries In Ref. [14], Tang, Chen and Lu studied the adaptive synchronization problem between two coupled complex networks. In Ref. [14], the drive complex network is described by x˙i (t ) = f (xi (t )) +
N X
cij Axj (t ),
i = 1, 2, . . . , N
(1)
j =1
where xi (t ) = (xi1 (t ), . . . , xin (t ))T ∈ Rn is the state vector of the ith node in the drive network; f : R × Rn → Rn is a nonlinear vector function, and x˙i (t ) = f (xi (t )) represents the dynamics of a single node; A is the inner connecting matrix between two connected nodes; C = (cij )N ×N is the coupling configuration matrix of the complex network, in which cij > 0 if there is a link from node j to node i (i 6= j), and cij = 0 (i 6= j) otherwise; and the diagonal elements of C are given by cii = − j=1,j6=i cij , i = 1, 2, . . . , N. Driven by network (1), the response network with an adaptive controller is given by
PN
y˙i (t ) = f (yi (t )) +
N X
dij Ayj (t ) + ui (t ),
i = 1, 2, . . . , N
(2)
j =1
where yi (t ) = (yi1 (t ), . . . , yin (t ))T ∈ Rn is the state vector of the ith node in the response network, D = (dij )N ×N is the coupling configuration matrix of the response network, and following ui is the controller designed for node i: ui =
N X
bij Ayj − gi ei ,
g˙i = ki |ei |2 , b˙ ij = −eTi Ayj ,
(3)
j =1
where ei = yi − xi is the synchronization error for node i between drive network and response network. Based on the drive–response network system (1)–(3), we will study a more general networks model by simultaneously considering network coupling delays and circumstance noises. We will consider the following drive–response network system:
" dxi (t ) =
f (xi (t )) +
N X
# cij Axj (t − τ ) dt + σ (t , xi (t ), X (t − τ ))dw(t ),
i = 1, 2, . . . , N
(4)
j =1
" dyi (t ) =
f (yi (t )) +
N X
# dij Ayj (t − τ ) + ui (t ) dt + σ (t , yi (t ), Y (t − τ ))dw(t ),
i = 1, 2, . . . , N
(5)
j =1
where xi , yi , f , A, C , D have been defined in (1) and (2), X (t ) = (x1 (t ), x2 (t ), . . . , xN (t )), Y (t ) = (y1 (t ), y2 (t ), . . . , yN (t )), τ denotes the time delay of the networks coupling, w(t ) is one-dimensional Brown motion, σ (t , xi , X (t − τ )) ∈ Rn is the noise intensity function, and following ui is the adaptive controller designed for node i: ui =
N X
bij Ayj (t − τ ) − gi ei (t ),
g˙i = ki |ei |2 , b˙ ij = −eTi (t )Ayj (t − τ )
(6)
j =1
In this paper, the configuration matrices C and D of network (4) and (5) are not assumed to be symmetric and/or irreducible. Moreover, the inner connecting matrix A is not necessary symmetric or diagonal.
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The following two assumptions are needed for the derivation of our main results: (H1) f (x) satisfies the Lipschitz condition, i.e. there exists constant L > 0 such that
|f (x) − f (y)| ≤ L|x − y|,
∀x, y ∈ Rn . (H2) The noise intensity function σ (t , x, V ) satisfies the Lipschitz condition with respect to x, V , i.e. ∀x, y ∈ Rn , V = {V1 , . . . , VN }, W = {W1 , . . . , WN } ∈ Rn×N there exists constants λ > 0, µ > 0 such that N X |σ (t , x, V ) − σ (t , y, W )|2 ≤ λ|x − y|2 + µ |Vi − Wi |2 . i =1
Moreover, σ (t , 0, 0) ≡ 0. Remark 1. If we take σ (t , x, V ) = l0 x + holds.
PN
i=1 li Vi ,
where li , i = 0, 1, . . . , N are real numbers, then it is easy to see that (H2)
Consider a general n-dimensional stochastic differential delay equation dx(t ) = Φ (t , x(t ), x(t − τ ))dt + Ψ (t , x(t ), x(t − τ ))dw(t )
(7)
on t ≥ 0 with initial condition ξ ∈ CF0 ([−τ , 0]; R ), where w(t ) = (w1 (t ), . . . , wn (t )) is an m-dimensional Brownian motion defined on the complete probability space (Ω , F , {Ft }t ≥0 , P) with filtration {Ft }t ≥0 satisfying the usual conditions (i.e., right continuous and F0 containing all P-null sets). By assuming that Φ and Ψ both satisfy the local Lipschitz condition and the linear growth condition, Mao [22] established a LaSalle-type theorem for stochastic differential delay equation (7) as follows: n
b
T
Lemma 1. Assume that there are functions V ∈ C 1,2 (R+ × Rn ; R+ ), γ ∈ L1 (R+ ; R+ ), and ω1 , ω2 ∈ C (Rn ; R+ ) such that 1 trace[Ψ T (t , x, y)Vxx (t , x)Ψ (t , x, y)] 2 (t , x, y) ∈ Rn × Rn × R+
LV (t , x, y) := Vt (t , x) + Vx (t , x)Φ (t , x, y) +
≤ γ (t ) − ω1 (x) + ω2 (y), ω1 (x) ≥ ω2 (x), x ∈ Rn , and lim
inf V (t , x) = ∞.
|x|→∞ 0≤t <∞
Then, for every ξ ∈ CFb 0 ([−τ , 0]; Rn ), lim [ω1 (x(t ; ξ )) − ω2 (x(t ; ξ ))] = 0 a.s.
t →∞
Moreover, if Ker (ω1 − ω2 ) = {0}, then for every ξ ∈ CFb 0 ([−τ , 0]; Rn ), lim x(t ; ξ ) = 0 a.s.
t →∞
The main purpose of this paper is to prove that the networks (4) and (5) can be synchronized using adaptive controller (6). In the next section, it can be seen that Lemma 1 plays a key role in the proof of the main results about the synchronization of two networks. 3. Synchronization criteria In this section, outer synchronization between networks (4) and (5) is investigated, and the main results are given in the following theorem and corollaries. Theorem 2. Suppose that assumptions (H1) and (H2) hold. Under nonidentical topological configurations, i.e. C 6= D, the drive network (4) and the response network (5) can be almost surely synchronized using the adaptive controller (6). Proof. The synchronization error between (4) and (5) can be written as:
" dei (t ) =
f (yi (t )) − f (xi (t )) +
N X
# (dij Ayj (t − τ ) − cij Axj (t − τ )) + ui (t ) dt
j =1
+ h(t , ei (t ), E (t − τ ))dw(t ),
i = 1, 2, . . . , N
(8)
where h(t , ei (t ), E (t − τ )) = σ (t , yi (t ), Y (t − τ )) − σ (t , xi (t ), X (t − τ )). It can be observed that the stability of the zero solution of synchronization error (8) implies the out synchronization of networks (4) and (5). Consider the following Lyapunov functional: V (t , e(t )) =
N X j =1
eTi (t )ei (t ) +
N X N N X X 1 (dij + bij − cij )2 + (gi − g¯ )2 i=1 j=1
j =1
ki
(9)
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where g¯ is a sufficiently large positive constant to be determined. Employing Itô’s formula, one has dV (t ) = LV (t )dt + VeT (t )H (t , e(t ), E (t − τ ))dw(t )
(10)
where H (t , e(t ), E (t − τ )) = (h(t , e1 (t ), E (t − τ )) , . . . , h(t , eN (t ), E (t − τ )) ) , e(t ) = ( (t ), T
LV (t ) =
N X
" 2eTi (t ) f (yi (t )) − f (xi (t )) +
i =1
+
T T
eT1
eT2
(t ), . . . ,
eTN
(t )) and T
N X (dij Ayj (t − τ ) − cij Axj (t − τ )) j =1
N X
# bij Ayj (t − τ ) − gi ei (t ) − 2
j =1
N X N X (dij + bij − cij )eTi (t )Ayj (t − τ ) i=1 j=1
N
+2
X (gi − g¯ )|ei (t )|2 + trace(H T (t , e(t ), E (t − τ ))H (t , e(t ), E (t − τ ))) i =1
≤
N X
2eTi (t )[f (yi (t )) − f (xi (t ))] +
i =1
cij eTi (t )Aej (t − τ )
i =1 j =1 N
N
−2
N X N X
X
g¯ |ei (t )|2 + λ
X
eTi (t )ei (t ) + µN
i =1
i =1
N X
eTi (t − τ )ei (t − τ ).
(11)
i=1
Noting that 2eTi (t )Aej (t − τ ) ≤ eTi (t )AAT ei (t ) + eTi (t − τ )ei (t − τ ),
(12)
one can obtain that
LV (t ) ≤ 2
N X
L+
i =1
1 2
λ − g¯ |ei (t )|2 + eT (t )(C ⊗ AAT )e(t ) + eT (t − τ )(C ⊗ In + µNInN )e(t − τ )
= −e (t )Q1 e(t ) + eT (t − τ )Q2 e(t − τ ) , −ω1 (e(t )) + ω2 (e(t − τ )) T
(13)
where Q1 = (P + P T )/2, P = (2g¯ − 2L − λ)InN − C ⊗ AAT , Q2 = C ⊗ In + µNInN . Referring to the above calculations, it can be observed that for a sufficiently large positive constant g¯ , the following inequality holds:
ω1 (e) > ω2 (e),
∀e 6= 0.
(14)
inf V (t , e) = ∞.
(15)
Moreover, lim
|e|→∞ 0≤t <∞
From Lemma 1, we have lim e(t ) → 0 , a.s.
t →∞
(16)
Hence the almost sure synchronization between the drive network (4) and the response network (5) can be realized using the adaptive controller (6). This completes the proof of Theorem 2. Remark 2. Note that when kek → ∞, one has V (e) → ∞. Then, we can conclude that the Lyapunov function V (e) is radically unbounded. Hence, according to the Lyapunov stability theory, the synchronization between networks (4) and (5) is globally realized. When the drive–response network system tends to a synchronous state, it can be seen that e → 0, and then g → g ∗ and B → B∗ , where g ∗ = (g1∗ , g2∗ , . . . , gN∗ ) and B∗ = (b∗ij )N ×N are constant vector and matrix. Such observation will be further illustrated in the following examples. Remark 3. The constants ki (i = 1, 2, . . . 10) can be properly chosen to adjust the synchronization speed of the corresponding ith node. Generally, a suitably large adaptive gain ki will lead to fast synchronization, while a small ki will need longer time to achieve synchronization of the drive–response system. Remark 4. We assume that the two complex networks are functioning in the same noise circumstance. However, the noise intensity of a specific node is related to the corresponding nodes’ states in the complex network, and has no relation to the another complex network. Hence, we can observe that, when outer synchronization between two complex networks has
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not been achieved, the noise intensity of one complex network is different from that of another one. However, when two complex networks are synchronized, the noise intensities of these two networks become exactly the same. Based on Theorem 2, following corollaries can be directly obtained: Corollary 3. Suppose that assumptions (H1) and (H2) hold. The drive network (4) and response network (5) can achieve almost sure synchronization using the following adaptive controller: ui =
N X
bij Ayj (t − τ ) − gi ei (t ),
(17)
j =1
g˙i = ki |ei |2 ,
(18)
b˙ ij = −eTi (t )Ayj (t − τ ), bij = 0, cij = dij ,
cij 6= dij
(19)
where ei (t ) = yi (t ) − xi (t ) is the synchronization error, ki is any positive constant for i = 1, 2, . . . , N. Corollary 4. Suppose that assumptions (H1) and (H2) hold. If the two complex networks have the same configuration matrices, i.e. C = D, then the driving network (4) and response network (5) can achieve almost sure synchronization using the following adaptive controller: ui = −gi ei (t ),
g˙i = ki |ei |2 ,
(20)
where ei (t ) = yi (t ) − xi (t ) is the synchronization error, ki is any positive constant for i = 1, 2, . . . , N. Remark 5. For networks with nonidentical topological structure, Corollary 3 can be used to simplify the designing of the controller, in which the number of control variables can be greatly reduced by setting bij = 0 for the case of cij = dij . The effectiveness of the method is shown in Example 2. When the networks have the same topological structures, a very simple adaptive controller can be used to synchronize each other from Corollary 4. 4. Numerical simulations In this section, two examples are given to show the effectiveness of the proposed synchronization scheme. The networks are composed of 10 coupled nodes, and each node takes the chaotic Lorenz system as its dynamics. A chaotic Lorenz system can be described by
−a
x˙ i = f (xi ) =
c 0
a −1 0
0 0 −b
!
xi1 xi2 xi3
! +
0 −xi1 xi3 xi1 xi2
!
, Hxi + W (xi ),
(21)
where a, b and c are parameters. When a = 10, b = 8/3 and c = 28, the system has a chaotic attractor. For any two state vectors xi and xj , there exists a positive constant r such that
|W (xj ) − W (xi )| ≤ r |xj − xi |,
(22)
hence the assumption (H1) is satisfied (please refer to Ref. [14] for more details). Example 1. In the first example, the drive network and the response network are assumed to have the same configuration matrices. By Corollary 4, two complex networks can be synchronized using the adaptive controller (20). For the sake of illustration, the drive network is composed of ten Lorenz nodes coupled via following configuration matrix C :
−3 C =
1 0 0 0 0 0 1 0 1
1
−2 1 0 0 0 0 0 0 0
0 1 −2 1 0 0 0 0 0 0
0 0 1 −2 1 0 0 0 0 0
0 0 0 1 −3 1 0 0 0 1
0 0 0 0 1 −2 1 0 0 0
0 0 0 0 0 1 −2 1 0 0
1 0 0 0 0 0 1 −3 1 0
0 0 0 0 0 0 0 1 −2 1
1 0 0 0 1 . 0 0 0 1 −3
(23)
The response network is also composed of ten nodes with the same topological structure as the drive network, i.e. D = C . The inner connection matrix is taken as an identity matrix of dimension 3, i.e. A = I3 . In simulation, it is assumed that the coupling delay of the network is τ = 0.5, and the noise intensity function P10 is σ (t , xi (t ), X (t − τ )) = 0.1xi (t ) + 0.02 k=1 xk (t − τ ). The initial state of the ith node of the drive network is
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Fig. 1. Trajectories of the synchronization error between networks (4) and (5) with C = D.
(xi1 (s), xi2 (s), xi3 (s)) = (10 + 0.1i, −6 + 0.1i, 3 + 0.1i), s ∈ [−τ , 0], while the initial state of the ith node of the response network is (yi1 (s), yi2 (s), yi3 (s)) = (1 − sin(i), 2 − 0.2 cos(i), −0.1i), s ∈ [−τ , 0]. The adaptive gains are taken as ki = 0.1(i = 1, 2, . . . 10). Fig. 1 shows the trajectories of synchronization errors eik (t )(k = 1, 2, 3; i = 1, 2, . . . 10) between the two complex networks. It can be observed that all of the state errors eik (t )(k = 1, 2, 3; i = 1, 2, . . . 10) tend to zero, which implies that two networks achieve synchronization. Fig. 2 shows the updated feedback strength gi (t )(i = 1, 2, . . . 10), which reaches a certain constant when the two network are synchronized. Fig. 3 plots the total synchronization error between complex networks (4) and (5). Here, the norm of the total synchronization errors is defined by
v u N uX |e(t )| = t [(yi1 (t ) − xi1 (t ))2 + (yi2 (t ) − xi2 (t ))2 + (yi3 (t ) − xi3 (t ))2 ].
(24)
i =1
Example 2. The second example considers the case that the drive network and the response network have different configuration matrices. Let the configuration matrix C of the drive network be (23), and the configuration matrix for the response network is given as follows:
−2 D=
0 0 0 1 0 0 1 0 0
0
−3 1 0 0 0 1 0 0 1
0 1 −1 0 0 0 0 0 0 0
0 0 0 −2 0 1 1 0 0 0
1 0 0 0 −3 0 1 0 0 1
0 0 0 1 0 −2 0 1 0 0
0 1 0 1 1 0 −4 1 0 0
1 0 0 0 0 1 1 −3 0 0
0 0 0 0 0 0 0 0 −1 1
0 1 0 0 1 . 0 0 0 1 −3
(25)
From Corollary 3, the two complex dynamical networks can be synchronized using the adaptive controller (17)–(19). P10 Similarly, let τ = 0.5, σ (t , x(t )) = x(t ), σ (t , xi (t ), X (t − τ )) = 0.1xi (t ) + 0.02 k=1 xk (t − τ ), take the initial conditions as (xi1 (s), xi2 (s), xi3 (s)) = (10 + 0.1i, −6 + 0.1i, 3 + 0.1i), s ∈ [−τ , 0], (yi1 (s), yi2 (s), yi3 (s)) = (1 − sin(i), 2 − 0.2 cos(i), −0.1i), s ∈ [−τ , 0]. Some simulation results are given in Figs. 4–7. Fig. 4 plots the trajectories of synchronization errors eik (t )(k = 1, 2, 3; i = 1, 2, . . . 10) between the two complex networks, which shows the realization of synchronization between the two networks. Figs. 5 and 6 show the updated feedback strength gi (i = 1, 2, . . . 10) and adaptive parameter bij (6=0; i, j = 1, 2, . . . 10). Fig. 7 plots the total synchronization error between complex network (4) and (5). Remark 6. By the comparison between Examples 1 and 2, we can observe that the complex networks with different topological structures need more control variables to synchronize with each other. From the simulation results, it can also be obtained that the drive–response network system with different topological structures needs a little more time to reach synchronous state compared with networks with identical structures. The exact impact of time delay on the synchronization speed has not been studied in this paper. In the near future, we will try to find out the optimal time-delay value which maximizes the speed of synchronization by referring to the results in Ref. [23].
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Fig. 2. Feedback strength gi (i = 1, 2, . . . , 10) of adaptive controller (20) for networks (4) and (5) with C = D.
Fig. 3. Total synchronization error between networks (4) and (5) with C = D.
Fig. 4. Trajectories of the synchronization error between networks (4) and (5) with C 6= D.
5. Conclusions In this paper, the adaptive synchronization problem for two complex network with noise perturbation has been investigated. By designing a suitable adaptive controller, the almost sure synchronization is realized referring to the
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Fig. 5. Feedback strength gi (i = 1, 2, . . . , 10) of adaptive controller (17)–(19) for networks (4) and (5) with C 6= D.
Fig. 6. Adaptive parameters bij (6=0; i, j = 1, 2, . . . 10) of controller (17)–(19) for networks (4) and (5) with C 6= D.
Fig. 7. Total synchronization error between network (4) and (5) with C 6= D.
stochastic LaSalle invariance principle. We simultaneously consider the effects of coupling delays of complex networks and circumstance noises on the outer synchronization between two networks. Our results are more general and more applicable comparing with the results in Ref. [14].
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Some simulations are also provided to show the effectiveness of the proposed synchronization method. From the simulations, we can observe that the drive–response network system can be synchronized using the developed adaptive controllers. More simulation results show that structure difference of the complex networks may affect the synchronization speed of the networks. However, such an effect is very limited since more control variables can be introduced when the networks are more different. Similarly, coupling delays of the complex networks can also affect the synchronization speed. However, simulation results show that this effect is not very significant either when the time delays vary in a certain range. Acknowledgements The authors would like to thank the editor and the reviewers for their valuable comments and suggestions, which have considerably improved the presentation of the paper. This work was jointly supported by the National Natural Science Foundation of China under Grant Nos. 10801032 and 60874088, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20070286003, the Natural Science Foundation of Jiangsu Province of China under Grant BK2009271. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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Physica A journal homepage: www.elsevier.com/locate/physa
Outer synchronization between two nonidentical networks with circumstance noise Guanjun Wang ∗ , Jinde Cao, Jianquan Lu Department of Mathematics, Southeast University, Nanjing 210096, China
article
info
Article history: Received 17 July 2009 Received in revised form 29 November 2009 Available online 22 December 2009 Keywords: Complex network Outer synchronization Adaptive controller Stochastic LaSalle theorem Lyapunov function
abstract This paper regards the outer synchronization between two delay-coupled complex dynamical networks with nonidentical topological structures and a noise perturbation. Considering one network as the drive network and the other one as the response network, the drive–response system achieves synchronous states through a suitably designed adaptive controller. The stochastic LaSalle invariance principle is employed to theoretically prove the almost sure synchronization between two networks. Finally, two numerical examples are examined in order to illustrate the proposed synchronization scheme. © 2009 Elsevier B.V. All rights reserved.
1. Introduction A complex network is a large set of interconnected nodes, in which a node is a fundamental unit with specific contents. Many real world systems can be described by complex networks, such as the World Wide Web, telephone call graphs, neural networks, scientific citation web, etc. Random graph theory is the traditional mathematical method in studying complex networks which was proposed by Erdös and Rényi [1] about fifty years ago. Recently, the development of modern computer provided a powerful computational tool to deal with large scale data from networks, which led to the discovery of the ‘‘small-world’’ and ‘‘scale-free’’ properties of complex networks [2,3]. Consequently, complex networks become a focus point of research which has attracted increasing attention from various fields of science and engineering. In particular, special attention has been focused on the synchronization problem for dynamical complex networks, in which each node is regarded as a dynamical element. Much work has been done for the synchronization of complex networks in the literature, see e.g. Refs. [4–11]. Wang and Chen [4,5] studied the synchronization problem for two specific complex networks with small-world and scale-free properties, respectively. In Ref. [6], sufficient conditions are provided to synchronize a class of delay-coupled complex dynamical networks by using the Lyapunov method. In Ref. [7], Zhou, Lu and Lü proposed an adaptive synchronization scheme for uncertain dynamical networks. Lu and Ho investigated the local and global synchronization for complex dynamical networks without assuming the symmetry and irreducibility of the coupling configuration matrices [8]. A review on synchronization of complex networks can be found in Ref. [11]. The synchronization within one network mentioned in the above references is named ‘‘inner synchronization’’ [12], which is concerned with the synchronization among the nodes within a network. Different from the ‘‘inner synchronization’’, the synchronization between two or more complex networks is called ‘‘outer synchronization’’, which was firstly studied by Li et al. [12]. And then outer synchronization was investigated by several other researchers in Refs. [13–16], etc. In Ref. [12], Li, Sun and Kurths studied the synchronization problem for two complex networks with identical topological
∗
Corresponding author. E-mail addresses: [email protected] (G. Wang), [email protected] (J. Cao), [email protected] (J. Lu).
0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.12.014
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structures by using an open-plus-closed-loop controller. Later on, the outer synchronization for two coupled complex networks was extended to the discrete time case [13]. In Ref. [14], adaptive controllers were designed to synchronize two complex networks with identical or nonidentical topological structures. In Ref. [16], Wu, Zheng and Zhou synchronized two different complex dynamical networks by constructing a nonlinear controller. For large scale networks, time delays is unavoidable and should be taken into account due to the finite information transmission and processing speeds among the network nodes. Time-delayed coupling extensively exists in many biological and physical systems such as gene regulatory networks, communication networks, and electrical power grids etc. It has also been discovered that time delays frequently have great influence on the behavior of dynamical systems [17,18]. Hence, it is very important to study the synchronization of complex networks with delay coupling. Noise is another factor which may affect the behavior of dynamical systems [19]. During the evolving process of complex dynamical networks, noises from circumstance are often inevitable, which may destroy the synchronization of complex network. Goldobina and Pikovsky [20, 21] even proved that self-sustained oscillators can be synchronized by common noise. Motivated by above discussions, we will study the outer synchronization problem for two delay-coupled complex dynamical networks with the consideration of a circumstance noise. Adaptive controllers are designed for the global outer synchronization of the drive–response network system, in which the networks have identical and nonidentical topological structures. Numerical simulations are provided to show the effectiveness of the proposed synchronization scheme. 2. Network modeling and preliminaries In Ref. [14], Tang, Chen and Lu studied the adaptive synchronization problem between two coupled complex networks. In Ref. [14], the drive complex network is described by x˙i (t ) = f (xi (t )) +
N X
cij Axj (t ),
i = 1, 2, . . . , N
(1)
j =1
where xi (t ) = (xi1 (t ), . . . , xin (t ))T ∈ Rn is the state vector of the ith node in the drive network; f : R × Rn → Rn is a nonlinear vector function, and x˙i (t ) = f (xi (t )) represents the dynamics of a single node; A is the inner connecting matrix between two connected nodes; C = (cij )N ×N is the coupling configuration matrix of the complex network, in which cij > 0 if there is a link from node j to node i (i 6= j), and cij = 0 (i 6= j) otherwise; and the diagonal elements of C are given by cii = − j=1,j6=i cij , i = 1, 2, . . . , N. Driven by network (1), the response network with an adaptive controller is given by
PN
y˙i (t ) = f (yi (t )) +
N X
dij Ayj (t ) + ui (t ),
i = 1, 2, . . . , N
(2)
j =1
where yi (t ) = (yi1 (t ), . . . , yin (t ))T ∈ Rn is the state vector of the ith node in the response network, D = (dij )N ×N is the coupling configuration matrix of the response network, and following ui is the controller designed for node i: ui =
N X
bij Ayj − gi ei ,
g˙i = ki |ei |2 , b˙ ij = −eTi Ayj ,
(3)
j =1
where ei = yi − xi is the synchronization error for node i between drive network and response network. Based on the drive–response network system (1)–(3), we will study a more general networks model by simultaneously considering network coupling delays and circumstance noises. We will consider the following drive–response network system:
" dxi (t ) =
f (xi (t )) +
N X
# cij Axj (t − τ ) dt + σ (t , xi (t ), X (t − τ ))dw(t ),
i = 1, 2, . . . , N
(4)
j =1
" dyi (t ) =
f (yi (t )) +
N X
# dij Ayj (t − τ ) + ui (t ) dt + σ (t , yi (t ), Y (t − τ ))dw(t ),
i = 1, 2, . . . , N
(5)
j =1
where xi , yi , f , A, C , D have been defined in (1) and (2), X (t ) = (x1 (t ), x2 (t ), . . . , xN (t )), Y (t ) = (y1 (t ), y2 (t ), . . . , yN (t )), τ denotes the time delay of the networks coupling, w(t ) is one-dimensional Brown motion, σ (t , xi , X (t − τ )) ∈ Rn is the noise intensity function, and following ui is the adaptive controller designed for node i: ui =
N X
bij Ayj (t − τ ) − gi ei (t ),
g˙i = ki |ei |2 , b˙ ij = −eTi (t )Ayj (t − τ )
(6)
j =1
In this paper, the configuration matrices C and D of network (4) and (5) are not assumed to be symmetric and/or irreducible. Moreover, the inner connecting matrix A is not necessary symmetric or diagonal.
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The following two assumptions are needed for the derivation of our main results: (H1) f (x) satisfies the Lipschitz condition, i.e. there exists constant L > 0 such that
|f (x) − f (y)| ≤ L|x − y|,
∀x, y ∈ Rn . (H2) The noise intensity function σ (t , x, V ) satisfies the Lipschitz condition with respect to x, V , i.e. ∀x, y ∈ Rn , V = {V1 , . . . , VN }, W = {W1 , . . . , WN } ∈ Rn×N there exists constants λ > 0, µ > 0 such that N X |σ (t , x, V ) − σ (t , y, W )|2 ≤ λ|x − y|2 + µ |Vi − Wi |2 . i =1
Moreover, σ (t , 0, 0) ≡ 0. Remark 1. If we take σ (t , x, V ) = l0 x + holds.
PN
i=1 li Vi ,
where li , i = 0, 1, . . . , N are real numbers, then it is easy to see that (H2)
Consider a general n-dimensional stochastic differential delay equation dx(t ) = Φ (t , x(t ), x(t − τ ))dt + Ψ (t , x(t ), x(t − τ ))dw(t )
(7)
on t ≥ 0 with initial condition ξ ∈ CF0 ([−τ , 0]; R ), where w(t ) = (w1 (t ), . . . , wn (t )) is an m-dimensional Brownian motion defined on the complete probability space (Ω , F , {Ft }t ≥0 , P) with filtration {Ft }t ≥0 satisfying the usual conditions (i.e., right continuous and F0 containing all P-null sets). By assuming that Φ and Ψ both satisfy the local Lipschitz condition and the linear growth condition, Mao [22] established a LaSalle-type theorem for stochastic differential delay equation (7) as follows: n
b
T
Lemma 1. Assume that there are functions V ∈ C 1,2 (R+ × Rn ; R+ ), γ ∈ L1 (R+ ; R+ ), and ω1 , ω2 ∈ C (Rn ; R+ ) such that 1 trace[Ψ T (t , x, y)Vxx (t , x)Ψ (t , x, y)] 2 (t , x, y) ∈ Rn × Rn × R+
LV (t , x, y) := Vt (t , x) + Vx (t , x)Φ (t , x, y) +
≤ γ (t ) − ω1 (x) + ω2 (y), ω1 (x) ≥ ω2 (x), x ∈ Rn , and lim
inf V (t , x) = ∞.
|x|→∞ 0≤t <∞
Then, for every ξ ∈ CFb 0 ([−τ , 0]; Rn ), lim [ω1 (x(t ; ξ )) − ω2 (x(t ; ξ ))] = 0 a.s.
t →∞
Moreover, if Ker (ω1 − ω2 ) = {0}, then for every ξ ∈ CFb 0 ([−τ , 0]; Rn ), lim x(t ; ξ ) = 0 a.s.
t →∞
The main purpose of this paper is to prove that the networks (4) and (5) can be synchronized using adaptive controller (6). In the next section, it can be seen that Lemma 1 plays a key role in the proof of the main results about the synchronization of two networks. 3. Synchronization criteria In this section, outer synchronization between networks (4) and (5) is investigated, and the main results are given in the following theorem and corollaries. Theorem 2. Suppose that assumptions (H1) and (H2) hold. Under nonidentical topological configurations, i.e. C 6= D, the drive network (4) and the response network (5) can be almost surely synchronized using the adaptive controller (6). Proof. The synchronization error between (4) and (5) can be written as:
" dei (t ) =
f (yi (t )) − f (xi (t )) +
N X
# (dij Ayj (t − τ ) − cij Axj (t − τ )) + ui (t ) dt
j =1
+ h(t , ei (t ), E (t − τ ))dw(t ),
i = 1, 2, . . . , N
(8)
where h(t , ei (t ), E (t − τ )) = σ (t , yi (t ), Y (t − τ )) − σ (t , xi (t ), X (t − τ )). It can be observed that the stability of the zero solution of synchronization error (8) implies the out synchronization of networks (4) and (5). Consider the following Lyapunov functional: V (t , e(t )) =
N X j =1
eTi (t )ei (t ) +
N X N N X X 1 (dij + bij − cij )2 + (gi − g¯ )2 i=1 j=1
j =1
ki
(9)
G. Wang et al. / Physica A 389 (2010) 1480–1488
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where g¯ is a sufficiently large positive constant to be determined. Employing Itô’s formula, one has dV (t ) = LV (t )dt + VeT (t )H (t , e(t ), E (t − τ ))dw(t )
(10)
where H (t , e(t ), E (t − τ )) = (h(t , e1 (t ), E (t − τ )) , . . . , h(t , eN (t ), E (t − τ )) ) , e(t ) = ( (t ), T
LV (t ) =
N X
" 2eTi (t ) f (yi (t )) − f (xi (t )) +
i =1
+
T T
eT1
eT2
(t ), . . . ,
eTN
(t )) and T
N X (dij Ayj (t − τ ) − cij Axj (t − τ )) j =1
N X
# bij Ayj (t − τ ) − gi ei (t ) − 2
j =1
N X N X (dij + bij − cij )eTi (t )Ayj (t − τ ) i=1 j=1
N
+2
X (gi − g¯ )|ei (t )|2 + trace(H T (t , e(t ), E (t − τ ))H (t , e(t ), E (t − τ ))) i =1
≤
N X
2eTi (t )[f (yi (t )) − f (xi (t ))] +
i =1
cij eTi (t )Aej (t − τ )
i =1 j =1 N
N
−2
N X N X
X
g¯ |ei (t )|2 + λ
X
eTi (t )ei (t ) + µN
i =1
i =1
N X
eTi (t − τ )ei (t − τ ).
(11)
i=1
Noting that 2eTi (t )Aej (t − τ ) ≤ eTi (t )AAT ei (t ) + eTi (t − τ )ei (t − τ ),
(12)
one can obtain that
LV (t ) ≤ 2
N X
L+
i =1
1 2
λ − g¯ |ei (t )|2 + eT (t )(C ⊗ AAT )e(t ) + eT (t − τ )(C ⊗ In + µNInN )e(t − τ )
= −e (t )Q1 e(t ) + eT (t − τ )Q2 e(t − τ ) , −ω1 (e(t )) + ω2 (e(t − τ )) T
(13)
where Q1 = (P + P T )/2, P = (2g¯ − 2L − λ)InN − C ⊗ AAT , Q2 = C ⊗ In + µNInN . Referring to the above calculations, it can be observed that for a sufficiently large positive constant g¯ , the following inequality holds:
ω1 (e) > ω2 (e),
∀e 6= 0.
(14)
inf V (t , e) = ∞.
(15)
Moreover, lim
|e|→∞ 0≤t <∞
From Lemma 1, we have lim e(t ) → 0 , a.s.
t →∞
(16)
Hence the almost sure synchronization between the drive network (4) and the response network (5) can be realized using the adaptive controller (6). This completes the proof of Theorem 2. Remark 2. Note that when kek → ∞, one has V (e) → ∞. Then, we can conclude that the Lyapunov function V (e) is radically unbounded. Hence, according to the Lyapunov stability theory, the synchronization between networks (4) and (5) is globally realized. When the drive–response network system tends to a synchronous state, it can be seen that e → 0, and then g → g ∗ and B → B∗ , where g ∗ = (g1∗ , g2∗ , . . . , gN∗ ) and B∗ = (b∗ij )N ×N are constant vector and matrix. Such observation will be further illustrated in the following examples. Remark 3. The constants ki (i = 1, 2, . . . 10) can be properly chosen to adjust the synchronization speed of the corresponding ith node. Generally, a suitably large adaptive gain ki will lead to fast synchronization, while a small ki will need longer time to achieve synchronization of the drive–response system. Remark 4. We assume that the two complex networks are functioning in the same noise circumstance. However, the noise intensity of a specific node is related to the corresponding nodes’ states in the complex network, and has no relation to the another complex network. Hence, we can observe that, when outer synchronization between two complex networks has
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not been achieved, the noise intensity of one complex network is different from that of another one. However, when two complex networks are synchronized, the noise intensities of these two networks become exactly the same. Based on Theorem 2, following corollaries can be directly obtained: Corollary 3. Suppose that assumptions (H1) and (H2) hold. The drive network (4) and response network (5) can achieve almost sure synchronization using the following adaptive controller: ui =
N X
bij Ayj (t − τ ) − gi ei (t ),
(17)
j =1
g˙i = ki |ei |2 ,
(18)
b˙ ij = −eTi (t )Ayj (t − τ ), bij = 0, cij = dij ,
cij 6= dij
(19)
where ei (t ) = yi (t ) − xi (t ) is the synchronization error, ki is any positive constant for i = 1, 2, . . . , N. Corollary 4. Suppose that assumptions (H1) and (H2) hold. If the two complex networks have the same configuration matrices, i.e. C = D, then the driving network (4) and response network (5) can achieve almost sure synchronization using the following adaptive controller: ui = −gi ei (t ),
g˙i = ki |ei |2 ,
(20)
where ei (t ) = yi (t ) − xi (t ) is the synchronization error, ki is any positive constant for i = 1, 2, . . . , N. Remark 5. For networks with nonidentical topological structure, Corollary 3 can be used to simplify the designing of the controller, in which the number of control variables can be greatly reduced by setting bij = 0 for the case of cij = dij . The effectiveness of the method is shown in Example 2. When the networks have the same topological structures, a very simple adaptive controller can be used to synchronize each other from Corollary 4. 4. Numerical simulations In this section, two examples are given to show the effectiveness of the proposed synchronization scheme. The networks are composed of 10 coupled nodes, and each node takes the chaotic Lorenz system as its dynamics. A chaotic Lorenz system can be described by
−a
x˙ i = f (xi ) =
c 0
a −1 0
0 0 −b
!
xi1 xi2 xi3
! +
0 −xi1 xi3 xi1 xi2
!
, Hxi + W (xi ),
(21)
where a, b and c are parameters. When a = 10, b = 8/3 and c = 28, the system has a chaotic attractor. For any two state vectors xi and xj , there exists a positive constant r such that
|W (xj ) − W (xi )| ≤ r |xj − xi |,
(22)
hence the assumption (H1) is satisfied (please refer to Ref. [14] for more details). Example 1. In the first example, the drive network and the response network are assumed to have the same configuration matrices. By Corollary 4, two complex networks can be synchronized using the adaptive controller (20). For the sake of illustration, the drive network is composed of ten Lorenz nodes coupled via following configuration matrix C :
−3 C =
1 0 0 0 0 0 1 0 1
1
−2 1 0 0 0 0 0 0 0
0 1 −2 1 0 0 0 0 0 0
0 0 1 −2 1 0 0 0 0 0
0 0 0 1 −3 1 0 0 0 1
0 0 0 0 1 −2 1 0 0 0
0 0 0 0 0 1 −2 1 0 0
1 0 0 0 0 0 1 −3 1 0
0 0 0 0 0 0 0 1 −2 1
1 0 0 0 1 . 0 0 0 1 −3
(23)
The response network is also composed of ten nodes with the same topological structure as the drive network, i.e. D = C . The inner connection matrix is taken as an identity matrix of dimension 3, i.e. A = I3 . In simulation, it is assumed that the coupling delay of the network is τ = 0.5, and the noise intensity function P10 is σ (t , xi (t ), X (t − τ )) = 0.1xi (t ) + 0.02 k=1 xk (t − τ ). The initial state of the ith node of the drive network is
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Fig. 1. Trajectories of the synchronization error between networks (4) and (5) with C = D.
(xi1 (s), xi2 (s), xi3 (s)) = (10 + 0.1i, −6 + 0.1i, 3 + 0.1i), s ∈ [−τ , 0], while the initial state of the ith node of the response network is (yi1 (s), yi2 (s), yi3 (s)) = (1 − sin(i), 2 − 0.2 cos(i), −0.1i), s ∈ [−τ , 0]. The adaptive gains are taken as ki = 0.1(i = 1, 2, . . . 10). Fig. 1 shows the trajectories of synchronization errors eik (t )(k = 1, 2, 3; i = 1, 2, . . . 10) between the two complex networks. It can be observed that all of the state errors eik (t )(k = 1, 2, 3; i = 1, 2, . . . 10) tend to zero, which implies that two networks achieve synchronization. Fig. 2 shows the updated feedback strength gi (t )(i = 1, 2, . . . 10), which reaches a certain constant when the two network are synchronized. Fig. 3 plots the total synchronization error between complex networks (4) and (5). Here, the norm of the total synchronization errors is defined by
v u N uX |e(t )| = t [(yi1 (t ) − xi1 (t ))2 + (yi2 (t ) − xi2 (t ))2 + (yi3 (t ) − xi3 (t ))2 ].
(24)
i =1
Example 2. The second example considers the case that the drive network and the response network have different configuration matrices. Let the configuration matrix C of the drive network be (23), and the configuration matrix for the response network is given as follows:
−2 D=
0 0 0 1 0 0 1 0 0
0
−3 1 0 0 0 1 0 0 1
0 1 −1 0 0 0 0 0 0 0
0 0 0 −2 0 1 1 0 0 0
1 0 0 0 −3 0 1 0 0 1
0 0 0 1 0 −2 0 1 0 0
0 1 0 1 1 0 −4 1 0 0
1 0 0 0 0 1 1 −3 0 0
0 0 0 0 0 0 0 0 −1 1
0 1 0 0 1 . 0 0 0 1 −3
(25)
From Corollary 3, the two complex dynamical networks can be synchronized using the adaptive controller (17)–(19). P10 Similarly, let τ = 0.5, σ (t , x(t )) = x(t ), σ (t , xi (t ), X (t − τ )) = 0.1xi (t ) + 0.02 k=1 xk (t − τ ), take the initial conditions as (xi1 (s), xi2 (s), xi3 (s)) = (10 + 0.1i, −6 + 0.1i, 3 + 0.1i), s ∈ [−τ , 0], (yi1 (s), yi2 (s), yi3 (s)) = (1 − sin(i), 2 − 0.2 cos(i), −0.1i), s ∈ [−τ , 0]. Some simulation results are given in Figs. 4–7. Fig. 4 plots the trajectories of synchronization errors eik (t )(k = 1, 2, 3; i = 1, 2, . . . 10) between the two complex networks, which shows the realization of synchronization between the two networks. Figs. 5 and 6 show the updated feedback strength gi (i = 1, 2, . . . 10) and adaptive parameter bij (6=0; i, j = 1, 2, . . . 10). Fig. 7 plots the total synchronization error between complex network (4) and (5). Remark 6. By the comparison between Examples 1 and 2, we can observe that the complex networks with different topological structures need more control variables to synchronize with each other. From the simulation results, it can also be obtained that the drive–response network system with different topological structures needs a little more time to reach synchronous state compared with networks with identical structures. The exact impact of time delay on the synchronization speed has not been studied in this paper. In the near future, we will try to find out the optimal time-delay value which maximizes the speed of synchronization by referring to the results in Ref. [23].
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Fig. 2. Feedback strength gi (i = 1, 2, . . . , 10) of adaptive controller (20) for networks (4) and (5) with C = D.
Fig. 3. Total synchronization error between networks (4) and (5) with C = D.
Fig. 4. Trajectories of the synchronization error between networks (4) and (5) with C 6= D.
5. Conclusions In this paper, the adaptive synchronization problem for two complex network with noise perturbation has been investigated. By designing a suitable adaptive controller, the almost sure synchronization is realized referring to the
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Fig. 5. Feedback strength gi (i = 1, 2, . . . , 10) of adaptive controller (17)–(19) for networks (4) and (5) with C 6= D.
Fig. 6. Adaptive parameters bij (6=0; i, j = 1, 2, . . . 10) of controller (17)–(19) for networks (4) and (5) with C 6= D.
Fig. 7. Total synchronization error between network (4) and (5) with C 6= D.
stochastic LaSalle invariance principle. We simultaneously consider the effects of coupling delays of complex networks and circumstance noises on the outer synchronization between two networks. Our results are more general and more applicable comparing with the results in Ref. [14].
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Some simulations are also provided to show the effectiveness of the proposed synchronization method. From the simulations, we can observe that the drive–response network system can be synchronized using the developed adaptive controllers. More simulation results show that structure difference of the complex networks may affect the synchronization speed of the networks. However, such an effect is very limited since more control variables can be introduced when the networks are more different. Similarly, coupling delays of the complex networks can also affect the synchronization speed. However, simulation results show that this effect is not very significant either when the time delays vary in a certain range. Acknowledgements The authors would like to thank the editor and the reviewers for their valuable comments and suggestions, which have considerably improved the presentation of the paper. This work was jointly supported by the National Natural Science Foundation of China under Grant Nos. 10801032 and 60874088, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20070286003, the Natural Science Foundation of Jiangsu Province of China under Grant BK2009271. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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