Cluster projective synchronization between community networks with nonidentical nodes

Physica A 391 (2012) 6190–6198

Contents lists available at SciVerse ScienceDirect

Physica A journal homepage: www.elsevier.com/locate/physa

Cluster projective synchronization between community networks with nonidentical nodes Zhaoyan Wu a , Xinchu Fu b,∗ a

College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China

b

Department of Mathematics, Shanghai University, Shanghai 200444, China

article

info

Article history: Received 17 October 2011 Available online 14 July 2012 Keywords: Network Community Cluster Projective synchronization Control

abstract In this paper, cluster projective synchronization between community networks with nonidentical nodes is investigated. Outer synchronization between two identical or nonidentical complex networks has been extensively studied, in which all the nodes synchronized each other in a common manner. However, in real community networks, different communities in networks usually synchronize with each other in a different manner, i.e., achieving cluster projective synchronization. Based on Lyapunov stability theory, sufficient conditions for achieving cluster projective synchronization are derived through designing proper controllers. Numerical simulations are provided to verify the correctness and effectiveness of the derived theoretical results. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Complex networks have been used to model real complex systems, in which the nodes denote the individuals and the connections between nodes denote the interactions between individuals. To describe the real world better, lots of network models are introduced, e.g., weighted networks [1,2], directed networks [3,4], hierarchical networks [5,6], community networks [1,7,8], and so on. Synchronization as an important and interesting collective behavior of complex networks has been extensively studied. And many kinds of synchronization have been introduced, such as complete synchronization [9–11], projective synchronization [12,13], impulsive synchronization [14,15], cluster synchronization [8,16,17], and so on. All the above research focused on the synchronization of one complex network, which can be called inner synchronization. Recently, synchronization between two coupled networks, which is called outer synchronization, has been extensively investigated, and many interesting and valuable results have been obtained [18–23]. In Ref. [18], Li et al. studied synchronization of two unidirectionally coupled networks with identical topology, and derived analytically a criterion for the synchronization. In Ref. [19], Wu et al. investigated the generalized outer synchronization between two completely different complex dynamical networks and derived a sufficient criterion for this generalized outer synchronization based on Barbalat’s lemma. In Ref. [20], Li et al. studied the synchronization between two discrete-time networks with identical connection topologies and derived analytically a sufficient criterion for this outer synchronization. In Ref. [21], Li and Xue studied the outer synchronization of two coupled networks with balanced structure topology using arbitrary coupling strength. In Ref. [22], Wang et al. regarded the outer synchronization between two delay-coupled complex dynamical networks with nonidentical topological structures and a noise perturbation. In Ref. [23], Wang et al. investigated the mixed outer synchronization between two complex networks with the same topological structure and time-varying coupling delay. In the above papers, all the nodes of one network synchronize with the corresponding nodes of another network in a common manner.

∗

Corresponding author. Tel.: +86 21 66134843; fax: +86 21 66133292. E-mail addresses: [email protected] (Z. Wu), [email protected] (X. Fu).

0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.06.070

Z. Wu, X. Fu / Physica A 391 (2012) 6190–6198

6191

However, for community networks, different communities in networks usually synchronize with each other in a different manner, i.e., achieving cluster projective synchronization. To the best of our knowledge, there are few papers having investigated the cluster projective synchronization between two networks. Motivated by the above discussions, this paper investigates cluster projective synchronization between two community networks. The node dynamics of different communities in the same community network are assumed to be nonidentical. For two community networks, their node dynamics can be identical or nonidentical. Both the drive-response community networks with identical and nonidentical node dynamics are considered. Based on Lyapunov stability theory, sufficient conditions for achieving cluster projective synchronization are derived through designing proper controllers. The paper is organized as follows. Section 2 introduces some preliminaries and the network models. Section 3 considers the cluster projective synchronization between the drive-response community networks. Section 4 provides several numerical simulations to verify the correctness and effectiveness of the derived results. Section 5 concludes the paper. Notation. Throughout this paper, for symmetric matrix P, the notation P > 0 (P < 0) means that the matrix P is positive definite (negative definite). IN denotes the N × N identity matrix. Q T denotes the transpose of the matrix Q . 2. Model description and preliminaries Consider a community network consisting of N nodes with l (2 ≤ l < N ) communities, which can be described by x˙ i (t ) = Aφi xi (t ) + fφi (xi (t )) + ε

l  

cij Γ xj (t ),

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

(1)

k=1 j∈Vk

where xi (t ) = (xi1 (t ), xi2 (t ), . . . , xin (t ))T ∈ Rn is the state variable of node i. Aφi is an n × n constant matrix, fφi : Rn → Rn is a nonlinear vector-valued function, and ε > 0 is the coupling strength. Vk (k = 1, 2, . . . , l) denotes the set of nodes in the kth community. φ : {1, 2, . . . , N } → {1, 2, . . . , l}, if node i ∈ Vk then φi = k. The matrix C = (cij )N ×N is the zero-row-sum outer coupling matrix, which denotes the network topology and is defined as, if there is a connection from node i to node j (i ̸= j), then cij ̸= 0, otherwise, cij = 0. Regard the network (1) as the drive network; the response network consisting of N nodes with l communities can be described by y˙ i (t ) = Bφi yi (t ) + gφi (yi (t )) + ε

l  

cij yj (t ),

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

(2)

k=1 j∈Vk

where yi (t ) = (yi1 (t ), yi2 (t ), . . . , yin (t ))T ∈ Rn is the state variable of node i. Bφi are n × n constant matrices, and gφi : Rn → Rn are nonlinear vector-valued functions. Remark 1. The constant matrices Aφi and Bφi can be identical or nonidentical as well as the nonlinear vector-valued functions fφi and gφi in network (1) and (2). Definition 1. The drive-response networks are said to realize cluster projective synchronization (CPS), if there exist l different scaling factors αφi (i = 1, 2, . . . , N ) such that limt →∞ ∥yi (t ) − αφi xi (t )∥ = 0. Our objective here is to make the drive-response community networks achieve cluster projective synchronization by applying proper controllers to the response network. Then, the controlled network can be described by y˙ i (t ) = Bφi yi (t ) + gφi (yi (t )) + ε

l  

cij yj (t ) + ui (t ),

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

(3)

k=1 j∈Vk

where ui (t ) are controllers to be designed. 3. The mail results In this section, the CPS of the drive-response community networks (1) and (3) will be investigated in three cases. Firstly, assume that the nodes dynamics in both community networks are identical, i.e., the constant matrices Aφi = Bφi = A and nonlinear vector-valued functions fφi = gφi = f for all 1 ≤ i ≤ N. Then, the drive-response networks (1) and (3) can be written as x˙ i (t ) = Axi (t ) + f (xi (t )) + ε

l  

cij xj (t ),

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

(4)

k=1 j∈Vk

y˙ i (t ) = Ayi (t ) + f (yi (t )) + ε

l   k=1 j∈Vk

cij yj (t ) + ui (t ),

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

(5)

6192

Z. Wu, X. Fu / Physica A 391 (2012) 6190–6198

Let V¯ φi denote the nodes in the φi th community which have direct connections to the nodes in other communities and ei (t ) = yi (t ) − αφi xi (t ) be the synchronization errors, then one has e˙ i (t ) = Aei (t ) + f (yi (t )) − αφi f (xi (t )) + ε

N 

cij ej (t ) + ε

j =1

N 

cij (αφj − αφi )xj (t ) + ui (t ),

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

(6)

j =1

N It is clear that j=1 cij (αφj − αφi )xj (t ) = 0 for i ̸∈ V¯ φi . For achieving the CPS of the drive-response community networks, the controllers are designed as follows



ui (t ) = −δi ei (t ) + αφi f (xi (t )) − f (yi (t )) − ε

N 

cij (αφj − αφi )xj (t ),

i ∈ V¯ φi ,

j =1

ui (t ) = −δi ei (t ) + αφi f (xi (t )) − f (yi (t )),

(7)

i ̸∈ V¯ φi ,

where δi > 0 are the feedback gains. Theorem 1. The drive-response community networks (4) and (5) with controllers (7) can achieve cluster projective synchronization if the following condition is satisfied

(λA − δ ∗ )IN + ε C˜ < 0,

(8)

where C˜ = (C + C T )/2, λA is the largest eigenvalue of (A + AT )/2 and δ ∗ = min1≤i≤N {δi }. Proof. Combining (6) and (7), one has e˙ i (t ) = Aei (t ) + ε

N 

cij ej (t ) − δi ei (t ),

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

(9)

j =1

Consider the following Lyapunov function V (t ) =

N 1

2 i =1

eTi (t )ei (t ).

Let e˜ j (t ) = (e1j (t ), e2j (t ), . . . , ekj (t ))T , then the derivative of V (t ) along the trajectories of (9) is V˙ (t ) =

k 

eTi (t )˙ei (t )

i=1

=

N 

eTi (t )Aei (t ) + ε

i=1

=

N 

≤

eTi (t )

=



A + AT



2

λA eTi (t )ei (t ) + ε

i =1 n 

cij eTi (t )ej (t ) −

i =1 j =1

i=1 N 

N  N 

ei (t ) + ε e˜ Tj (t )

δi eTi (t )ei (t )

i=1 n 

e˜ Tj (t )C e˜ j (t ) −

i =1 n 

N 



δi eTi (t )ei (t )

i=1

C + CT

i =1

N 

2



e˜ j (t ) −

N 

δ ∗ eTi (t )ei (t )

i=1

e˜ Tj (t )((λA − δ ∗ )IN + ε C˜ )˜ej (t ).

j=1

From (8), we have V˙ (t ) < 0 and limt →∞ ∥ei (t )∥ = 0 according to Lyapunov stability theory, i.e., the cluster projective synchronization between drive-response networks is achieved. This completes the proof.  Let λC be the largest eigenvalue of (C + C T )/2. The feedback gain δi can be estimated by the following corollary. Corollary 1. If the minimum feedback gain is such that

δ ∗ > λA + ελC , then the drive-response networks (4) and (5) with controllers (7) can achieve cluster projective synchronization. The minimum feedback gain given in Corollary 1 is too conservative and much larger than the value needed. From a practical view, the feedback gains are desired to be as small as possible. Hence, the adaptive technique is adopted to design the controllers.

Z. Wu, X. Fu / Physica A 391 (2012) 6190–6198

6193

Theorem 2. The drive-response community networks (4) and (5) can achieve cluster projective synchronization with the following controllers ui (t ) = −δ(t )ei (t ) + αφi f (xi (t )) − f (yi (t )) − ε

N 

cij (αφj − αφi )xj (t ),

i ∈ V¯ φi ,

j =1

i ̸∈ V¯ φi ,

ui (t ) = −δ(t )ei (t ) + αφi f (xi (t )) − f (yi (t )),

˙ t) = θ δ(

N 

(10)

eTi (t )ei (t ),

i =1

where θ is a small positive constant. Proof. Combining (6) and (10), one has e˙ i (t ) = Aei (t ) + ε

N 

cij ej (t ) − δ(t )ei (t ),

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

j =1

δ˙ (t ) = θ

N 

(11)

eTi (t )ei (t ).

i =1

Consider the following Lyapunov function V (t ) =

N 1

2 i=1

eTi (t )ei (t ) +

1 2θ

ˆ 2, (δ(t ) − δ)

where δˆ is an arbitrary positive constant to be determined. Then, the derivative of V (t ) along the trajectories of (11) is V˙ (t ) =

N 

eTi (t )˙ei (t ) +

i=1

=

N 

1

θ

eTi (t )Aei (t ) + ε

i=1

=

N  i=1

≤

N  i=1

=

n 

ˆ δ( ˙ t) (δ(t ) − δ) N  N 

cij eTi (t )ej (t ) −

i=1 j=1

eTi (t )



A + AT



2

λA eTi (t )ei (t ) + ε

ei (t ) + ε

N 

δˆ eTi (t )ei (t )

i =1 n 

e˜ Tj (t )C e˜ j (t ) −

i =1 n 

e˜ Tj (t )



δˆ eTi (t )ei (t )

i=1

C + CT 2

i=1

N 



e˜ j (t ) −

N 

δˆ eTi (t )ei (t )

i=1

ˆ IN + ε C˜ )˜ej (t ). e˜ Tj (t )((λA − δ)

j =1

ˆ IN + ε C˜ < 0 when δˆ is sufficiently large, i.e., V˙ (t ) < 0. According to Lyapunov stability theory, one Then, one has (λA − δ) can derive limt →∞ ∥ei (t )∥ = 0, i.e., the cluster projective synchronization between drive-response networks is achieved. This completes the proof.  Remark 2. In the above discussion, all the nodes dynamics are assumed to be identical. However, in many real community networks, the nodes in the same community often have identical node dynamics and different communities have different node dynamics. Secondly, assume that different communities have different node dynamics and the nodes within a community have identical node dynamics, i.e., Aφi = Bφi = Aφj = Bφj and fφi = gφi = fφj = gφj in community networks (1) and (3) if φi = φj , otherwise, Aφi = Bφi ̸= Aφj = Bφj and fφi = gφi ̸= fφj = gφj . Then rewrite the drive-response community networks (1) and (3) as x˙ i (t ) = Aφi xi (t ) + fφi (xi (t )) + ε

l  

cij xj (t ),

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

(12)

k=1 j∈Vk

y˙ i (t ) = Aφi yi (t ) + fφi (yi (t )) + ε

l   k=1 j∈Vk

cij yj (t ) + ui (t ),

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

(13)

6194

Z. Wu, X. Fu / Physica A 391 (2012) 6190–6198

Theorem 3. Suppose that the controllers in (13) are designed as follows ui (t ) = −δi ei (t ) + αφi fφi (xi (t )) − fφi (yi (t )) − ε

N 

i ∈ V¯ φi ,

cij (αφj − αφi )xj (t ),

(14)

j =1

i ̸∈ V¯ φi .

ui (t ) = −δi ei (t ) + αφi fφi (xi (t )) − fφi (yi (t )),

Then the drive-response networks (12) and (13) can achieve cluster projective synchronization if the following condition is satisfied

(λmax (Aφ ) − δ ∗ )IN + ε C˜ < 0,

(15)

where λmax (Aφ ) is the largest eigenvalue of (Aφi + Aφi )/2 for all i ∈ {1, 2, . . . , N }. Proof. Similar to the proof of Theorem 1, one can easily give the proof, thus it is omitted here. T



Theorem 4. The drive-response community networks (12) and (13) can achieve cluster projective synchronization with the following controllers ui (t ) = −δ(t )ei (t ) + αφi fφi (xi (t )) − fφi (yi (t )) − ε

N 

cij (αφj − αφi )xj (t ),

i ∈ V¯ φi ,

j =1

ui (t ) = −δ(t )ei (t ) + αφi fφi (xi (t )) − fφi (yi (t )),

˙ t) = θ δ(

N 

i ̸∈ V¯ φi ,

eTi (t )ei (t ),

i=1

where θ is a small positive constant. Proof. Similar to the proof of Theorem 2, the proof can be easily given, thus it is omitted here.



Finally, in the real world, different complex networks usually have different node dynamics. Therefore, consider the following drive-response community networks with different nodes dynamics x˙ i (t ) = Aφi xi (t ) + fφi (xi (t )) + ε

l  

cij xj (t ),

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

(16)

k=1 j∈Vk

y˙ i (t ) = Bφi yi (t ) + gφi (yi (t )) + ε

l  

cij yj (t ) + ui (t ),

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

(17)

k=1 j∈Vk

where Aφi xi (t ) + fφi (xi (t )) ̸= Bφi xi (t ) + gφi (xi (t )) for all i ∈ {1, 2, . . . , N }. Theorem 5. Suppose that the controllers in (17) are designed as follows ui (t ) = −δi ei (t ) + αφi fφi (xi (t )) − gφi (yi (t )) − (Bφi − Aφi )αφi xi (t ) − ε

N 

cij (αφj − αφi )xj (t ),

i ∈ V¯ φi ,

j=1

ui (t ) = −δi ei (t ) + αφi fφi (xi (t )) − gφi (yi (t )) − (Bφi − Aφi )αφi xi (t ),

i ̸∈ V¯ φi .

Then the drive-response networks (16) and (17) can achieve cluster projective synchronization if the following condition is satisfied

(λmax (Bφ ) − δ ∗ )IN + ε C˜ < 0,

(18)

where λmax (Bφ ) is the largest eigenvalue of (Bφi + Bφi )/2 for all i ∈ {1, 2, . . . , N }. T

Proof. The proof is similar to that of Theorem 1, it is therefore omitted here.



Theorem 6. The drive-response community networks (16) and (17) can achieve cluster projective synchronization with the following controllers ui (t ) = −δ(t )ei (t ) + αφi fφi (xi (t )) − gφi (yi (t )) − (Bφi − Aφi )αφi xi (t ) − ε

N 

cij (αφj − αφi )xj (t ),

j =1

ui (t ) = −δ(t )ei (t ) + αφi fφi (xi (t )) − gφi (yi (t )) − (Bφi − Aφi )αφi xi (t ),

˙ t) = θ δ(

N 

eTi (t )ei (t ),

i=1

where θ is a small positive constant. Proof. The proof is similar to that of Theorem 2, so it is omitted here.



i ̸∈ V¯ φi ,

i ∈ V¯ φi ,

Z. Wu, X. Fu / Physica A 391 (2012) 6190–6198

6195

Fig. 1. The network with three communities and the coupling matrix C , if there exists a link between node i and node j, then cij = 1, otherwise, cij = 0.

Fig. 2. The synchronization errors of three communities, ei (t ) = yi (t ) − xi (t ), i = 1, 2, . . . , 6, ej (t ) = yj (t ) − 2xj (t ), j = 7, 8, . . . , 12, ek (t ) = yk (t ) − 3xk (t ), k = 13, 14, . . . , 19.

4. Numerical simulation In this section, several numerical examples are provided to verify the derived results. A community network consisting of 19 nodes with three communities is shown in Fig. 1. Firstly, verify Theorems 1 and 2, and choose the nodes dynamics of drive-response community networks as the wellknown Lorenz chaotic system [24]

   x˙ 1 −10 x˙ 2 = 28 x˙ 3 0

10 −1 0

0 0 −8/3

x1 x2 x3

 

 +

0 −x 1 x 3 x1 x2

 .

The network topology is shown in Fig. 1; then the coupling C is a symmetry matrix and its largest eigenvalue is zero. The largest eigenvalue of (A + AT )/2 can be calculated as λA = 14.0256; then according to Corollary 1, one can choose δi = 15 for all i ∈ {1, 2, . . . , N }. In numerical simulations, the coupling strength ε = 0.1, the scaling factors (α1 , α2 , α3 ) = (1, 2, 3), and the initial values of state variables are chosen randomly. Fig. 2 shows the synchronization errors of three communities respectively. Moreover, choose the initial value of δ(t ) as δ(0) = 5 and the adaptive gain θ = 0.001. Fig. 3 shows the synchronization errors of three communities and the orbit of δ(t ) respectively. Secondly, verify Theorems 3 and 4, and choose the nodes dynamics of the three communities as the above Lorenz chaotic system, the well-known Chen system [25]

   x˙ 1 −35 x˙ 2 = −7 x˙ 3 0

35 28 0

0 0 −3

x1 x2 x3

 

 +

0 −x 1 x 3 x1 x2

 ,

6196

Z. Wu, X. Fu / Physica A 391 (2012) 6190–6198

Fig. 3. The left shows the synchronization errors of three communities, ei (t ) = yi (t ) − xi (t ), i = 1, 2, . . . , 6, ej (t ) = yj (t ) − 2xj (t ), j = 7, 8, . . . , 12, ek (t ) = yk (t ) − 3xk (t ), k = 13, 14, . . . , 19; the right shows the orbit of δ(t ).

Fig. 4. The synchronization errors of three communities, ei (t ) = yi (t ) + xi (t ), i = 1, 2, . . . , 6, ej (t ) = yj (t ) − xj (t ), j = 7, 8, . . . , 12, ek (t ) = yk (t ) − 2xk (t ), k = 13, 14, . . . , 19.

and the well-known Lü system [26]

   x˙ 1 −36 x˙ 2 = 0 x˙ 3 0

36 15 0

0 0 −3

x1 x2 x3

 

0 −x 1 x 3 x1 x2

 +

 .

The topology is also chosen as C in Fig. 1. The largest eigenvalue of (Aφi + ATφi )/2 (i = 1, 2, . . . , N ) can be calculated as λmax (Aφ ) = 30.9710, then one can choose δi = 35 for all i ∈ {1, 2, . . . , N }. In numerical simulations, the coupling strength ε = 0.1, the scaling factors (α1 , α2 , α3 ) = (−1, 1, 2), and the initial values of state variables are chosen randomly. Fig. 4 shows the synchronization errors of three communities respectively. For Theorem 4, choose the initial value of δ(t ) as δ(0) = 5 and adaptive gain θ = 0.001. Fig. 5 shows the synchronization errors of three communities and the orbit of δ(t ) respectively. Finally, verify Theorems 5 and 6, and choose the node dynamics of the drive network as the Chen system and that of the response network as the Lorenz system. The topology is also chosen as C in Fig. 1. The largest eigenvalue can be calculated as λmax (Bφ ) = 14.0256; then one can choose δi = 15 for all i ∈ {1, 2, . . . , N }. In numerical simulations, the coupling strength ε = 0.1, the scaling factors (α1 , α2 , α3 ) = (−1, −2, −3), and the initial values of state variables are chosen randomly. Fig. 6 shows the synchronization errors of three communities respectively. For Theorem 6, choose the initial value of δ(t ) as δ(0) = 5 and adaptive gain θ = 0.001. Fig. 7 shows the synchronization errors of three communities and the orbit of δ(t ) respectively. 5. Conclusions and discussions Cluster projective synchronization of drive-response community networks have been studied in this paper. By designing proper controllers, the CPS of the drive-response networks with identical or nonidentical nodes can be achieved. According to Lyapunov stability theory, several simple and useful criteria for synchronization are obtained. In this paper, we assume

Z. Wu, X. Fu / Physica A 391 (2012) 6190–6198

6197

Fig. 5. The left shows the synchronization errors of three communities, ei (t ) = yi (t )+ xi (t ), i = 1, 2, . . . , 6, ej (t ) = yj (t )− xj (t ), j = 7, 8, . . . , 12, ek (t ) = yk (t ) − 2xk (t ), k = 13, 14, . . . , 19; the right shows the orbit of δ(t ).

Fig. 6. The synchronization errors of three communities, ei (t ) = yi (t ) + xi (t ), i = 1, 2, . . . , 6, ej (t ) = yj (t ) + 2xj (t ), j = 7, 8, . . . , 12, ek (t ) = yk (t ) + 3xk (t ), k = 13, 14, . . . , 19.

Fig. 7. The left shows the synchronization errors of three communities, ei (t ) = yi (t )+ xi (t ), i = 1, 2, . . . , 6, ej (t ) = yj (t )− xj (t ), j = 7, 8, . . . , 12, ek (t ) = yk (t ) − 2xk (t ), k = 13, 14, . . . , 19; the right shows the orbit of δ(t ).

that the drive-response community networks have the same number of nodes and each community has the same number of nodes as well. As for the CPS between those drive-response networks with different numbers of nodes, we will discuss them in our future research. Numerical simulations are provided to verify the correctness and effectiveness of the derived theoretical results.

6198

Z. Wu, X. Fu / Physica A 391 (2012) 6190–6198

Acknowledgments This research is jointly supported by the NSFC grant 11072136, the Shanghai Univ. Leading Academic Discipline Project (A.13-0101-12-004), the Youth Scientific Fund of Jiangxi Normal Univ. (3911), and the Startup Fund for Ph.D. of Jiangxi Normal Univ. (3087). 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]

J.M. Kumpula, J.P. Onnela, J. Saramäi, K. Kaski, J. Kertész, Phys. Rev. Lett. 99 (2007) 228701. M. Chavez, D.U. Hwang, A. Amann, H.G.E. Hentschel, S. Boccaletti, Phys. Rev. Lett. 94 (2005) 218701. J. Lu, D.W.C. Ho, J. Kurths, Phys. Rev. E 80 (2009) 066121. M. Yang, Y. Liu, Z. You, P. Sheng, Nonlinear Anal. RWA 11 (2010) 2127. R.N. Mantegna, Eur. Phys. J. B 11 (1999) 193. B.J. Chang, R.H. Hwang, Inform. Sci. 176 (2006) 522. J. Wang, B. Wu, L. Wang, F. Fu, Phys. Rev. E 78 (2008) 051923. K. Wang, X. Fu, K. Li, Chaos 19 (2009) 023106. C. Li, G. Chen, Physica A 343 (2004) 263. W. Guo, S. Chen, F. Austin, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1631. Y. Xu, W. Zhou, J. Fang, W. Sun, Phys. Lett. A 374 (2010) 1673. S. Zheng, Q. Bi, G. Cai, Phys. Lett. A 373 (2009) 1553. M. Hu, Z. Xu, Y. Yang, Physica A 387 (2008) 3759. S. Zheng, G. Dong, Q. Bi, Phys. Lett. A 373 (2009) 4255. K. Li, C.H. Lai, Phys. Lett. A 372 (2008) 1601. E. Guirey, M. Bees, A. Martin, M. Srokosz, Phys. Rev. E 81 (2010) 051902. W. Lu, B. Liu, T. Chen, Chaos 20 (2010) 013120. C. Li, W. Sun, J. Kurths, Phys. Rev. E 76 (2007) 046204. X. Wu, W. Zheng, J. Zhou, Chaos 19 (2009) 013109. C. Li, C. Xu, W. Sun, J. Xu, J. Kurths, Chaos 19 (2009) 013106. Z. Li, X. Xue, Chaos 20 (2010) 023106. G. Wang, J. Cao, J. Lu, Physica A 389 (2010) 1480. J. Wang, Q. Ma, L. Zeng, M.S. Abd-Elouahab, Chaos 21 (2011) 013121. E. Lorenz, J. Atmospheric Sci. 20 (1963) 130. G. Chen, T. Ueta, Int. J. Bifurcation Chaos 9 (1999) 1465. J. Lü, G. Chen, Int. J. Bifurcation Chaos 12 (2002) 659.