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Cluster projective synchronization of fractional-order complex network via pinning control.

1. Introduction

In the past few decades, complex networks behaviors have attracted a great deal of attention in a variety of fields due to their wide and potential applications. Typical complex networks include the Internet, the World Wide Web, neural networks, and so on [1-5]. Synchronization, as a typical collective dynamical behavior of coupled dynamical systems, has been widely studied. Until now, several types of synchronization have been investigated, such as complete synchronization [6], phase synchronization [7], and generalized synchronization [8]. Projective synchronization (PS) [9], which was first proposed by Mainieri and Rehacek that the drive state vector and the response state vectors synchronize up to a constant scaling factor. The proportional feature of PS can be employed to extend binary digital to M-nary communication for achieving fast communication. Recently, Li and Chen [10] discussed projective synchronization of the random networks. After that, the authors studied the projective synchronization of time-delayed chaotic systems in a driven-response complex network. In [11], they investigated the projective and lag synchronization between general complex networks via impulsive control.

In particular, in many social and biological networks, which can be divided naturally into communities, nodes in the same community often have the same type of function. Cluster synchronization is an exact phrase that describes this important phenomenon [12]. Cluster synchronization is achieved when the dynamical nodes reach complete synchronization in each subgroup called cluster but no synchronization among the different clusters. Many results have been available for cluster synchronization of complex networks. For example, Ayati and Khaki-Sedigh studied cluster synchronization of a connected chaotic network and a star-like complex network in [13]. Wang and Song [14] investigated the cluster synchronization problem for linearly coupled networks. More recently, Hu et al. [15] studied the cluster synchronization for directed complex dynamical networks via pinning control. In [16, 17], cluster synchronization in community networks is studied with integer-order system nodes, and several sufficient conditions for synchronization are obtained analytically.

On the other hand, as we know, the well-studied integer-order complex networks are the special cases of the fractional-order ones. It has been revealed that, in interdisciplinary fields, various systems have been found to exhibit fractional dynamics. For example, viscoelasticity, dielectric polarization, quantum evolution of complex system, fractional kinetics, and anomalous attenuation can be described by fractional differential equations. To the best of our knowledge, most studies to date have been concerned with integer-order complex networks, and the corresponding research on fractional-order complex networks has received very little attention despite its practical significance [18-23]. Therefore, it is of great interest to investigate the synchronization in complex colored networks consisting of nodes with fractional-order dynamics.

Motivated by the above discussions, the cluster projective synchronization in complex dynamical networks with fractional-order dynamical nodes by pinning control is investigated in this paper. This method decreases the control cost to some extent by reducing the number of nodes. The pinning controllers are designed according to the nodes property, respectively. We derive some simple and useful criteria for cluster synchronization for any initial values through an effective control scheme.

This paper is organized as follows. In Section 2, the network model of fractional-order network and mathematical preliminaries is introduced. Section 3 is devoted to investigating the cluster projective synchronization of the complex coupled networks. In Section 4, illustrative examples are shown to support the theory results. Conclusions are drawn in Section 5.

2. Mathematical Preliminaries and Model for Community Networks

2.1. Fractional-Order Derivative. Fractional calculus is a generalization of integration and differentiation to a noninteger-order integrodifferential operator [sub.a][D.sup.q.sub.t] which is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where q is the fractional order which can be a complex number and R(q) is the real part of q. The numbers a and t are the limits of the operator. There are many definitions for fractional differential equations. Three most frequently used definitions for general fractional differential equations are the Grunwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definitions.

The Grunwald-Letnikov definition (GL) derivative with fractional-order q is described by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where the symbol [*] means the integer part.

The Riemann-Liouville (RL) definition of fractional derivatives is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [GAMMA](*) is the gamma function.

The Caputo (C) fractional derivative is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

It is well known that the initial conditions for the fractional differential equations with Caputo derivatives take on the same form as those for the integer-order ones, which is very suitable for practical problems. Therefore, we will use the Caputo definition for the fractional derivatives in this paper.

2.2. Model Description for Community Networks. Consider that the complex networks considered in this paper consist of N nodes and m communities, where 2 [less than or equal to] m [less than or equal to] N. Let {[C.sub.1], [C.sub.2],..., [C.sub.m]} denote m communities of complex networks and [[union].sup.m.sub.i=1] [C.sub.i] = {1,2,..., N}. Without loss of generality, let [C.sub.1] = {1,2,...,[r.sub.1]}, [C.sub.1] = {[r.sub.1] + 1,...,[r.sub.1] + [r.sub.2]},..., [C.sub.m] = {[r.sub.1] + ... + [r.sub.m-1] + 1,...,[r.sub.1] + ... + [r.sub.m-1] + [r.sub.m]} with r + ... + [r.sub.m-1] + [r.sub.m] = N. If node i belongs to the jth community then let [[delta].sub.i] = j. Denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the set of all nodes in the [[delta].sub.i]th cluster. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represents all the nodes in the [[delta].sub.i] cluster, which have direct connections with the nodes in other communities.

A complex network consisting of N coupled identical nodes, with each node being a n-dimensional fractional-order dynamical system, can be described by

[D.sup.q.sub.*][x.sub.i](t) = f([x.sub.i](t)) + [epsilon] [N.summation over (i=1)] [c.sub.ij][GAMMA][x.sub.j](t) i = 1,2,...,N, (5)

where 0 < q < 1 and [x.sub.i] = [([x.sub.i1], [x.sub.i2],..., [x.sub.in]).sup.T] [member of] [R.sup.n] is the state variable of the node i. f : [R.sup.n] [right arrow] [R.sup.n] describes the dynamics of nodes and is differential and capable of performing abundant dynamical behaviors. [epsilon] > 0 is the coupling strength and [GAMMA] = diag([[beta].sub.1], [[beta].sub.2],..., [[beta].sub.n]) is the inner-coupling matrix. The matrix C = [([c.sub.ij]).sub.NxN] is the coupling configuration diffusive matrix, which is defined as follows: if there is a connection between nodes i and j is connected, then [c.sub.ij] = [c.sub.ji] > 0 (i [not equal to] j); otherwise [c.sub.ij] = [c.sub.ji] = 0 (i [not equal to] j); let [c.sub.ii] = -[[summation].sup.N.sub.j-1,i[not equal to]j] [c.sub.ij]. Further, assume that there are no isolated clusters in the network and the network is connected, so the coupling configuration C is symmetrical and irreducible. [u.sub.i] [member of] [R.sup.n] are controllers to be designed later. For simplicity of further discussion, decompose the function f([x.sub.i](t)) into two parts, A([x.sub.i](t)) + F([x.sub.i](t)), where A is an n x n constant matrix and F : [R.sup.n] [right arrow] [R.sup.n] are nonlinear vector-valued functions.

Then the controlled network can be rewritten as

[D.sup.q.sub.*][x.sub.i](t) = A([x.sub.i](t)) + F([x.sub.i](t)) + [epsilon] [N.summation over (j=1)] [c.sub.ij][GAMMA][x.sub.j](t) + [u.sub.i] i = 1,2,...,N, (6)

Definition 1. Let {1,2,..., N} be the N nodes of the networks and let {[C.sub.1], [C.sub.2],..., [C.sub.m]} be the m communities, respectively. A network with m communities is said to realize cluster projective synchronization if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Define the error variables

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where M = diag([lambda], [lambda],...,[lambda]) is the scaling matrix and [S.sub.[delta]i] (t) is a solution of an isolated node in the [[delta].sub.i]th community; we assume that the smooth goal dynamics can be described by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] maybe an equilibrium point, a periodic orbit, or even a chaotic orbit. Where B [member of] [R.sup.nxn] is a constant matrix, g(s(t)) is the nonlinear part of the isolated reference node dynamics, respectively.

Remark 2. It is easy to see that the definition of the cluster projective synchronization encompasses the cluster synchronization and cluster antisynchronization when the scaling matrix is selected to take the corresponding specific values, respectively.

Throughout this paper, the following assumption and lemma are needed to prove our main results.

Lemma 3 (see [22]). For a given autonomous fractional-order linear system

[D.sup.q.sub.*]x = Ax, with x(0) = [x.sub.0], (10)

where x [member of] [R.sup.n] is the state vector, A [member of] [R.sup.nxn] is a constant matrix with eigenvalues [[lambda].sub.1], [[lambda].sub.2],..., [[lambda].sub.n], the fractional order q [member of] (0,1), and system (9) is asymptotically stable if and only if [absolute value of arg([[lambda].sub.i])] > q[pi]/2, i = 1,2,...,n.

Lemma 4 (see [24]). Assume A [member of] [R.sup.NxN] satisfies the following conditions:

(1) [a.sub.ij] [greater than or equal to] 0 (i [not equal to] j), [a.sub.ii] = - [[summation].sup.N.sub.j=1,j[not equal to]i] [a.sub.ij], i, j,...,N,

(2) A is irreducible.

Then (i) real parts of the eigenvalues of A are all negatives except an eigenvalue 0 with multiplicity 1; (ii) A has the right eigenvector [(1,1,...,1).sup.T] corresponding to the eigenvalue 0; (iii) let [xi] = [([[xi].sub.1], [[xi].sub.2],...,[[xi].sub.N]).sup.T] be the left eigenvector of A corresponding to the eigenvalue 0; then we can let [[xi].sub.i] > 0 hold for all i = 1,2,...,N.

Lemma 5 (see [24]). If the matrix C is defined as in Lemma 4 and diagonal matrix K = diag([k.sub.1], [k.sub.2],...,[k.sub.n]) with [k.sub.i] [greater than or equal to] 0 (i = 1,2,...,N), then all eigenvalues of the matrix C - K are negative.

3. Main Results

In this section, we present a scheme to make a complex network achieve cluster projective synchronization. First of all, we propose the concept of interlink and intralink nodes. Node i is said to be the interlink node if i belongs to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], while i is said to be the intralink node if i belongs to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. According to the diffusive coupling condition of matrix A, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

So the error dynamical system is described as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

In order to make the network achieve cluster synchronization, the control input [u.sub.i] is designed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII, (13)

where [k.sub.i] > 0 is the feedback control gain, which can adjust the synchronization speed.

Theorem 6. For a certain fractional-order q, the fractional-order complex network (6) can achieve the cluster projective synchronization with controller (13) if

[absolute value of arg(A + [[lambda].sub.i][epsilon][GAMMA])] > [q[pi]/2], i = 1,2,...,N. (14)

Proof. Combining (6) and (13), one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Denote K = diag([k.sub.1], [k.sub.2],,..., [k.sub.N]) with [k.sub.i] = 0 for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; thus one can obtain the following equation:

[D.sup.q.sub.*] e(t) = Ae(t) + [epsilon](C - K)e(t)[GAMMA]. (16)

Since C is an irreducible matrix and K = diag([k.sub.1], [k.sub.2],..., [k.sub.N]) with [k.sub.i] [greater than or equal to] 0 (i = 1,2,..., N), according to Lemmas 4 and 5, there exists unitary [xi] = ([[xi].sub.1], [[xi].sub.2],..., [[xi].sub.N]) such that

(C - K)[xi] = [xi][LAMBDA], (17)

where [LAMBDA] = diag([[lambda].sub.1], [[lambda].sub.2],..., [[lambda].sub.N]) and [[lambda].sub.i] is the eigenvalue of matrix C-K and satisfies 0 > [[lambda].sub.1] [greater than or equal to] [[lambda].sub.2] [greater than or equal to] ... [greater than or equal to] [[lambda].sub.N] = [[lambda].sub.min] (C-K).

It follows from (16) and (17) that

[D.sup.q.sub.*] e(t) = (Ae(t)) [xi] + [epsilon][GAMMA]e(t)[xi][LAMBDA]. (18)

Denote [beta](t) = e(t)[xi], [beta](t) = ([[beta].sub.1](t), [[beta].sub.2](t),..., [[beta].sub.N](t)); thus

[D.sup.q.sub.*] [beta](t) = A[beta](t) + [epsilon][GAMMA][beta](t)[LAMBDA]; (19)

that is,

[D.sup.q.sub.*][[beta].sub.i](t) = (A + [[lambda].sub.i][epsilon][GAMMA]) [[beta].sub.i](t), i = 1,2,...,N. (20)

According to Lemma 3, system (19) is asymptotically stable if and only if all the eigenvalues of A + [[lambda].sub.i][epsilon][GAMMA] satisfy [absolute value of arg(A + [[lambda].sub.i][epsilon][GAMMA])] > q[pi]/2, which implies that system (6) can achieve the cluster projective synchronization.

Based on Theorem 6, a corollary can be easily derived as follows.

Corollary 7. If matrix C - K is irreducible, for a certain fractional-order q, the fractional-order complex network (6) can achieve cluster projective synchronization via controller (13) if

[absolute value of arg([[bar.[lambda]].sub.ik] (A + [[lambda].sub.i][epsilon][GAMMA]))] > q[pi]/2, k = 1, 2,..., n, i = 1, 2,..., N, (21)

where [[lambda].sub.ik](*) is the eigenvalue of the ith node matrix A + [[lambda].sub.i][epsilon][GAMMA].

4. Numerical Example

In what follows, we take a representative example to demonstrate the effectiveness of the proposed approach for cluster projective synchronization.

Example 1. We arbitrarily set the network size as N = 16. Suppose this network consists of three communities. The size of three communities is [N.sub.1] = 5, [N.sub.2] = 5, and [N.sub.3] = 6. The local dynamics of nodes are described by Lorenz system. The fractional-order Lorenz system is given by

[D.sup.q.sub.*][x.sub.1] = a([x.sub.2] - [x.sub.1]), [D.sup.q.sub.*][x.sub.2] = b[x.sub.1] - [x.sub.2] - [x.sub.1][x.sub.3], [D.sup.q.sub.*][x.sub.3] = [x.sub.1][x.sub.2] - c[x.sub.3], (22)

where a = 10, b = 28, c = 8/3, parameters for which the system exhibits chaotic behavior. The chaotic attractor is depicted by Figure 1.

Without loss of generality, we suppose that [C.sub.1] = {1,2, 3,4,5}, [C.sub.2] = {6, 7, 8, 9, 10}, and [C.sub.3] = {11, 12, 13, 14, 15, 16}.

In the following simulations, inner-coupling matrix [GAMMA] is chosen as identity matrix. The following quantities are used to investigate the process of cluster projective synchronization:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

where E(t) is the total error of cluster projective synchronization for all communities and [E.sub.12](t), [E.sub.13](t), [E.sub.23](t) are the errors between two different communities. S is the states of Lorenz system. All the initial conditions are chosen randomly in [0, 2]. The results of simulation can be seen in Figures 2 and 3.

As can be seen from Figure 2, the synchronization errors E(t) converge to zero. Figure 3 shows that the synchronization errors [E.sub.12](t), [E.sub.13](t), [E.sub.23](t) do not converge to zero as t [right arrow] [infinity]. That is to say that the nodes in the same cluster reach synchronization and there is no synchronization among the different clusters, which implies that the desired cluster projective synchronization is achieved.

5. Conclusions

In this paper, we have investigated the cluster projective synchronization of complex networks with community structure. Based on the stability theory of the fractional-order differential system, the controllers are designed differently for the nodes in one community, which have direct connections to the nodes in the other communities and the nodes without direct connections to the nodes in the other communities. Several sufficient conditions for the network to achieve cluster projective synchronization are derived. Finally, a representative numerical example is provided to illustrate the effectiveness of the derived theoretical results.

http://dx.doi.org/10.1155/2014/314742

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions in helping them improve this paper.

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Li-xin Yang, Wan-sheng He, Fan-di Zhang, and Jin-ping Jia

School of Mathematic and Statistics, Tianshui Normal University, Tianshui 741001, China

Correspondence should be addressed to Li-xin Yang; jiaodayanglixin@163.com

Received 10 March 2014; Revised 20 April 2014; Accepted 22 April 2014; Published 12 May 2014

Academic Editor: Jun Hu
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Title Annotation:Research Article
Author:Yang, Li-xin; He, Wan-sheng; Zhang, Fan-di; Jia, Jin-ping
Publication:Abstract and Applied Analysis
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Date:Jan 1, 2014
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