Cluster Projective Synchronization of Fractional-Order Complex Network via Pinning Control

and Applied Analysis 3 A network with m communities is said to realize cluster projective synchronization if lim t→∞ 󵄩󵄩󵄩󵄩 x i −Mx j 󵄩󵄩󵄩󵄩 = 0, ∀i, j ∈ C n , n = 1, 2, . . . , m, lim t→∞ 󵄩󵄩󵄩󵄩 x i −Mx j 󵄩󵄩󵄩󵄩 ̸ = 0, i ∈ C n1 , j ∈ C n2 , n 1 ̸ = n 2 = 1, 2, . . . , m. (7) Define the error variables e i (t) = x i (t) − MS δi (i = 1, 2, . . . , N) , (8) where M = diag(λ, λ, . . . , λ) is the scaling matrix and S δi (t) is a solution of an isolated node in the δ i th community; we assume that the smooth goal dynamics can be described by


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][2][3][4][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][19][20][21][22][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.

Mathematical Preliminaries and Model for Community Networks
2.1.Fractional-Order Derivative.Fractional calculus is a generalization of integration and differentiation to a nonintegerorder integrodifferential operator     which is defined by where  is the fractional order which can be a complex number and () is the real part of .The numbers  and  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  is described by where the symbol [⋅] means the integer part.The Riemann-Liouville (RL) definition of fractional derivatives is given by where Γ(⋅) is the gamma function.
The Caputo () fractional derivative is defined as follows: 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.

Model
If node  belongs to the th community then let   = .Denote by    the set of all nodes in the   th cluster.Ũ  represents all the nodes in the   cluster, which have direct connections with the nodes in other communities.
A complex network consisting of  coupled identical nodes, with each node being a -dimensional fractionalorder dynamical system, can be described by where 0 <  < 1 and   = ( 1,  2 , . . .,   )  ∈   is the state variable of the node . :   →   describes the dynamics of nodes and is differential and capable of performing abundant dynamical behaviors. > 0 is the coupling strength and Γ = diag( 1 ,  2 , . . .,   ) is the inner-coupling matrix.The matrix  = (  ) × is the coupling configuration diffusive matrix, which is defined as follows: if there is a connection between nodes  and  is connected, then   =   > 0 ( ̸ = ); otherwise Further, assume that there are no isolated clusters in the network and the network is connected, so the coupling configuration  is symmetrical and irreducible.  ∈   are controllers to be designed later.For simplicity of further discussion, decompose the function (  ()) into two parts, (  ()) + (  ()), where  is an  ×  constant matrix and  :   →   are nonlinear vectorvalued functions.
Then the controlled network can be rewritten as Define the error variables where  = diag(, , . . ., ) is the scaling matrix and   () is a solution of an isolated node in the   th community; we assume that the smooth goal dynamics can be described by where    () may be an equilibrium point, a periodic orbit, or even a chaotic orbit.Where  ∈  × is a constant matrix, (()) 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 where ∈   is the state vector,  ∈  × is a constant matrix with eigenvalues  1 ,  2 , . . .,   , the fractional order  ∈ (0, 1), and system (9) is asymptotically stable if and only if | arg(  )| > /2,  = 1, 2, . . ., .

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  is said to be the interlink node if  belongs to Ũ  , while  is said to be the intralink node if  belongs to    − Ũ  .
According to the diffusive coupling condition of matrix , we have So the error dynamical system is described as follows: In order to make the network achieve cluster synchronization, the control input   is designed as follows: where   > 0 is the feedback control gain, which can adjust the synchronization speed.
It follows from ( 16) and ( 17 where   (⋅) is the eigenvalue of the ℎ node matrix  +   Γ.

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  = 16.Suppose this network consists of three communities.The size of three communities is  1 = 5,  2 = 5, and  3 = 6.The local dynamics of nodes are described by Lorenz system.The fractional-order Lorenz system is given by where  = 10,  = 28,  = 8/3, parameters for which the system exhibits chaotic behavior.The chaotic attractor is depicted by Figure 1.
As can be seen from Figure 2, the synchronization errors () converge to zero. Figure 3 shows that the synchronization errors  12 (),  13 (),  23 () do not converge to zero as  → ∞.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.

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.

3 Figure 1 :
Figure 1: The chaotic orbit of the Lorenz system.