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

Abstract

Synchronization is the strongest form of collective phenomena in complex systems of interacting components. In this paper, the problem of cluster projective synchronization of complex networks with fractional-order nodes based on the fractional-order differential equation stability theory is investigated. Only the nodes in one community which have direct connections to the nodes in other communities are controlled. Some sufficient synchronization conditions are derived via pinning control. Numerical simulations are provided to show the effectiveness of the theoretical results.

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 _𝑎𝐷^𝑞_𝑡 which is defined by _𝑎𝐷^𝑞_𝑡=𝑑^𝑞𝑑𝑡^𝑞𝑅𝑞𝑞>01𝑅=0_𝑡𝑎(𝑑𝜏)^−𝑞𝑅𝑞<0, (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 _𝑎^GL𝐷^𝑞_𝑡𝑓(𝑡)=lim_ℎ→0𝑓_ℎ^{(𝑞)[(𝑡−𝑎)/ℎ]}_𝑗=0(−1)^𝑗𝑞𝑗𝑓𝑡−𝑗ℎ, (2) where the symbol [⋅] means the integer part.

The Riemann-Liouville (RL) definition of fractional derivatives is given by _𝑎^GL𝐷^𝑞_𝑡𝑓(𝑡)=1Γ×𝑛−𝑞_𝑡𝑎𝑓^(𝑛)(𝜏)(𝑡−𝜏)^{𝑞−𝑛+1}𝑑𝜏,𝑛−1<𝑞<𝑛, (3) where Γ(⋅) is the gamma function.

The Caputo (𝐶) fractional derivative is defined as follows: ^𝐶_𝑎𝐷^𝑎_𝑡𝑓(𝑡)=1Γ𝑛−𝑞^𝑡_𝑎𝑓^(𝑛)(𝜏)(𝑡−𝜏)^{𝑞−𝑛+1}𝑑𝜏,𝑛−1<𝑞<𝑛. (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≤𝑚≤𝑁. Let {𝐶₁,𝐶₂,…,𝐶_𝑚} denote 𝑚 communities of complex networks and ∪^𝑚_𝑖=1𝐶_𝑖={1,2,…,𝑁}. Without loss of generality, let 𝐶₁={1,2,…,𝑟₁}, 𝐶₁={𝑟₁+1,…,𝑟₁+𝑟₂},…,𝐶_𝑚={𝑟₁+⋯+𝑟_𝑚−1+1,…,𝑟₁+⋯+𝑟_𝑚−1+𝑟_𝑚} with 𝑟+⋯+𝑟_𝑚−1+𝑟_𝑚=𝑁. If node 𝑖 belongs to the 𝑗th community then let 𝛿_𝑖=𝑗. Denote by 𝑈_𝛿𝑖 the set of all nodes in the 𝛿_𝑖th cluster. 𝑈_𝛿𝑖 represents all the nodes in the 𝛿_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^𝑞_∗𝑥_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+𝜀^𝑁_𝑖=1𝑐_𝑖𝑗Γ𝑥_𝑗(𝑡)𝑖=1,2,…,𝑁, (5) where 0<𝑞<1 and 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛)^𝑇∈𝑅^𝑛 is the state variable of the node 𝑖. 𝑓∶𝑅^𝑛→𝑅^𝑛 describes the dynamics of nodes and is differential and capable of performing abundant dynamical behaviors. 𝜀>0 is the coupling strength and Γ=diag(𝛽₁,𝛽₂,…,𝛽_𝑛) is the inner-coupling matrix. The matrix 𝐶=(𝑐_𝑖𝑗)_𝑁×𝑁 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_𝑖𝑗=𝑐_𝑗𝑖>0(𝑖≠𝑗); otherwise 𝑐_𝑖𝑗=𝑐_𝑗𝑖=0(𝑖≠𝑗); let 𝑐_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑖≠𝑗}𝑐_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_𝑖∈𝑅^𝑛 are controllers to be designed later. For simplicity of further discussion, decompose the function 𝑓(𝑥_𝑖(𝑡)) into two parts, 𝐴(𝑥_𝑖(𝑡))+𝐹(𝑥_𝑖(𝑡)), where 𝐴 is an 𝑛×𝑛 constant matrix and 𝐹∶𝑅^𝑛→𝑅^𝑛 are nonlinear vector-valued functions.

Then the controlled network can be rewritten as 𝐷^𝑞_∗𝑥_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝐹𝑥_𝑖(𝑡)+𝜀^𝑁_𝑗=1𝑐_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑢_𝑖𝑖=1,2,…,𝑁. (6)

Definition 1. Let {1,2,…,𝑁} be the 𝑁 nodes of the networks and let {𝐶₁,𝐶₂,…,𝐶_𝑚} be the 𝑚 communities, respectively. 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,𝑛₁≠𝑛₂=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_𝑖th community; we assume that the smooth goal dynamics can be described by 𝐷^𝑞_∗𝑆_𝛿𝑖(𝑡)=𝐵𝑆_𝛿𝑖(𝑡)+𝑔𝑆_𝛿𝑖(𝑡), (9) where 𝑆_𝛿𝑖(𝑡) may be an equilibrium point, a periodic orbit, or even a chaotic orbit. Where 𝐵∈𝑅^𝑛×𝑛 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^𝑞_∗𝑥=𝐴𝑥,𝑤𝑖𝑡ℎ𝑥(0)=𝑥₀, (10) where 𝑥∈𝑅^𝑛 is the state vector, 𝐴∈𝑅^𝑛×𝑛 is a constant matrix with eigenvalues 𝜆₁,𝜆₂,…,𝜆_𝑛, the fractional order 𝑞∈(0,1), and system (9) is asymptotically stable if and only if |arg(𝜆_𝑖)|>𝑞𝜋/2, 𝑖=1,2,…,𝑛.

Lemma 4 (see [24]). Assume 𝐴∈𝑅^𝑁×𝑁 satisfies the following conditions:(1)𝑎_𝑖𝑗≥0(𝑖≠𝑗), 𝑎_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_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)^𝑇corresponding to the eigenvalue 0; (iii) let 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑁)^𝑇 be the left eigenvector of 𝐴 corresponding to the eigenvalue 0; then we can let 𝜉_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(𝑘₁,𝑘₂,…,𝑘_𝑛) with 𝑘_𝑖≥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 𝑈_𝛿𝑖, while 𝑖 is said to be the intralink node if 𝑖 belongs to 𝑈_𝛿𝑖−𝑈_𝛿𝑖.

According to the diffusive coupling condition of matrix 𝐴, we have 𝜀^𝑁_𝑗=1𝑐_𝑖𝑗Γ𝑆_𝛿𝑖(𝑡)=0,𝑈_𝛿𝑖−𝑈_𝛿𝑖. (11)

So the error dynamical system is described as follows: 𝐷^𝑞_∗𝑒_𝑖(𝑡)=𝐴𝑒_𝑖(𝑡)+𝐹𝑥_𝑖(𝑡)+𝑀(𝐴−𝐵)𝑆_𝛿𝑖(𝑡)−𝑀𝑔𝑆_𝛿𝑖(𝑡)+𝜀^𝑁_𝑗=1𝑐_𝑖𝑗Γ𝑒_𝑗(𝑡)+𝑢_𝑖(𝑡). (12) In order to make the network achieve cluster synchronization, the control input 𝑢_𝑖 is designed as follows: 𝑢_𝑖=−𝐹𝑥_𝑖(𝑡)−𝑀(𝐴−𝐵)𝑆_𝛿𝑖+𝑀𝑔𝑆_𝛿𝑖(𝑡)−𝑘_𝑖𝑒_𝑖(𝑡)𝑖∈𝑈_𝛿𝑖,0𝑖∈𝑈_𝛿𝑖−𝑈_𝛿𝑖, (13) where 𝑘_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 arg𝐴+𝜆_𝑖𝜀Γ>𝑞𝜋2,𝑖=1,2,…,𝑁. (14)

Proof. Combining (6) and (13), one has 𝐷^𝑞_∗𝑒_𝑖(𝑡)=𝐴𝑒_𝑖(𝑡)+𝜀^𝑁_𝑖=1𝑐_𝑖𝑗Γ𝑒_𝑗(𝑡)−𝑘_𝑖𝑒_𝑖(𝑡),1≤𝑖≤𝑁. (15) Denote 𝐾=diag(𝑘₁,𝑘₂,,…,𝑘_𝑁) with 𝑘_𝑖=0 for all 𝑈_𝛿𝑖−𝑈_𝛿𝑖, 𝑒(𝑡)=(𝑒₁(𝑡),𝑒₂(𝑡),…,𝑒_𝑁(𝑡))∈𝑅^𝑛×𝑁; thus one can obtain the following equation: 𝐷^𝑞_∗𝑒(𝑡)=𝐴𝑒(𝑡)+𝜀(𝐶−𝐾)𝑒(𝑡)Γ. (16) Since 𝐶 is an irreducible matrix and 𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑁) with 𝑘_𝑖≥0(𝑖=1,2,…,𝑁), according to Lemmas 4 and 5, there exists unitary 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑁) such that (𝐶−𝐾)𝜉=𝜉Λ, (17) where Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑁) and 𝜆_𝑖 is the eigenvalue of matrix 𝐶−𝐾and satisfies 0>𝜆₁≥𝜆₂≥⋯≥𝜆_𝑁=𝜆_min(𝐶−𝐾).
It follows from (16) and (17) that 𝐷^𝑞_∗𝑒(𝑡)=(𝐴𝑒(𝑡))𝜉+𝜀Γ𝑒(𝑡)𝜉Λ. (18) Denote 𝛽(𝑡)=𝑒(𝑡)𝜉, 𝛽(𝑡)=(𝛽₁(𝑡),𝛽₂(𝑡),…,𝛽_𝑁(𝑡)); thus 𝐷^𝑞_∗𝛽(𝑡)=𝐴𝛽(𝑡)+𝜀Γ𝛽(𝑡)Λ; (19) that is, 𝐷^𝑞_∗𝛽_𝑖(𝑡)=𝐴+𝜆_𝑖𝜀Γ𝛽_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+𝜆_𝑖𝜀Γ satisfy |arg(𝐴+𝜆_𝑖𝜀Γ)|>𝑞𝜋/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 arg𝜆_𝑖𝑘𝐴+𝜆_𝑖𝜀Γ>𝑞𝜋2,𝑘=1,2,…,𝑛,𝑖=1,2,…,𝑁, (21) where 𝜆_𝑖𝑘(⋅) is the eigenvalue of the 𝑖𝑡ℎ node matrix 𝐴+𝜆_𝑖𝜀Γ.

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 𝑁₁=5, 𝑁₂=5, and 𝑁₃=6. The local dynamics of nodes are described by Lorenz system. The fractional-order Lorenz system is given by 𝐷^𝑞_∗𝑥₁=𝑎𝑥₂−𝑥₁,𝐷^𝑞_∗𝑥₂=𝑏𝑥₁−𝑥₂−𝑥₁𝑥₃,𝐷^𝑞_∗𝑥₃=𝑥₁𝑥₂−𝑐𝑥₃, (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.

Figure 1 

The chaotic orbit of the Lorenz system.

Without loss of generality, we suppose that 𝐶₁={1,2,3,4,5}, 𝐶₂={6,7,8,9,10}, and 𝐶₃={11,12,13,14,15,16}.

In the following simulations, inner-coupling matrix Γ is chosen as identity matrix. The following quantities are used to investigate the process of cluster projective synchronization: 𝐸(𝑡)=^𝑁_𝑖=1𝑥_𝑖(𝑡)−𝑀𝑆_𝛿𝑖(𝑡,𝐸₁₂)(𝑡)=^𝑁_𝑖=1𝑥_𝑢(𝑡)−𝑥_𝑣(𝑡),𝑢∈𝐶₁,𝑣∈𝐶₂,𝐸₁₃(𝑡)=𝑥_𝑢(𝑡)−𝑥_𝑣(𝑡),𝑢∈𝐶₁,𝑣∈𝐶₃,𝐸₂₃(𝑡)=𝑥_𝑢(𝑡)−𝑥_𝑣(𝑡),𝑢∈𝐶₂,𝑣∈𝐶₃, (23) where 𝐸(𝑡) is the total error of cluster projective synchronization for all communities and 𝐸₁₂(𝑡),𝐸₁₃(𝑡),𝐸₂₃(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.

Figure 2 

Time evolution of synchronization error 𝐸(𝑡).

Figure 3 

Time evolution of synchronization error 𝐸(𝑡).

As can be seen from Figure 2, the synchronization errors 𝐸(𝑡) converge to zero. Figure 3 shows that the synchronization errors 𝐸₁₂(𝑡),𝐸₁₃(𝑡),𝐸₂₃(𝑡) do not converge to zero as 𝑡→∞. 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.

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.