Adaptive Cluster Synchronization for Nondelayed and Delayed Coupling Complex Networks with Nonidentical Nodes

Abstract

We focus on the cluster synchronization problem for a kind of general networks with nondelayed and delayed coupling. Based on the pinning control scheme, a small fraction of the nodes in each cluster are pinned for successful control, and the states of the whole dynamical networks can be globally forced to the objective cluster states. Sufficient conditions are derived to guarantee the realization of the cluster synchronization pattern for all initial values by means of the Lyapunov stability theorem and linear matrix inequalities (LMIs). By using the adaptive update law, relative smaller control gains are obtained, and hence the control cost can be substantially lower. Numerical simulations are also exploited to demonstrate the effectiveness and validity of the main result.

1. Introduction

In the past few decades, synchronization and control problems are being widely studied [1–9]. Synchronization can be understood as the adjustment of rhythms or coherence of states by interaction [8] and realized via a sufficient information exchange among the nodes’ interconnection in dynamical networks. The meaning of synchronization can interpret the essence of the collective behavior in nature and it has many potential applications in such areas as biological systems [1], secure communications [3], information processing, mechanical engineering, identification and pattern recognition [4], neuronal networks [2, 6, 7, 9], and so forth. In literature, there are many widely studied synchronization patterns, which define the correlated in-time behaviors among the nodes in a dynamical network, such as phase synchronization [3], impulsive synchronization [4], and complete synchronization [5].

Cluster synchronization has been well studied in recent works [10–14]. It requires that the coupled oscillators split into subgroups called clusters, such that the oscillators synchronized with one another in the same cluster, but there is no synchronization among different clusters. In [11], Wu et al. acquired a general network to a selected cluster synchronization pattern by means of a pinning control strategy. Wang et al. in [12] considered the dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community, a way to design the controllers is very far-reaching, which motivates the present study. In [13], the cluster synchronization in an array of hybrid coupled neural networks with delay was investigated by Cao and Li. A method to realize cluster synchronization by constructing a special coupling matrix was proposed and several sufficient conditions for cluster synchronization based on the Lyapunov stability theory and the linear matrix inequality (LMI) technique were derived.

Some networks could achieve synchronization by utilizing information exchanges among local connections, while most networks in reality are generated randomly, which means they could not fulfill synchronization only depending on the coupling without using external forces. Hence, in many cases, negative state feedback control schemes are necessary to be designed to force the final synchronous state into a certain required objective trajectory. In the previous work [11], Wu et al. have proved that one can always pin a coupled complex network by adding some negative feedback controllers if the control strength is large enough. However, in practice, it is not allowed that the control gains are arbitrarily large. For synchronization, it was pointed out that theoretical values of the control gains are much larger than needed in practice. Therefore, one may ask: Can we find the sharp bound of the control gains? Similarly, in a pinning process, it is also important to make the control strength as small as possible. Some results about the adaptive adjustment control were presented in [15–19]. In [15], Lu and Cao proposed a simple scheme for the synchronization of an uncertain complex dynamical network with delayed coupling. The adaptive controllers to ensure the states of uncertain dynamical network with coupling delays to globally asymptotically synchronize were designed. A class of chaotic neural networks with known or unknown parameters by using the adaptive control and linear feedback with update law was investigated by He and Cao in [16]. They derived an adaptive feedback scheme to achieve synchronization of two coupled neural networks with a time-varying delay.

It is worth noticing that most of the studies on cluster synchronization for complex dynamical networks have been performed under some implicit assumptions that there exists the information communication of nodes via the edges only at time 𝑡 or at time 𝑡−𝜏. However, in many circumstances, this simplification does not match satisfactorily the peculiarities of real networks: there exists the information communication of nodes not only at time 𝑡 or 𝑡−𝜏 but also at time 𝑡, 𝑡−𝜏. This general complex dynamical network model with time nondelayed and delayed coupling was investigated in [14, 20–23] and so forth. In [21], Xu and Su studied the general complex dynamical networks for synchronization and pinning control, they obtained less conservative criteria for both continuous-time and discrete-time complex dynamical networks with time delayed and nondelayed coupling by linearizing the system, and hence their results were established locally. Lou et al. discussed the adaptive synchronization of two complex networks with nondelayed and delayed couplings in [22]. They found that the coupling configuration matrices were not necessarily symmetric or irreducible and presented several criteria for synchronization of the two complex networks based on the Lyapunov stability theory. The exponential cluster synchronization of coupled impulsive genetic oscillators with external disturbances and communication delay was investigated by Zhang et al. in [14]. Some cluster synchronization criteria for the delayed and nondelayed coupling impulsive genetic oscillators were presented and the maximal allowable bound of time delay was obtained as well. However, the cluster synchronization problems for the nondelayed and delayed coupling networks in [14] were established based on the strong assumption that all of the nodes have the same dynamical behaviors.

In this paper, we study the cluster synchronization problem for a kind of general complex dynamical networks with time nondelayed and delayed coupling. We emphasize that the dynamical behaviors of the nodes in different clusters are distinct too. Some global synchronization criteria are derived by means of the pinning control method. The adaptive adjustment controllers are designed skillfully as they have the potential ability to obtain more suitable control gains, which are more important in reality. Moreover, we apply the Lyapunov stability theorem to ensure the realization of cluster synchronization. The required global synchronization criteria then follows immediately from the LMIs.

The rest of the paper is organized as follows. In Section 2, we present some preparations and the general complex dynamical network model. In Section 3, we analyze the cluster synchronization of the network with time nondelayed and delayed coupling under the adaptive control. Numerical simulations are given to verify our theoretical results in Section 4. Concluding remarks are drawn in Section 5.

Notations. The mark 𝐴^𝑇 denotes the transport of the matrix 𝐴. 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space. 𝑅^𝑛×𝑛 are 𝑛×𝑛 real matrices. diag{⋯} stands for a diagonal matrix. The sign ‖⋅‖ stands for the Euclid norm of the matrix or the vector. A symmetric real matrix 𝐴 is positive definite (semidefinite) if 𝑥^𝑇𝐴𝑥>0(≥0) for all nonzero 𝑥 we denote this as 𝐴>0(𝐴≥0). 𝐼 stands for the identity matrix with proper dimension. The dimension of these vectors and matrices will be cleared in the context.

2. Preliminaries and Model Description

We first make some assumptions for a network with cluster structure as follows. Suppose that these networks have 𝑁 nodes and 𝑚 clusters with 𝑁>𝑚≥2. Let 𝜇_𝑖=𝑗 if node 𝑖 belongs to the 𝑗th cluster. Denote 𝑈_𝑗 as the set of all nodes in the 𝑗th cluster and 𝑈_𝑗 as the set consisting of all nodes in the 𝑗th cluster which have direct connections to the nodes in other clusters. Thus we have (1)𝑈_𝑖⋂𝑈_𝑗=𝜙, for 𝑖≠𝑗 and 𝑖,𝑗=1,2,…,𝑚; (2)⋃^𝑚_𝑖=1𝑈_𝑖={1,2,…,𝑁}.

Some preliminaries such as definitions and lemmas will be given in the following, which are necessary throughout the paper.

Definition 1 (see [24]). Synchronization manifold 𝑀=𝑥^𝑇₁,…,𝑥^𝑇_𝑁^𝑇∈𝑅^𝑁×𝑛∶𝑥_𝑖=𝑥_𝑗,𝑖,𝑗=1,2,…,𝑁, (1) where 𝑥_𝑖=(𝑥_𝑖1,…,𝑥_𝑖𝑛)^𝑇∈𝑅^𝑛 for 𝑖=1,2,…,𝑁.

Definition 2 (see [12]). A network with 𝑁 oscillators is said to achieve cluster synchronization if it satisfies lim_𝑡→+∞𝑥_𝑖(𝑡)−𝑠_𝜇𝑖(𝑡)=0,𝑖=1,2,…,𝑁, (2) where 𝑠_𝜇𝑖(𝑡)∈𝑅^𝑛 is a solution of an isolate node and satisfies 𝑠̇_𝜇𝑖(𝑡)=𝑓_𝜇𝑖𝑠_𝜇𝑖(𝑡),𝑖=1,2,…,𝑁, (3) which describes the identical local dynamics for the nodes in the 𝜇_𝑖th cluster (community).

Definition 3 (see [11]). Function class QUAD(Δ,𝑃,𝜂): let Δ=diag{𝛿₁,…,𝛿_𝑛} be a diagonal matrix and 𝑃=diag{𝑝₁,…,𝑝_𝑛} a positive-definite diagonal matrix. QUAD(Δ,𝑃,𝜂) denotes a class of continuous functions 𝑓(𝑥,𝑡)∶𝑅^𝑛×[0,+∞)→𝑅^𝑛 satisfying 𝑥−𝑦^𝑇𝑃𝑓(𝑥,𝑡)−𝑓𝑦,𝑡−Δ𝑥−𝑦≤−𝜂𝑥−𝑦^𝑇𝑥−𝑦 (4) for some 𝜂>0, all 𝑥, 𝑦∈𝑅^𝑛 and all 𝑡≥0.

Lemma 4 (Schur complement [25]). The following linear matrix inequality (LMI) 𝑄(𝑥)𝑆(𝑥)𝑆(𝑥)^𝑇𝑅(𝑥)>0 (5) is equivalent to one of the following conditions:(1)𝑄(𝑥)>0, 𝑅(𝑥)−𝑆(𝑥)^𝑇𝑄(𝑥)⁻¹𝑆(𝑥)>0;(2)𝑅(𝑥)>0, 𝑄(𝑥)−𝑆(𝑥)𝑅(𝑥)⁻¹𝑆(𝑥)^𝑇>0,where 𝑄(𝑥) and 𝑅(𝑥) are symmetric matrices and 𝑆(𝑥) is a matrix with suitable dimensions.

In general, the 𝜇_𝑖th cluster equation with time nondelayed and delayed coupling under pinning control scheme 𝑢_𝑖(𝑡) can be described as 𝑥̇_𝑖(𝑡)=𝑓_𝜇𝑖𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑥_𝑗(𝑡−𝜏)+𝑢_𝑖(𝑡),𝑖=1,…,𝑁, (6) where 𝑁 is the number of nodes and 𝑥_𝑖(𝑡)=(𝑥_𝑖1,…,𝑥_𝑖𝑛)^𝑇∈𝑅^𝑛 are the state variables of the 𝑖th node, 𝑡∈[0,+∞). 𝑓_𝜇𝑖∶𝑅^𝑛→𝑅^𝑛 is a continuously differentiable function describes the local dynamics of nodes in the 𝜇_𝑖th cluster. For any pair of indices 𝑖 and 𝑗, if node 𝑖 and node 𝑗 belong to different clusters, that is, 𝜇_𝑖≠𝜇_𝑗, then 𝑓_𝜇𝑖≠𝑓_𝜇𝑗. Γ=diag{𝑟₁,𝑟₂,…,𝑟_𝑛}∈𝑅^𝑛×𝑛 is the inner-coupling matrix with 𝑟_𝑖≥0, 𝑖=1,2,…,𝑛. 𝐴=(𝑎_𝑖𝑗) and 𝐵=(𝑏_𝑖𝑗)∈𝑅^𝑁×𝑁 are the coupling matrices with zero-sum rows 𝑎_𝑖𝑖=−∑^𝑁_{𝑗≠𝑖,𝑗=1}𝑎_𝑖𝑗, 𝑏_𝑖𝑖=−∑^𝑁_{𝑗≠𝑖,𝑗=1}𝑏_𝑖𝑗, 𝑎_𝑖𝑗>0,𝑏_𝑖𝑗>0 for 𝑖,𝑗=1,…,𝑁, which are determined by the topological structure of the networks. 𝑢_𝑖(𝑡)(𝑖=1,2,…,𝑁) are some local negative feedback controllers to be designed later. And we assume that 𝐴 and 𝐵 are symmetric and irreducible matrices. That means there is no isolate node in the system. This diffusivity condition implies that the coupling dynamical network would be decoupled when the array of nodes is synchronized.

3. The Stability Analysis for Cluster Synchronization under Pinning Adaptive Control

In this section, the cluster synchronization problem amounts to proving the stability of the error system. By using a suitably chosen control scheme, that is, pinning adaptive controllers that control only a fraction of the nodes in every cluster which have direct connections with the nodes in other clusters will be designed to globally stabilize the dynamical network with nondelayed and delayed coupling. We present the main result of this paper in the form of Theorem 6, then give the detailed proof process.

Let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠_𝜇𝑖(𝑡) (7) be the error state of node 𝑖 between the current state 𝑥_𝑖(𝑡) and the cluster objective state 𝑠_𝜇𝑖(𝑡), where 𝑠_𝜇𝑖(𝑡) are defined as in Definition 2 for 𝑖=1,…,𝑁; they are the cluster objective trajectories that the dynamical network (6) will be forced to.

Transform the system (6) yields the following error dynamical system: 𝑒_𝑖̇(𝑡)=𝑓_𝜇𝑖𝑥_𝑖(𝑡)−𝑓_𝜇𝑖𝑠_𝜇𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑗(𝑡−𝜏)+𝑢_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑠_𝜇𝑖(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑠_𝜇𝑖(𝑡−𝜏),𝑖=1,2,…,𝑁. (8)

According to the zero-row-sum condition, it is not difficult to get ^𝑁_𝑗=𝑖𝑎_𝑖𝑗Γ𝑠_𝜇𝑖(𝑡)=0,^𝑁_𝑗=𝑖𝑏_𝑖𝑗Γ𝑠_𝜇𝑖(𝑡−𝜏)=0 (9) if and only if 𝑖∈𝑈_𝜇𝑖⧵𝑈_𝜇𝑖 for 𝑖=1,2,…,𝑁.

In view of this special property, we design the local linear negative feedback controllers as 𝑢_𝑖(𝑡)=−𝑑_𝑖(𝑡)𝑒_𝑖(𝑡)−∑^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑠_𝜇𝑖(𝑡)−∑^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑠_𝜇𝑖(𝑡−𝜏),𝑖∈𝑈_𝜇𝑖,0,𝑖∈𝑈_𝜇𝑖⧵𝑈_𝜇𝑖, (10) where 𝑑_𝑖(𝑡)>0, 𝑖=1,2,…,𝑁, are the time-varying adaptive control gains that can be suitably chosen by the cluster synchronization system (6). To guarantee negative feedback, the adaptive gains are designed as 𝑑̇_𝑖(𝑡)=𝛼_𝑖𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡),𝑖=1,2,…,𝑁, (11) where 𝛼_𝑖, 𝑖=1,2,…,𝑁, are positive constants to be determined.

Remark 5. Through the above controllers, the first term in (10) is used to synchronize all nodes in the same cluster, while the remainder term in the controller is to weaken the mutual influences among clusters at the intersection nodes. By the way, we design the adaptive gains as (11) (adaptive update law) in order to make the control strength as small as possible, and hence the control cost can be substantially lower and our strategy can be easier to implement.

In the following, the main result of cluster synchronization criteria for nondelayed and delayed coupling under the pinning adaptive control scheme (10) and the adaptive update law (11) will be given.

Theorem 6. Suppose 𝑓_𝜇𝑖(⋅)∈QUAD(𝑃,Δ,𝜂), 𝑖=1,…,𝑁, such that 𝑟_𝑘𝐴+𝛿_𝑘1𝐼−𝐷+𝑄2𝑟_𝑘𝐵12𝑟_𝑘𝐵−𝑄<0 (12) hold for all 𝑘=1,2,…,𝑛, then the cluster synchronization manifold 𝑀=𝑠_𝜇1,…,𝑠_𝜇𝑁 (13) of the network (6) is globally stable for any time delay 𝜏>0 and for any initial values under the pinning state feedback controllers (10) and the adaptive update law (11), where 𝑄∈𝑅^𝑛×𝑛 is a positive definite diagonal matrix and 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑁}.

Proof. For notational convenience, we denote 𝑒_𝑖̃(𝑡)=𝑒_1𝑖(𝑡),…,𝑒_𝑁𝑖(𝑡)^𝑇,𝑒_𝑖̃(𝑡−𝜏)=𝑒_1𝑖(𝑡−𝜏),…,𝑒_𝑁𝑖(𝑡−𝜏)^𝑇. (14) Select the following Lyapunov functional candidate as 𝑉(𝑡)=12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)+^𝑁_𝑖=112𝛼_𝑖𝑑_𝑖(𝑡)−𝑑_𝑖²+^𝑛_𝑘=1𝑝_𝑘^𝑡_𝑡−𝜏𝑒_𝑘̃(𝑠)^𝑇𝑄𝑒_𝑘̃(𝑠)𝑑𝑠, (15) where 𝑑_𝑖, 𝑖=1,2,…,𝑁, are positive constants to be determined. The derivative of 𝑉(𝑡) along the trajectories of the error system (8) with considering the pinning adaptive control scheme (10) and the adaptive update law (11) as described above is 𝑉̇(𝑡)=^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖̇(𝑡)+^𝑁_𝑖=11𝛼_𝑖𝑑_𝑖(𝑡)−𝑑_𝑖𝑑̇_𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝑄𝑒_𝑘̃(𝑡)−𝑒_𝑘̃(𝑡−𝜏)^𝑇𝑄𝑒_𝑘̃(=𝑡−𝜏)_𝑁𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓_𝜇𝑖𝑥_𝑖(𝑡)−𝑓_𝜇𝑖𝑠_𝜇𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑒_𝑗(𝑡−𝜏)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑠_𝜇𝑖(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑠_𝜇𝑖(𝑡−𝜏)+𝑢_𝑖(𝑡)+^𝑁_𝑖=11𝛼_𝑖𝑑_𝑖(𝑡)−𝑑_𝑖𝛼_𝑖𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝑄𝑒_𝑘̃(𝑡)−𝑒_𝑘̃(𝑡−𝜏)^𝑇𝑄𝑒_𝑘̃(=𝑡−𝜏)_𝑁𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓_𝜇𝑖𝑥_𝑖(𝑡)−𝑓_𝜇𝑖𝑠_𝜇𝑖(𝑡)−Δ𝑒_𝑖(𝑡+_𝑁𝑖=1)𝑒_𝑖(𝑡)^𝑇𝑃Δ𝑒_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑒_𝑗(𝑡−𝜏)−𝑑_𝑖(𝑡)𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑑_𝑖(𝑡)−𝑑_𝑖𝑃𝑒_𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝑄𝑒_𝑘̃(𝑡)−𝑒_𝑘̃(𝑡−𝜏)^𝑇𝑄𝑒_𝑘̃(𝑡−𝜏). (16) According to Definition 3, we have 𝑉̇(𝑡)≤−𝜂^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝑟_𝑘𝑒_𝑘̃(𝑡)^𝑇𝐴𝑒_𝑘̃(𝑡)+𝑒_𝑘̃(𝑡)^𝑇𝐵𝑒_𝑘̃(+𝑡−𝜏)_𝑛𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝛿_𝑘𝐼−𝐷𝑒_𝑘̃(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝑄𝑒_𝑘̃(𝑡)−^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡−𝜏)^𝑇𝑄𝑒_𝑘̃(𝑡−𝜏)=−𝜂^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝑟_𝑘𝐴+𝛿_𝑘𝐼−𝐷+𝑄𝑒_𝑘̃(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝑟_𝑘𝐵𝑒_𝑘̃()−𝑡−𝜏_𝑛𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡−𝜏)^𝑇𝑄𝑒_𝑘̃(𝑡−𝜏). (17) With an eye to Lemma 4, we could know that if the condition (12) hold for 𝑘=1,…,𝑛, then we can get that 𝑉̇(𝑡)≤−𝜂^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡). (18) According to the Lyapunov stability theorem, one has 𝑒_𝑖(𝑡)→0(𝑡→+∞) (19) whcih equals to lim_𝑡→+∞𝑒_𝑖(𝑡)=lim_𝑡→+∞𝑥_𝑖(𝑡)−𝑠_𝜇𝑖(𝑡)=0,∀𝑖=1,2,…,𝑁. (20) Combine with the designing of the adaptive controllers. It can be derived that 𝑑̇_𝑖(𝑡)→0(𝑡→+∞)⟺𝑑_𝑖(𝑡)→𝑑_𝑖(𝑡→+∞), (21) where 𝑑_𝑖(𝑖=1,…,𝑁) are positive constants, it means that the nodes within 𝜇_𝑖th cluster are fully synchronized to dynamics state 𝑠_𝜇𝑖, while nodes in separate clusters behave independently. Therefore, we get the global stability of the cluster synchronization manifold 𝑀=𝑠_𝜇1,…,𝑠_𝜇𝑁. (22) That completes the proof of the main result.

Remark 7. The main theory explicitly showed a kind of effective and relatively cheap control strategy control scheme designed in this paper for the cluster synchronization of complex dynamical networks. That is, throughout the above proof, we can get some relative proper and practical control gains by using the pinning control scheme (10) as well as the adaptive control gains update law (11). Therefore, we can save much resources and costs for the smaller control gains than the previous works got in [11, 12]. Although one can pin a coupled complex network by adding some negative feedback controllers if the control strength is large enough, it is not practical that the control gains are arbitrarily large.

In the following, we give two corollaries, which are the investigation of simplified complex networks (6). Those results could come to the conclusion that our main result in this paper is more comprehensive and meaningful than the previous ones.

Corollary 8. Suppose 𝑓_𝜇𝑖(⋅)∈QUAD(𝑃,Δ,𝜂), 𝑖=1,…,𝑁, such that 𝛿_𝑘𝐼+𝑟_𝑘𝐴−𝐷<0 (23) hold for 𝑘=1,2,…,𝑛, then the cluster synchronization manifold 𝑀=𝑠_𝜇1,…,𝑠_𝜇𝑁 (24) of the following dynamical complex network 𝑥̇_𝑖(𝑡)=𝑓_𝜇𝑖𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑢_𝑖(𝑡),𝑖=1,…,𝑁 (25) is globally stable for all initial values under the pinning state feedback control scheme 𝑢_𝑖(𝑡)=−𝑑_𝑖(𝑡)𝑒_𝑖(𝑡)−∑^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑠_𝜇𝑖(𝑡),𝑖∈𝑈_𝜇𝑖,0,𝑖∈𝑈_𝜇𝑖⧵𝑈_𝜇𝑖 (26) and the adaptive update law (11), where 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑁}.

Remark 9. The similar proof process of Corollary 8 could be thought of as the simplified process of Theorem 6. Similar conclusion also can be found in [12]. In this special case, we investigate the complex network without time delay.

Corollary 10. Suppose 𝑓_𝜇𝑖(⋅)∈QUAD(𝑃,Δ,𝜂), 𝑖=1,…,𝑁, such that 𝛿_𝑘1𝐼−𝐷+𝑄2𝑟_𝑘𝐵12𝑟_𝑘𝐵−𝑄<0 (27) hold for 𝑘=1,2,…,𝑛, then the cluster synchronization manifold 𝑀=𝑠_𝜇1,…,𝑠_𝜇𝑁 (28) of the time delayed coupling dynamical network 𝑥̇_𝑖(𝑡)=𝑓_𝜇𝑖𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑥_𝑗(𝑡−𝜏)+𝑢_𝑖(𝑡),𝑖=1,…,𝑁 (29) under the pinning control scheme 𝑢_𝑖(𝑡)=−𝑑_𝑖(𝑡)𝑒_𝑖(𝑡)−∑^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑠_𝜇𝑖(𝑡−𝜏),𝑖∈𝑈_𝜇𝑖,0,𝑖∈𝑈_𝜇𝑖⧵𝑈_𝜇𝑖 (30) and the adaptive update law (11) is globally stable for any time delay 𝜏>0, where 𝑄∈𝑅^𝑛×𝑛 is a positive definite diagonal matrix and 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑁}.

Remark 11. The proof of Corollary 10 could be thought of as another simplification of the proof of Theorem 6; this corollary, of course, is another special situation of the main result, that is, the network with time delayed coupling. This corollary can tells the result in this paper including the previous works, so we did a more comprehensive work.

4. Numerical Simulation

The above criteria can be applied to networks with different topologies and sizes. In this section, we put an effective example to illustrate the validity of the theory given in previous sections.

Example 12. We use a chaotic Lorenz oscillator and a chaotic Chua’s oscillator with different parameters, respectively, as the example to illustrate the main results (Theorem 6) of our paper. The chaotic Lorenz oscillator model could be described as follows: 𝑥̇₁𝑥̇₂𝑥̇₃=10𝑥₂−𝑥₁28𝑥₁−𝑥₂−𝑥₁𝑥₃𝑥₁𝑥₂−83𝑥₃, (31) and the chaotic Chua’s oscillator model is 𝑥̇₁𝑥̇₂𝑥̇₃=15𝑥₂−𝑥₁+𝑔𝑥₁𝑥₁−𝑥₂+𝑥₃−25.5𝑥₂, (32) where 𝑔𝑦=573𝑦+−.14𝑦+1𝑦−1 (33) Their vector field functions are denoted by 𝑓₁, 𝑓₂, and 𝑓₃, respectively. According to Theorem 6, the validity of the result must satisfy the QUAD condition, so we should take the first step to verify Definition 3 to guarantee cluster synchronization for the network (6) under the pinning controller (10). To achieve the purpose, Wang et al. did a good job. The detailed proof process could be found in [12]; based on that paper analysis, we can set 𝑃=𝐼₃,Δ=52𝐼₃,Γ=𝐼₃. (34)

The network consists of 6 nodes that are divided into two clusters 𝑈₁={1,2,3} and 𝑈₂={4,5,6} (see Figure 1(a)). From the network structure shown in Figure 1, we can get that the time nondelayed coupling matrix 𝐴 is 𝐴=[−1,1,0,0,0,0;0,−1,1,0,0,0;1,0,−1,0,0,0;0,0,1,−2,1,0;0,0,0,0,−1,1;0,0,0,1,0,−1], (35)and the time delayed coupling matrix we assume is 𝐵=0.5𝐴; both of the matrices satisfy the diffusive condition. Cluster 𝑈₁={1,2,3} follows the dynamics of (30) while cluster 𝑈₂={4,5,6} follows the dynamics of (31). We add the adaptive adjustment controllers on node 3 and 4 as they have direct connection between two clusters.

(a)

(b)

(c)

(d)

Figure 1 

A network with size 𝑁=6 and two clusters shown in (a). (b, c) stand for the cluster synchronization error 𝐸(𝑡) in each cluster and (d) represents the error 𝐸12(𝑡) between the two communities.

By using MATLAB LMI TOOLBOX on condition (11), we derive 𝑄=diag(0.1963,0.4452,0.7425,0.2687), (36) where 𝑄 is a positive definite diagonal matrix which meets the requirement in Theorem 6.

Now define the error 𝐸 of cluster synchronization for this controlled network as 𝐸(𝑡)=1𝑙^𝑙_𝑖=1𝑥_𝑖(𝑡)−𝑠_𝜇𝑖(𝑡), (37) where 𝑙 is the number of nodes in each cluster. Obviously, here 𝑙=3. Define 𝐸₁₂(𝑡)=𝑥_𝑖0(𝑡)−𝑥_𝑗0(𝑡),𝑖₀∈𝑈₁,𝑗₀∈𝑈₂ (38) as the error between two communities.

According to the conditions in Theorem 6, we know that the cluster synchronization manifold 𝑀 is globally stable for any positive time delay 𝜏. We plot the curves of the cluster synchronization errors of each cluster in Figure 1 (Figures 1(b) and 1(c) stand for cluster synchronization errors in cluster 1 and 2, resp., and Figure 1(d) represents for the error between two communities). It is clear that the cluster synchronization is achieved if 𝐸(𝑡) converges to zero and 𝐸₁₂(𝑡) do not as 𝑡→+∞. The pinning adaptive controllers give a successful realization of cluster synchronization with the varying control gain 𝑑(𝑡); in this situation, we derive the control gain matrix 𝐷=24𝐼. It is more practical in real applications than we got in the view of theories.

5. Conclusion

This paper considered a class of complex dynamical networks with time nondelayed and delayed coupling. Instead of controlling all nodes, pinning adaptive control was used to pin a small fraction of the nodes in each cluster for successfully control, and the states of the whole dynamical networks can be globally exponentially forced to the objective cluster states. We have derived sufficient criteria for cluster synchronization under pinning control as the situation of dynamics are different in every cluster, which is more practical and meaningful in realistic. The adaptive method to adjust the control gains suitably rather than as large as possible was introduced and global synchronization of the general complex networks were achieved by using the Lyapunov stability theorem and linear matrix inequality. The numerical simulation was exploited to demonstrate the effectiveness of the pinning adaptive strategy proposed in this paper.

Acknowledgments

The authors thank the referees and the editors for their valuable comments on this paper. The work is supported by the Natural Science Foundation of China (Grant no. 61273220), Guangdong Education University Industry Cooperation Projects (Grant no. 2009B090300355), and the Shenzhen Basic Research Project (JC201006010743A, JCYJ20120613105730482).