Cluster Synchronization of Nonlinearly Coupled Complex Networks via Pinning Control

Abstract

We consider a method for driving general complex networks into prescribed cluster synchronization patterns by using pinning control. The coupling between the vertices of the network is nonlinear, and sufficient conditions are derived analytically for the attainment of cluster synchronization. We also propose an effective way of adapting the coupling strengths of complex networks. In addition, the critical combination of the control strength, the number of pinned nodes and coupling strength in each cluster are given by detailed analysis cluster synchronization of a special topological structure complex network. Our theoretical results are illustrated by numerical simulations.

1. Introduction

Complex networks synchronization is an important phenomenon in both mathematical and physical sciences because of its myriad applications to diverse problems such as communications security, seismology, parallel image processing as well as many others [1–7]. Loosely speaking, synchronization is the process in which two (or more) dynamical systems seek to adjust a certain prescribed property of their motion to a common behavior in the limit as time tends to infinity either by virtue of coupling or by forcing [8]. Some common synchronization patterns that have been widely studied are complete synchronization [9], lag synchronization [10], cluster synchronization [11], phase synchronization [12], and partial synchronization [13, 14].

Since Pecora and Carroll [15] found the chaos synchronization in 1990, synchronization has been widely studied because of its potential application in many different areas. Rulkov et al. [16] explored generalized synchronization of chaos in directionally coupled chaotic systems. (i.e., a response system is driven with the output of a driving system, but there is no feedback to the driver.) Hramov and Koronovskii [17] proposed an approach to the synchronization of chaotic oscillators based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators and the quantitative measure of chaotic oscillator synchronous behavior.

In applications, many coupled oscillators are organized into subgroups called clusters, and cluster synchronization is the phenomenon in which all the nodes in one cluster are synchronized and those in different clusters are not. Cluster synchronization is an interesting phenomenon that has great application potentials, and many results have recently appeared that are of importance to our understanding of this problem. Ma et al. [18], for example, constructed a new coupling scheme that is capable of stabilizing every cluster synchronization pattern in a connected network with identical nodes by using cooperative and competitive coupling weights. Ma also derived a sufficient condition for the global stability of the synchronization patterns. Similarly, Wu et al. [19] showed how to construct the coupling matrix and modify the control strengths of a linearly coupled complex network with identical nodes by using pinning control. Lu et al. [20] investigated the cluster synchronization of dynamical networks with community structure and nonidentical nodes in the presence of time delays by applying the conventional feedback control schemes. Then, Lu et al. [21] carried on to study the cluster synchronization of general bi-directed networks with non-identical clusters and demonstrated a possible connection between the feasibility of cluster synchronization and the ratio of intra-to-inter cluster links of the network. Zhang et al. [22] found that different topologies of intercluster couplings might lead to different synchronizability, and different synchronous phases were revealed by varying the intercluster and intracluster coupling strengths.

Many cluster synchronization studies that have been conducted so far, however, have been confined to the smaller and less applicable class of complex networks that are linearly coupled [18–21] thus leaving the more important and naturally occurring nonlinearly coupled ones virtually uninvestigated. Nonlinearly coupled networks abound in nature (common examples are neural and metabolic networks in which the coupling configurations are oscillate continuously between two fixed states), and this paper attempts to fill this gap in our knowledge by investigating the nonlinearly coupled problem using pinning control. Pinning control is a very effective scheme for synchronizing chaos in complex networks by controlling only a small number of the system nodes [23], and it is a method that has been successfully applied to many spatially extended systems in the 1990s.

The main aim of this paper is to (nontrivially) extend the results of Wu et al. [19] to the more general problem of cluster synchronizing nonlinearly coupled complex networks by transforming the nonlinear coupling function into a linear. Our results are anticipated to be applicable to many cluster synchronization problems in various fields of science and technology.

This paper is organized as follows. In Section 2, we propose a model for nonlinearly coupled complex networks and state some necessary definitions, lemmas, and assumptions. In Section 3, we study the global cluster synchronization problem of networks by using pinning control and derive a sufficient condition for its attainment. Some adaptive feedback algorithms are also proposed for the coupling strengths of practical real-world networks. Numerical simulation results are presented in Section 4. Section 5 concludes the paper.

2. Model and Preliminaries

First, we give the mathematical definition of cluster synchronization.

Definition 2.1 (see [18, 19]). Let 𝐺₁={1,2,…,𝑘₁}, 𝐺₂={𝑘₁+1,𝑘₁+2,…,𝑘₁+𝑘₂},…,𝐺_𝑑={𝑘₁+𝑘₂+⋯+𝑘_𝑑−1+1,…,𝑘₁+𝑘₂+⋯+𝑘_𝑑} be a partition of the set 𝐺={1,2,…,𝑚} for 1<𝑑<𝑚, 1<𝑘_𝑙<𝑚, and ∑^𝑑_𝑙=1𝑘_𝑙=𝑚, and, for every 𝑖∈𝐺, let 𝑖̃ be the counting index of the subset in which the number 𝑖 lies, that is, 𝑖∈𝐺_̃𝑖. A network with 𝑚 nodes is said to realize cluster synchronization with partition {𝐺₁,𝐺₂,…,𝐺_𝑑} if the state variables of the nodes satisfy lim_𝑡→+∞‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖=0 for 𝑖̃=𝑗̃ and lim_𝑡→+∞‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖≠0 for 𝑖̃≠𝑗̃ for all initial values.

Next, we state the problem under consideration. A general nonlinearly coupled complex dynamical network with 𝑚 identical nodes can be described as𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡+^𝑚_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝑖=1,2,…,𝑚, (2.1) where 𝑥_𝑖(𝑡)=[𝑥¹_𝑖(𝑡),𝑥²_𝑖(𝑡),…,𝑥^𝑛_𝑖(𝑡)]^𝑇∈𝑅^𝑛 is the state vector of the 𝑖th node, 𝑓∶𝑅^𝑛×𝑅⁺→𝑅^𝑛 is a vector-valued function representing the activity of an individual subsystem, 𝑔(𝑥_𝑖(𝑡))=[ℎ(𝑥¹_𝑖(𝑡)),ℎ(𝑥²_𝑖(𝑡)),…,ℎ(𝑥^𝑛_𝑖(𝑡))]^𝑇 where ℎ∶𝑅→𝑅 is a nonlinear function, Γ=[𝛾_𝑖𝑗]∈𝑅^𝑛×𝑛 is the inner coupling matrix, and 𝐴=[𝑎_𝑖𝑗]∈𝑅^𝑚×𝑚 is the diffusive coupling matrix with 𝑎_𝑖𝑗 being the coupling weight along the edge from vertex 𝑗 to vertex 𝑖 and 𝑎_𝑖𝑖=−∑^𝑚_{𝑗=1,𝑖≠𝑗}𝑎_𝑖𝑗 for 𝑖,𝑗=1,2,…,𝑚.

When control is introduced, the nonlinearly coupling network (2.1) becomes 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡+^𝑚_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡)+𝑢_𝑖(𝑡),𝑖∈𝐽⁰,𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡+^𝑚_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝑖∉𝐽⁰, (2.2) where 𝐽⁰ is a subset of {1,2,…,𝑚} denoting the controlled nodes set and 𝑢_𝑖(𝑡) denotes the control on the node 𝑖∈𝐽⁰.

In this paper, we use the following approach to realize the given cluster synchronization pattern: (1) select 𝑑 special solutions 𝑠₁(𝑡),𝑠₂(𝑡),…,𝑠_𝑑(𝑡) of the homogenous system𝑠̇(𝑡)=𝑓(𝑠(𝑡),𝑡) (2.3) with distinct initial values such that lim_𝑡→+∞‖𝑠_𝑢(𝑡)−𝑠_𝑣(𝑡)‖≠0 for 𝑢≠𝑣. Then cluster synchronization is equivalent to the synchronization of the states 𝑥_𝑖(𝑡) with the states 𝑠_̃𝑖(𝑡), that is, lim_𝑡→+∞‖𝑥_𝑖(𝑡)−𝑠_̃𝑖(𝑡)‖=0 for 𝑖=1,2,…,𝑚. (2) Pinning controllers are added to some of the nodes in every cluster, without loss of generality, let the first 𝑙_𝑢 (1≤𝑙_𝑢≤𝑘_𝑢) nodes be selected to be pinned in every cluster; so let the controlled nodes set 𝐽⁰ be 𝐽=⋃^𝑑_𝑢=1𝐽_𝑢, 𝐽_𝑢={𝑏_𝑢−1+1,𝑏_𝑢−1+2,…,𝑏_𝑢−1+𝑙_𝑢}, 𝑏₀=0, and 𝑏_𝑢=∑^𝑢_𝑟=1𝑘_𝑟. We introduce nonlinear feedback pinning controllers 𝑢_𝑖(𝑡)=−𝜀_̃𝑖Γ𝑔𝑥_𝑖(𝑡)−𝑔𝑠_̃𝑖(𝑡),𝑖∈𝐽, (2.4) where 𝜀_̃𝑖>0 with 𝑖∈𝐽 are the control strengths. Then the following coupling system is considered:𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡+^𝑚_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡)−𝜀_̃𝑖Γ𝑔𝑥_𝑖(𝑡)−𝑔𝑠_̃𝑖(𝑡𝑥̇),𝑖∈𝐽,_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡+^𝑚_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝑖∉𝐽. (2.5) (3) Find sufficient conditions for the attainment of cluster synchronization for any initial value by pinning control.

The following are some definitions, lemmas, and notations which will be used throughout this paper.

Definition 2.2. If 𝐴=[𝑎_𝑖𝑗]∈𝑅^𝑚×𝑚 is an irreducible matrix such that 𝑎_𝑖𝑗=𝑎_𝑗𝑖≥0 for all 𝑖≠𝑗 and ∑^𝑚_𝑗=1𝑎_𝑖𝑗=0 for all 𝑖=1,2,…,𝑚, then we say that 𝐴∈𝐀_𝟏.

Definition 2.3 (see [18, 19]). Let 𝐴𝐴=¹¹𝐴₁₂⋯𝐴_1𝑑𝐴₂₁𝐴₂₂⋯𝐴_2𝑑𝐴⋮⋮⋱⋮_𝑑1𝐴_𝑑2⋯𝐴_𝑑𝑑 (2.6) be an 𝑚×𝑚 symmetric matrix such that 𝐴_𝑢𝑣∈𝑅^{𝑘𝑢×𝑘𝑣}, 𝑢,𝑣=1,2,…,𝑑. If each block 𝐴_𝑢𝑣 is a zero-row-sum matrix, then we say that 𝐴∈𝐌_𝟏. Furthermore, if 𝐴_𝑢𝑢∈𝐀_𝟏, 𝑢=1,2,…,𝑑, then we say that 𝐴∈𝐌_𝟐. It also follows from the symmetry of 𝐴 that 𝐴_𝑢𝑣=𝐴^𝑇_𝑣𝑢 and 𝐴_𝑢𝑣,𝐴_𝑣𝑢 are zero-row-sum matrices and that 𝐴_𝑢𝑣 is a zero-column-sum matrix.

Definition 2.4 (see [24]). We denote 𝟏_𝑛=[1,1,…,1]^𝑇∈𝑅^𝑛, 𝐼_𝑛=diag{1,1,…,1}∈𝑅^𝑛×𝑛, and define 𝑄_𝑛=𝐼_𝑛−1𝑛𝟏_𝑛⋅𝟏^𝑇_𝑛∈𝑅^𝑛×𝑛. (2.7)

Obviously, −𝑄_𝑛∈𝐀_𝟏. Moreover, it is easy to check that if 𝑀∈𝑅^𝑚×𝑛 is a zero-row-sum matrix, then 𝑀𝑄_𝑛=𝑀. So we have𝑥^𝑇𝑀𝑦=𝑥^𝑇𝑀𝑄_𝑛1𝑦≤2𝑥^𝑇𝑀𝑀^𝑇𝑥+𝑦^𝑇Q_𝑛𝑦. (2.8) The eigenvalues of 𝑄_𝑛 are 0 with multiplicity 1 and 1 with multiplicity 𝑛−1.

Definition 2.5 (see [25, 26]). Let 𝑃=diag{𝑝₁,𝑝₂,…,𝑝_𝑛} and Δ=diag{𝛿₁,𝛿₂,…,𝛿_𝑛} be positive definite diagonal matrices. A function 𝑓∶𝑅^𝑛×𝑅⁺→𝑅^𝑛 is said to belong to QUAD(𝑃,Δ), denoted by 𝑓(𝑥,𝑡)∈QUAD(𝑃,Δ) if 𝑓 satisfies the inequality 𝑥−𝑦^𝑇𝑃𝑓(𝑥,𝑡)−𝑓𝑦,𝑡−Δ𝑥−𝑦≤−𝜂𝑥−𝑦^𝑇𝑥−𝑦 (2.9) for some 𝜂>0, 𝑥,𝑦∈𝑅^𝑛, and 𝑡>0.

Definition 2.5 is known to be a typical property that is possessed by many of the benchmark chaotic systems such as the Lorenz system, the Chen system, the Lü system, and the unified chaotic system.

Definition 2.6 (see [27]). A function ℎ∶𝑅→𝑅 is said to belong to UNI(ℎ,ℎ), denoted by ℎ(⋅)∈UNI(ℎ,ℎ), if ℎ satisfies ℎ(𝑥)𝑦ℎ≥−ℎ𝑥−𝑦≥ℎ>0 (2.10) for every 𝑥,𝑦∈𝑅 for some constants ℎ and ℎ.

Lemma 2.7 (see [25, 26]). If 𝐴=[𝑎_𝑖𝑗]∈𝑅^𝑚×𝑚 is such that 𝑎_𝑖𝑗=𝑎_𝑗𝑖 and 𝑎_𝑖𝑖=−∑^𝑚_{𝑗=1,𝑖≠𝑗}𝑎_𝑖𝑗,𝑖,𝑗=1,2,…,𝑚, then 𝑢^𝑇𝐴𝑣=^𝑚𝑚_𝑖=1𝑗=1𝑢_𝑖𝑎_𝑖𝑗𝑣_𝑗=−_𝑗>𝑖𝑎_𝑖𝑗𝑢_𝑖−𝑢_𝑗𝑣_𝑖−𝑣_𝑗 (2.11) for all vectors 𝑢=[𝑢₁,𝑢₂,…,𝑢_𝑚]^𝑇 and 𝑣=[𝑣₁,𝑣₂,…,𝑣_𝑚]^𝑇.

Lemma 2.8 (see [25, 26]). If 𝐴=[𝑎_𝑖𝑗]∈𝐀_𝟏 and 𝜀>0, then all the eigenvalues of the matrix 𝑎𝐴=¹¹−𝜀𝑎₁₂…𝑎_1𝑚𝑎₂₁𝑎₂₂…𝑎_2𝑚𝑎⋮⋮⋱⋮_𝑚1𝑎_𝑚2…𝑎_𝑚𝑚 (2.12) are negative.

3. Main Result

In the section, we derive a sufficient condition for the attainment of cluster synchronization by considering a coupling matrix under pinning control.

First, we prove the following theorem which follows directly from matrix theory and the Lyapunov function theorem.

Theorem 3.1. Suppose that the coupling matrix 𝐴 in system (2.5) satisfies 𝐴∈𝐌_𝟐 and the inner coupling matrix Γ=diag{𝛾₁,𝛾₂,…,𝛾_𝑛} with 𝛾_𝑖>0 (𝑖=1,2,…,𝑛). Let Δ=diag{𝛿₁,𝛿₂,…,𝛿_𝑛} and let 𝑃=diag{𝑝₁,𝑝₂,…,𝑝_𝑛} be positive definite diagonal matrices such that 𝑓(𝑥,𝑡)∈QUAD(𝑃,Δ); ℎ(⋅)∈UNI(ℎ,ℎ) with ℎ≥ℎ>0. If max_𝑘𝛿_𝑘𝐼_𝑘𝑢min_𝑘𝛾_𝑘+ℎ𝐴_𝑢𝑢−Ξ_𝑢+^𝑑_𝑣=𝑢+1𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣+(𝑢−1)ℎ²𝑄_𝑘𝑢≤0,𝑢=1,2,…,𝑑, (3.1) where 𝑄_𝑘𝑢∈𝑅^{𝑘𝑢×𝑘𝑢} is defined as in Definition 2.4 for 𝑢=1,2,…,𝑑, Ξ_𝑢=diag{𝜀_{𝑏𝑢−1+1},𝜀_{𝑏𝑢−1+2},…,𝜀_𝑏𝑢}∈𝑅^{𝑘𝑢×𝑘𝑢} with 𝜀_{𝑏𝑢−1+𝑖}=𝜀_𝑢>0 if 𝑏_𝑢−1+𝑖∈𝐽_𝑢, and 𝜀_{𝑏𝑢−1+𝑖}=0 otherwise. Then, the solutions 𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑚(𝑡) of system (2.5) satisfy lim_𝑡→+∞‖𝑥_𝑖(𝑡)−𝑠_̃𝑖(𝑡)‖=0 for 𝑖=1,2,…,𝑚, that is, the complex network (2.5) is a realized cluster synchronization with partition {𝐺₁,𝐺₂,…,𝐺_𝑑}.

Proof. Let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠_̃𝑖(𝑡), 𝑖=1,2,…,𝑚. Then, subtracting system (2.3) from system (2.5) yields the error system 𝑒_𝑖̇(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡−𝑓𝑠_̃𝑖(𝑡),𝑡+^𝑚_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡)−𝜀_̃𝑖Γ𝑔𝑥_𝑖(𝑡)−𝑔𝑠_̃𝑖(𝑡𝑒̇),𝑖∈𝐽,_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡−𝑓𝑠_̃𝑖(𝑡),𝑡+^𝑚_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡),𝑖∉𝐽. (3.2) Since 𝐴∈𝐌_𝟐, this implies that ^𝑚_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑠_̃𝑗(𝑡)=Γ^𝑑_{𝑢=1𝑗∈𝐺𝑢}𝑎_𝑖𝑗𝑔𝑠_𝑢(𝑡)=0. (3.3) For all 𝑢,𝑣=1,2,…,𝑑, 𝐴_𝑢𝑣 is both a zero-row-sum and a zero-column-sum matrix; therefore, _{𝑖∈𝐺𝑢𝑗∈𝐺𝑣}𝑎_𝑖𝑗𝑥^𝑘_𝑖(𝑡)ℎ𝑥^𝑘_𝑖(𝑡)=0,_{𝑖∈𝐺𝑢𝑗∈𝐺𝑣}𝑎_𝑖𝑗𝑠^𝑘_̃𝑖(𝑡)ℎ𝑥^𝑘_𝑖(𝑡)=0,_{𝑖∈𝐺𝑢𝑗∈𝐺𝑣}𝑎_𝑖𝑗𝑠^𝑘_̃𝑗(𝑡)ℎ𝑥^𝑘_𝑖(𝑡)=0,_{𝑖∈𝐺𝑢𝑗∈𝐺𝑣}𝑎_𝑖𝑗𝑥^𝑘_𝑖(𝑡)ℎ𝑠^𝑘_̃𝑖(𝑡)=0,_{𝑖∈𝐺𝑢𝑗∈𝐺𝑣}𝑎_𝑖𝑗𝑥^𝑘_𝑖(𝑡)ℎ𝑠^𝑘_̃𝑗(𝑡)=0,_{𝑖∈𝐺𝑢𝑗∈𝐺𝑣}𝑎_𝑖𝑗𝑠^𝑘_̃𝑖(𝑡)ℎ𝑠^𝑘_̃𝑖(𝑡)=0. (3.4) Now let 𝑒^𝑗̃(𝑡)=[𝑒^𝑗₁(𝑡),𝑒^𝑗₂(𝑡),…,𝑒^𝑗_𝑚(𝑡)]^𝑇 and 𝑒^𝑗_𝑙̃(𝑡)=[𝑒^𝑗_{𝑏𝑙−1+1}(𝑡),𝑒^𝑗_{𝑏𝑙−1+2}(𝑡),…,𝑒^𝑗_𝑏𝑙(𝑡)]^𝑇. Then, we obtain ^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃Δ𝑒_𝑖(𝑡)=^𝑚𝑛_𝑖=1𝑘=1𝑝_𝑘𝛿_𝑘𝑒^𝑘_𝑖(𝑡)²=^𝑛_𝑘=1𝑝_𝑘𝛿_𝑘^𝑚_𝑖=1𝑒^𝑘_𝑖(𝑡)²=^𝑛_𝑘=1𝑝_𝑘𝛿_𝑘𝑒^𝑘̃(𝑡)^𝑇𝑒^𝑘̃(𝑡)=^𝑛_𝑘=1𝑝_𝑘𝛿_𝑘^𝑑_𝑙=1𝑒^𝑘_𝑙̃(𝑡)^𝑇𝑒^𝑘_𝑙̃(𝑡). (3.5) Next, define a Lyapunov function by 𝑉(𝑡)=12^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡) (3.6) and differentiate 𝑉(𝑡) along system (3.2) to get 𝑑𝑉(𝑡)=𝑑𝑡^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓𝑥_𝑖(𝑡),𝑡−𝑓𝑠_̃𝑖(𝑡),𝑡−Δ𝑒_𝑖(𝑡)+^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃Δe_𝑖(𝑡+_𝑚𝑖=1)𝑒_𝑖(𝑡)^𝑇𝑃^𝑚_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡)−_𝑖∈𝐽𝜀_𝑖𝑒_𝑖(𝑡)^𝑇𝑃Γ𝑔𝑥_𝑖(𝑡)−𝑔𝑠_̃𝑖(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡). (3.7) Note that 𝑓(𝑥,𝑡)∈QUAD(𝑃,Δ), and from equality (3.5),we have 𝑉₁(𝑡)=^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓𝑥_𝑖(𝑡),𝑡−𝑓𝑠_̃𝑖(𝑡),𝑡−Δ𝑒_𝑖(𝑡)+^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃Δ𝑒_𝑖(𝑡)≤−𝜂^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝛿_𝑘^𝑑_𝑙=1𝑒^𝑘_𝑙̃(𝑡)^𝑇𝑒^𝑘_𝑙̃(𝑡). (3.8) For all 𝑥^𝑗(𝑡)=[𝑥^𝑗₁(𝑡),𝑥^𝑗₂(𝑡),…,𝑥^𝑗_𝑚(𝑡)]^𝑇, 𝑥^𝑗_𝑢(t)=[𝑥^𝑗_{𝑏𝑢−1+1}(𝑡),𝑥^𝑗_{𝑏𝑢−1+2}(𝑡),…,𝑥^𝑗_𝑏𝑢(𝑡)]^𝑇, ℎ(𝑥^𝑗_𝑢(𝑡))=[ℎ(𝑥^𝑗_{𝑏𝑢−1+1}(𝑡)),ℎ(𝑥^𝑗_{𝑏𝑢−1+2}(𝑡)),…,ℎ(𝑥^𝑗_𝑏𝑢(𝑡))]^𝑇, 𝑠^𝑗_𝑢̃(𝑡)=[𝑠^𝑗_𝑢(𝑡),𝑠^𝑗_𝑢(𝑡),…,𝑠^𝑗_𝑢(𝑡)]^𝑇∈𝑅^𝑘𝑢, 𝑢=1,2,…,𝑑, and 𝑒^𝑗_𝑢̃(𝑡)=𝑥^𝑗_𝑢(𝑡)−𝑠^𝑗_𝑢̃(𝑡), 𝑗=1,2,…,𝑛. It also follows that 𝑉₃(𝑡)=−_𝑖∈𝐽𝜀_̃𝑖𝑒_𝑖(𝑡)^𝑇𝑃Γ𝑔𝑥_𝑖(𝑡)−𝑔𝑠_̃𝑖(𝑡)≤−ℎ^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘^𝑑_𝑙=1𝑒^𝑘_𝑙̃(𝑡)^𝑇Ξ_𝑙𝑒^𝑘_𝑙̃(𝑡), (3.9) since ℎ(⋅)∈UNI(ℎ,ℎ). Finally, we handle the most difficult 𝑉₂(𝑡), by using equality (3.3) and Lemma 2.7. More precisely, we get 𝑉₂(𝑡)=^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃^𝑚_𝑗=1𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡=_{𝑚𝑚𝑖=1𝑗=1})𝑎_𝑖𝑗𝑒_𝑖(𝑡)^𝑇𝑃Γ𝑔𝑥_𝑗(𝑡)−𝑔𝑠_̃𝑗(𝑡)=^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐴𝑔𝑥^𝑘(𝑡)−𝑔𝑠^𝑘̃(𝑡)=−^𝑛_𝑘=1𝑝_𝑘𝛾_{𝑘𝑗>𝑖}𝑎_𝑖𝑗𝑒^𝑘_𝑖(𝑡)−𝑒^𝑘_𝑗(𝑡)ℎ𝑥^𝑘_𝑖(𝑡)−ℎ𝑠^𝑘_̃𝑖(𝑡)−ℎ𝑥^𝑘_𝑗(𝑡)−ℎ𝑠^𝑘_̃𝑗(𝑡). (3.10) Upon using the equalities (3.4), Lemma 2.7, and the fact that ℎ(⋅)∈UNI(ℎ,ℎ), we obtain 𝑉₂(𝑡)=−^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘^𝑑𝑏_𝑢=1^𝑢−1_{𝑖=𝑏𝑢−1}^𝑏₊₁^𝑢_𝑗=𝑖+1𝑎_𝑖𝑗𝑥^𝑘_𝑖(𝑡)−𝑥^𝑘_𝑗(𝑡)ℎ𝑥^𝑘_𝑖(𝑡)−ℎ𝑥^𝑘_𝑗(𝑡)+^𝑑−1𝑑_𝑢=1^𝑏_𝑣=𝑢+1^𝑢_{𝑖=𝑏𝑢−1}^𝑏₊₁^𝑣_{𝑗=𝑏𝑣−1+1}𝑎_𝑖𝑗𝑥^𝑘_𝑖(𝑡)−𝑠^𝑘_̃𝑖(𝑡)−𝑥^𝑘_𝑗(𝑡)−𝑠^𝑘_̃𝑗(𝑡)×ℎ𝑥^𝑘_𝑖(𝑡)−ℎ𝑠^𝑘_̃𝑖(𝑡)−ℎ𝑥^𝑘_𝑗(𝑡)−ℎ𝑠^𝑘_̃𝑗(𝑡)≤^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘^𝑑𝑏_𝑢=1^𝑢−1_{𝑖=𝑏𝑢−1}^𝑏₊₁^𝑢_𝑗=𝑖+1−ℎ𝑎_𝑖𝑗𝑥^𝑘_𝑖(𝑡)−𝑥^𝑘_𝑗(𝑡)²+^𝑑−1𝑑_𝑢=1^𝑏_𝑣=𝑢+1^𝑢_{𝑖=𝑏𝑢−1}^𝑏₊₁^𝑣_{𝑗=𝑏𝑣−1+1}2𝑎_𝑖𝑗𝑥^𝑘_𝑖(𝑡)ℎ𝑥^𝑘_𝑗(𝑡)=^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘^𝑑_𝑢=1ℎ𝑥^𝑘_𝑢(𝑡)^𝑇𝐴_𝑢𝑢𝑥^𝑘_𝑢(𝑡)+^𝑑−1𝑑_{𝑢=1𝑣=𝑢+1}2𝑥^𝑘_𝑢(𝑡)^𝑇𝐴_𝑢𝑣ℎ𝑥^𝑘_𝑣(𝑡)=^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘^𝑝_𝑢=1ℎ𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑢𝑒^𝑘_𝑢̃(𝑡)+^𝑑−1𝑑_{𝑢=1𝑣=𝑢+1}2𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣ℎ𝑥^𝑘_𝑣(𝑡), (3.11) where ^𝑑−1𝑑_{𝑢=1𝑣=𝑢+1}2𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣ℎ𝑥^𝑘_𝑣(𝑡)≤^𝑑−1𝑑_{𝑢=1𝑣=𝑢+1}𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣𝑒^𝑘_𝑢̃(𝑡)+ℎ²𝑒^𝑘_𝑣̃(𝑡)^𝑇𝑄_𝑘𝑣𝑒^𝑘_𝑣̃(𝑡≤_{𝑑−1𝑑𝑢=1𝑣=𝑢+1})𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣𝑒^𝑘_𝑢̃(𝑡)+^𝑑−1𝑑_{𝑢=1𝑣=𝑢+1}ℎ²𝑒^𝑘_𝑣̃(𝑡)^𝑇𝑄_𝑘𝑣𝑒^𝑘_𝑣̃(𝑡)=^𝑑−1𝑑_{𝑢=1𝑣=𝑢+1}𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣𝑒^𝑘_𝑢̃(𝑡)+^𝑑_𝑢=2^𝑢−1_𝑣=1ℎ²𝑒^𝑘_𝑢̃(𝑡)^𝑇𝑄_𝑘𝑢𝑒^𝑘_𝑢̃(𝑡), (3.12) for we have 2𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣ℎ𝑥^𝑘_𝑣(𝑡)=2𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣𝑄_𝑘𝑣ℎ𝑥^𝑘_𝑣(𝑡)≤𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣𝑒^𝑘_𝑢̃(𝑡)+ℎ𝑥^𝑘_𝑣(𝑡)^𝑇𝑄_𝑘𝑣ℎ𝑥^𝑘_𝑣(𝑡≤𝑒̃_𝑘𝑢)(𝑡)^𝑇𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣𝑒^𝑘_ũ(𝑡)+ℎ²𝑒^𝑘_𝑣̃(𝑡)^𝑇𝑄_𝑘𝑣𝑒^𝑘_𝑣̃(𝑡) (3.13) (from Lemma 2.7 and inequality (2.8)).
Combining inequalities (3.11) and (3.12), we get𝑉₂(𝑡)≤^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘^𝑑_𝑢=1𝑒^𝑘_𝑢̃(𝑡)^𝑇ℎ𝐴_𝑢𝑢+^𝑑_𝑣=𝑢+1𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣+(𝑢−1)ℎ²𝑄_𝑘𝑢𝑒^𝑘_𝑢̃(𝑡) (3.14) and (upon substituting inequalities (3.8), (3.9), and (3.14) into equality (3.7)), we have 𝑑𝑉(𝑡)𝑑𝑡≤−𝜂^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝛿_𝑘^𝑑_𝑢=1𝑒^𝑘_𝑢̃(𝑡)^𝑇𝑒^𝑘_𝑢̃(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘^𝑑_𝑢=1ℎ𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑢𝑒^𝑘_𝑢̃(𝑡)+^𝑑_𝑣=𝑢+1𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣𝑒^𝑘_𝑢̃(𝑡)+(𝑢−1)ℎ²_𝑘𝑒^𝑘_𝑢̃(𝑡)^𝑇𝑄_𝑘𝑢𝑒^𝑘_𝑢̃(𝑡−_𝑛𝑘=1)𝑝_𝑘𝛾_𝑘^𝑑_𝑢=1𝑒^𝑘_𝑢̃(𝑡)^𝑇Ξ_𝑢𝑒^𝑘_𝑢̃(𝑡)=−𝜂^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘^𝑑_𝑢=1𝑒^𝑘_𝑢̃(𝑡)^𝑇𝛿_𝑘𝐼_𝑘𝑢+ℎ𝛾_𝑘𝐴_𝑢𝑢−ℎ𝛾_𝑘Ξ_𝑢+^𝑑_{𝑣=𝑢−1}𝛾_𝑘𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣+(𝑢−1)ℎ²_𝑘𝛾_𝑘𝑄_𝑘𝑢𝑒^𝑘_𝑢̃(𝑡). (3.15) Hence, by combining inequality (3.15) and condition (3.1), it gives 𝑑𝑉(𝑡)𝑑𝑡≤−𝜂^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)≤−2𝜂max_𝑖𝑝_𝑖𝑉(𝑡) (3.16) and then the solution 𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑚(𝑡) of system (2.5) satisfies lim_𝑡→∞∑^𝑚_𝑖=1‖𝑥_𝑖(𝑡)−𝑠_̃𝑖(𝑡)‖=0 for all initial values.

Remark 3.2. It follows from Theorem 3.1 that every complex network with a coupling matrix satisfying inequality (3.1) can be cluster synchronized by pinning control. However, we should construct a coupling matrix 𝐴∈𝐌_𝟐 and select suitable control strengths 𝜀₁,𝜀₂,…,𝜀_𝑑>0, in order to make inequality (3.1) holds. It inspires that we can control some networks to achieve cluster synchronization by constructing a coupling matrix. For 𝑖≠𝑗, if 𝑎_𝑖𝑗≠0 with 𝑖,𝑗∈𝐺_𝑢, then the coupling between oscillators 𝑖 and 𝑗 is called intercluster coupling, which can be regarded as a mechanism to synchronize 𝑖 and 𝑗. Since 𝐴_𝑢𝑢−Ξ_𝑢<0 (from Lemma 2.8), the intercluster coupling strength should always be chosen to be controlled, and the following coupling matrix is constructed by adding partially connected (intercluster) coupling strengths 𝑐𝐴₁₁𝐴₁₂⋯𝐴_1𝑑𝐴₂₁𝑐𝐴₂₂⋯𝐴_2𝑑𝐴⋮⋮⋱⋮_𝑑1𝐴_𝑑2⋯𝑐𝐴_𝑑𝑑 (3.17) for some positive constant 𝑐 [19, 20]. The control strengths are then 𝑐𝜀_𝑙, 𝑙=1,2,…,𝑑. Furthermore, because it is usual for the theoretical value of 𝑐 to be much larger than its true value in applications (we shall show this in Section 4), it is important that 𝑐 be made as small as practically possible.

To make Theorem 3.1 more applicative, we give the following corollaries.

Corollary 3.3. Suppose 𝑓(𝑥,𝑡)∈QUAD(𝑃,Δ), ℎ(⋅)∈UNI(ℎ,ℎ), Γ=diag{𝛾₁,𝛾₂,…,𝛾_𝑛} is positive definite, (3.17) is the coupling matrix, and 𝑄_𝑘𝑢∈𝑅^{𝑘𝑢×𝑘𝑢} is defined as in Definition 2.4; if the coupling strength satisfies 𝑐≥max_{1≤𝑢≤𝑑}𝑐_𝑢, (3.18) where 𝑐_𝑢=max_𝑘−𝛿_𝑘+𝛾_𝑘𝜆_max∑^𝑑_{𝑣=𝑢−1}𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣+ℎ²(𝑢−1)𝑄_𝑘𝑢ℎ𝛾_𝑘𝜆_max𝐴_𝑢𝑢−Ξ_𝑢, (3.19) then system (2.5) can be cluster synchronized.

When ℎ=ℎ=1, ℎ(𝑥)=𝑥 is linear function; then system (2.5) become linearly coupling system; we have the following.

Corollary 3.4. Suppose 𝑓(𝑥,𝑡)∈QUAD(𝑃,Δ), ℎ(⋅)∈UNI(1,1), Γ=𝐼_𝑛, 𝑄_𝑘𝑢∈𝑅^{𝑘𝑢×𝑘𝑢} is defined as in Definition 2.4, and (3.17) is the coupling matrix; if the coupling strength satisfies 𝑐≥max_{1≤𝑢≤𝑑}𝑐_𝑢, (3.20) where 𝑐_𝑢=−max_𝑘𝛿_𝑘+𝜆_max∑^𝑑_{𝑣=𝑢−1}𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣+(𝑢−1)𝑄_𝑘𝑢𝜆_max𝐴_𝑢𝑢−Ξ_𝑢, (3.21) then system (2.5) can be cluster synchronized.

Remark 3.5. Corollary 3.4 gives a sufficient condition for achieving synchronization of linearly coupled complex networks under pinning control, which is different from that obtained by Wu et al. [19, Theorem 2].

As pointed out by Wu et al. [19], real-world complex networks have coupling strengths that are usually many orders of magnitude less than their theoretically assumed values, and minor modifications are, therefore, required on the above pinning scheme to accommodate for relatively small coupling strengths. Thus by using an adaptive feedback control technique, we have the following theorem.

Theorem 3.6. Suppose that 𝜀₁,𝜀₂,…,𝜀_𝑑 are 𝑑 positive constants and 𝐴𝐴=¹¹𝐴₁₂⋯𝐴_1𝑑𝐴₂₁𝐴₂₂⋯𝐴_2𝑑𝐴⋮⋮⋱⋮_𝑑1𝐴_𝑑2⋯𝐴_𝑑𝑑∈𝐌_𝟐. (3.22) Pick the coupling matrix 𝐴(𝑡)=𝑎_𝑖𝑗(𝑡)_𝑚×𝑚=𝑐(𝑡)𝐴₁₁𝐴₁₂⋯𝐴_1𝑑𝐴₂₁𝑐(𝑡)𝐴₂₂⋯𝐴_2𝑑𝐴⋮⋮⋱⋮_𝑑1𝐴_𝑑2⋯𝑐(𝑡)𝐴_𝑑𝑑 (3.23) with the control strengths 𝜀_𝑙(𝑡)=𝑐(𝑡)𝜀_𝑙,𝑙=1,2,…,𝑑, (3.24) where 𝑐(𝑡) is the adaptive coupling strength. Let Δ=diag{𝛿₁,𝛿₂,…,𝛿_𝑛} and 𝑃=diag{𝑝₁,𝑝₂,…,𝑝_𝑛} be positive definite diagonal matrices such that 𝑓(𝑥,𝑡)∈QUAD(𝑃,Δ); ℎ(⋅)∈UNI(ℎ,ℎ) with ℎ≥ℎ>0. Then the coupled system 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑡)Γ𝑔𝑥_𝑗(𝑡)−𝜀_̃𝑖(𝑡)Γ𝑔𝑥_𝑖(𝑡)−𝑔𝑠_̃𝑖(𝑡𝑥̇),𝑖∈𝐽,_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑡)Γ𝑔𝑥_𝑗(𝑡)𝑐̇(𝑡),𝑖∉𝐽,=𝛼^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡) (3.25) can be realized cluster synchronization for 𝑐(0)≥0 and 𝛼>0.

Proof. Choose a constant 𝛼>0, and let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠_̃𝑖(𝑡). Define a Lyapunov function by 𝑉(𝑡)=12^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)+1(𝑡2𝛼𝛽𝑐−𝛽𝑐)², (3.26) where 𝑐 and 𝛽 are constants that will be determined later. Differentiating 𝑉(𝑡) above gives 𝑑𝑉(𝑡)=𝑑𝑡^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖̇(𝑡)𝑐̇(𝑡)𝑐̇(𝑡)𝑐(𝑡)=−𝛼+𝛽_𝑚𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓𝑥_𝑖(𝑡),𝑡−𝑓𝑠_̃𝑖(𝑡),𝑡+^𝑚_𝑗=1𝑐(𝑡)𝑎_𝑖𝑗Γ𝑔𝑥_𝑗(𝑡)−𝑐−𝛽𝑐(𝑡)^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)−_𝑖∈𝐽𝑐(𝑡)𝜀_̃𝑖𝑒_𝑖(𝑡)^𝑇𝑃Γ𝑔𝑥_𝑖(𝑡)−𝑔𝑠_̃𝑖(𝑡)=^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓𝑥_𝑖(𝑡),𝑡−𝑓𝑠_̃𝑖(𝑡),𝑡−Δ𝑒_𝑖(𝑡)+^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃Δ𝑒_𝑖(𝑡)+^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃^𝑚_𝑗=1𝑎_𝑖𝑗(𝑡)Γ𝑔𝑥_𝑗(𝑡)−_𝑖∈𝐽𝑐(𝑡)𝜀_𝑖𝑒_𝑖(𝑡)^𝑇𝑃Γ𝑔𝑥_𝑖(𝑡)−𝑔𝑠_̃𝑖(𝑡)−𝑐−𝛽𝑐(𝑡)^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡), (3.27) and as in Theorem 3.1, we have ^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃^𝑚_𝑗=1𝑎_𝑖𝑗(𝑡)Γ𝑔𝑥_𝑗(𝑡≤_𝑛𝑘=1)𝑝_𝑘𝛾_𝑘^𝑑_𝑢=1ℎ𝑐(𝑡)𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑢𝑒^𝑘_𝑢̃(𝑡)+_𝑣>𝑢𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣𝑒^𝑘_𝑢̃(𝑡)+_𝑣<𝑢ℎ²𝑒^𝑘_𝑢̃(𝑡)^𝑇𝑄_𝑘𝑢𝑒^𝑘_𝑢̃(𝑡). (3.28)
Inequalities (3.27) and (3.28) imply that𝑑𝑉(𝑡)𝑑𝑡≤−𝜂^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝛿_𝑘^𝑑_𝑢=1𝑒^𝑘_𝑢̃(𝑡)^𝑇𝑒^𝑘_𝑢̃(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘^𝑑_𝑢=1ℎ𝑐(𝑡)𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑢𝑒^𝑘_𝑢̃(𝑡)+_𝑣>𝑢𝑒^𝑘_𝑢̃(𝑡)^𝑇𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣𝑒^𝑘_𝑢̃(𝑡)+_𝑣<𝑢ℎ²𝑒^𝑘_𝑢̃(𝑡)^𝑇𝑄_𝑘𝑢𝑒^𝑘_𝑢̃(𝑡(𝑡))−𝑐^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘^𝑑_𝑢=1𝑒^𝑘_𝑢̃(𝑡)^𝑇Ξ_𝑢𝑒^𝑘_𝑢̃(𝑡)−𝑐−𝛽𝑐(𝑡)^𝑛_𝑘=1𝑝_𝑘^𝑑_𝑢=1𝑒^𝑘_𝑢̃(𝑡)^𝑇𝑒^𝑘_𝑢̃(𝑡)=−𝜂^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘^𝑑_𝑢=1𝑒^𝑘_𝑢̃(𝑡)^𝑇𝛿_𝑘𝐼_𝑘𝑢+^𝑑_{𝑣=𝑢−1}𝛾_𝑘𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣+(𝑢−1)ℎ²𝛾_𝑘𝑄_𝑘𝑢−𝑐𝐼_𝑢+𝑐(𝑡)ℎ𝛾_𝑘𝐴_𝑢𝑢−ℎ𝛾_𝑘Ξ_𝑢+𝛽𝐼_𝑢𝑒^𝑘_𝑢̃(𝑡). (3.29) Thus by choosing 𝛽 and 𝑐 such that ℎ𝛾_𝑘𝐴_𝑢𝑢−ℎ𝛾_𝑘Ξ_𝑢+𝛽𝐼_𝑢≤0 and 𝛿_𝑘𝐼_𝑘𝑢+∑^𝑑_{𝑣=𝑢−1}𝛾_𝑘𝐴_𝑢𝑣𝐴^𝑇_𝑢𝑣+(𝑢−1)ℎ²𝛾_𝑘𝑄_𝑘𝑢−𝑐𝐼_𝑢≤0 for any 𝑘=1,2,…,𝑛, 𝑢=1,2,…,𝑑, we obtain 𝑑𝑉(𝑡)𝑑𝑡≤−𝜂^𝑚_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)<0. (3.30) Hence 𝑥_𝑖(𝑡)→𝑠_̃𝑖(𝑡) and 𝑐(̇𝑡)→0 (by Cauchy's convergence principle), and 𝑐(𝑡) converges to some final coupling strength 𝑐₀.

4. Numerical Simulation

In this section, we illustrate the theorems of the previous section by numerically simulating some selected cluster complex networks.

4.1. Cluster Synchronization of Chua'S Circuit by Pinning Control

The Chua's circuit has been built and used in many laboratories as a physical source of pseudorandom signals, and in numerous experiments on synchronization studies, such as secure communication systems and simulations of brain dynamics. It has also been used extensively in many numerical simulations and exploited in avant-garde music compositions [28] and in the evolution of natural languages [29]. Arrays of Chua’s circuits have been used to generate 2-dimensional spiral waves and 3-dimensional scroll waves [30].

The isolated Chua's circuit is described by𝑥̇₁=𝑝𝑥₂−𝑥₁−𝑔𝑥₁,𝑥̇₂=𝑥₁−𝑥₂+𝑥₃,𝑥̇₃=−𝑞𝑥₂, (4.1) where 𝑔(𝑥)=𝑚₀𝑥+((𝑚₁−𝑚₀)/2)(|𝑥+1|−|𝑥−1|). We take that 𝑝=10, 𝑞=14.87, 𝑚₀=−0.68, 𝑚₁=−1.27, and Chua's circuit is chaotic (Figure 1).

Figure 1 

Chua's circuit with 𝑥𝑖(0)=0.1, for 𝑖=1,2,3.

Consider a network with 3 clusters and 15 nodes (𝐺₁={1,…,5}, 𝐺₂={6,…,10}, 𝐺₃={11,…,15}) with the topology structure (Figure 2).

Figure 2 

A community network with size 𝑁=18. There are three communities, each of them is denoted by the dots with gray degree.

Let Ξ₁=Ξ₂=Ξ₃=diag{10,10,0,0,0,0}, 𝑐=15, and 𝑠₁(𝑡), 𝑠₂(𝑡), and 𝑠₃(𝑡) be the solutions of the noncoupled system 𝑠(̇𝑡)=𝑓(𝑠(𝑡),𝑡) with initial values 𝑠₁(0)=[0.1,0.1,0.1]^𝑇, 𝑠₂(0)=[0.2,0.2,0.2]^𝑇, and 𝑠₃(0)=[0.3,0.3,0.3]^𝑇. The following equality is used to investigate the process of cluster: 𝐸(𝑡)=118¹⁸_𝑖=1𝑥_𝑖(𝑡)−𝑠_̃𝑖(𝑡)². (4.2) If lim_𝑡→∞𝐸(𝑡)=0, then we say that the complex network achieves cluster synchronization. The state behavior of the clusters is shown in Figures 3(a), 3(b), 3(c) and the evolution of 𝐸(𝑡) in pinning Chua's circuit networks is shown in Figure 3(d). By using adaptive control scheme, it can be seen that the coupling strength decreases to 3.6 while the original coupling strength is 15 in Figure 4.

(a)

(b)

(c)

(d)

Figure 3 

Cluster synchronization of a nonlinearly coupled Chua's circuit network. (a, b, c) The trajectory of the nodes. (d) The error dynamics of the above Chua’s circuit network.

(a)

(b)

Figure 4 

Adaptive adjustment of the coupling strength of a pinned Chua’s circuit network.

4.2. Cluster Synchronization of Hopfield Neural Network by Pinning Control

The Hopfield neural network is a simple artificial network which is able to store certain memories or patterns in a manner rather similar to the brain—the full pattern can be recovered if the network is presented with only partial information.

Consider a Hopfield neural network𝑑𝑥(𝑥)𝑑𝑡=−𝐷𝑥+𝑇𝑝+𝐼, (4.3) where 𝑥=[𝑥₁,𝑥₂,𝑥₃]^𝑇∈𝑅³, 𝐷=𝐼₃, 𝑇=1.25−3.2−3.2−3.21.1−4.4−3.24.41. (4.4) And 𝑝(𝑥)=[𝑞(𝑥₁),𝑞(𝑥₂),𝑞(𝑥₃)]^𝑇, where 𝑞(𝑠)=(|𝑠+1|+|𝑠−1|)/2 and 𝐼=0. As in [31], 𝑓(𝑥,𝑡)∈QUAD(5.5682𝐼₃,𝐼₃), and we take the nonlinear coupling 𝑔(𝑥)=5𝑥+sin(𝑥) and inner coupling matrix Γ=𝐼₃.

Consider a network with 2 clusters and 5 nodes (𝐺₁={1,2}, 𝐺₂={3,4,5}) with coupling matrix 𝐴∈𝐌_𝟐, where 𝐴=−5510−15−5−1011−1−1055005−105−1155−10. (4.5)

Let Ξ₁=diag{9,0} and Ξ₂=diag{9,0,0}. Then, condition (3.1) of Theorem 3.1 is satisfied. Let 𝑠₁(𝑡) and 𝑠₂(𝑡) be the solutions of the noncoupled system 𝑠(̇𝑡)=𝑓(𝑠(𝑡),𝑡) with initial values 𝑠₁(0)=[0.1,0.2,0.3]^𝑇 and 𝑠₂(0)=[0.4,0.5,0.6]^𝑇. The following equality is used to investigate the process of cluster: 𝐸(𝑡)=15⁵_𝑖=1𝑥_𝑖(𝑡)−𝑠_̃𝑖(𝑡)². (4.6) If lim_𝑡→∞𝐸(𝑡)=0, then we say that the complex network achieves cluster synchronization. The state behavior of the first and second clusters is shown, respectively, by the blue and black curves in Figure 5(a), and the evolution of 𝐸(𝑡) in pinning Hopfield neural networks is shown in Figure 5(b). The state behavior of the Hopfield neural networks is shown in Figure 6(a), and the evolution of 𝐸(𝑡) in pinning Hopfield neural networks is shown in Figure 6(b). The signal is compared with the trajectory of a nonlinearly coupled Hopfield neural network without pinning control in Figure 7.

(a)

(b)

Figure 5 

Cluster synchronization of a nonlinearly coupled Hopfield neural network. (a) The states of the nodes. (b) The error dynamics of the above Hopfield neural network.

(a)

(b)

Figure 6 

The trajectory of a nonlinearly coupled Hopfield neural network without pinning control. (a) The states of the nodes. (b) The error dynamics of the above Hopfield neural network.

(a)

(b)

(c)

(d)

Figure 7 

(a) The trajectory of 𝑥1 in a nonlinearly coupled Hopfield neural network without pinning control. (b) The signal added on 𝑥1 in the above Hopfield neural network. (c) The trajectory of 𝑥4 in a nonlinearly coupled Hopfield neural network without pinning control. (d) The signal added on 𝑥4 in the above Hopfield neural network.

Then, we discuss the cluster synchronization of a Hopfield neural network by adaptive pinning control. In this simulation, the initial values of the network nodes are randomly drawn (and then fixed) from the standard uniform distribution on the open interval (0,1), and 𝑠₁(𝑡), 𝑠₂(𝑡), and 𝑠₃(𝑡) are the solutions of the uncoupled system 𝑠(̇𝑡)=𝑓(𝑠(𝑡),𝑡) with initial values 𝑠₁(0)=[0.1,0.2,0.3]^𝑇, 𝑠₂(0)=[0.4,0.5,0.6]^𝑇, and 𝑠₃(0)=[0.7,0.8,0.9]^𝑇. In the topological structure of the network (Figure 10(a)), we select 𝑙₁=9, 𝜀₁=37, 𝑙₂=8, 𝜀₂=26, 𝑙₃=9, and 𝜀₃=27; then we can calculate 𝑐₁=31.61545, 𝑐₂=31.7185, 𝑐₃=31.8915 from inequalities (3.19) in Corollary 3.3. Thus, the condition for achieving cluster synchronization is satisfied if 𝑐>31.8915. We choose the coupling strength 𝑐=32.

Let 𝐸(𝑡)=1300³⁰⁰_𝑖=1𝑥_𝑖(𝑡)−𝑠_̃𝑖(𝑡)². (4.7)

Figure 8 shows the time evolution of 𝐸(𝑡) and 𝑐(𝑡) for the pinned Hopfield neural network (4.3) with initial values that were randomly chosen from the interval [0,1]. The coupling strength is designed by using the adaptive technique given in Theorem 3.6, and the final coupling strength is 𝑐=12.8706.

(a)

(b)

Figure 8 

Adaptive adjustment of the coupling strength of a pinned Hopfield neural network.

Next, we present the cluster synchronization for the same systems with different coupling strength by numerical simulation. Figure 9(a) shows 𝐸(𝑡) with 𝑐=12.8706 and Figure 9(b) shows 𝐸(𝑡) with 𝑐=32 for the same initial values as those in Figure 8. Clearly, cluster synchronization is attained much faster for the network with 𝑐=32 than for the one with 𝑐=12.8706 in Figures 9(a) and 9(b). One can see that under the same network system as above, if we take 𝑐=5.3602, the cluster synchronization fails (Figure 9(c)).

(a) 𝑐=12.8706

(b) 𝑐=32

(c) 𝑐=5.3602

Figure 9 

The cluster synchronization error under different coupling strengths.

(a) Topological structure of the complex network

(b) 𝑙1−𝜀1−𝑐1 for 𝑢=1

(c) 𝑙2−𝜀2−𝑐2 for 𝑢=2

(d) 𝑙3−𝜀3−𝑐3 for 𝑢=3

Figure 10 

(Color and style online) 𝑙𝑢−𝜀𝑢−𝑐𝑢 expresses the critical combination of the coupling strength, the number of pinned nodes, and the control strength of the 𝑢th cluster. (b) The relationship between 𝜀, 𝑙 and 𝑐 of the first cluster. (c) The second cluster. (d) The third cluster.

Remark 4.1. By using the adaptive technique, we get a final coupling strength, with which the cluster synchronization is realized (Figure 9(a)). Obviously, it is smaller than theoretical coupling strength. In practice, the final coupling strength is very effective, since the synchronization cost is lower than the cluster synchronization with the theoretical coupling strength. If the coupling strength is smaller than it, the cluster synchronization probably fails (Figure 9(c)).

4.3. The Critical Combination of the Control Strength, the Number of Pinned Nodes, and the Coupling Strength

In this simulation, we consider a network with 3 clusters and 300 nodes (𝐺₁={1,2,…,100}, 𝐺₂={101,102,…,200}, and 𝐺₃={201,202,…,300}) with a topological structure that is shown in Figure 10(a). Specifically, the network is obtained by integrating three scale-free networks that are held together by common edges the exact number of which is proportional to the degree of the nodes. Using the topological structure of the network (Figure 10(a)), we construct the coupling matrix 𝐴∈𝐌_𝟐 such that 𝑎_𝑖𝑗>0 if the point (𝑖,𝑗) in Figure 10(a) is “⋅” and 𝑎_𝑖𝑗<0 if the point (𝑖,𝑗) is “×” (Style online). Following [32], no more than one quarter of the cluster nodes are pinned in every cluster in this experiment and we choose to pin those nodes with the largest degrees. Finally, we reject all coupling strengths that are greater than a certain fixed constant (chosen to be 75 in this experiment) because the coupling strengths cannot be too large in practice. A critical value for the coupling strength c_𝑢 is then calculated (using Corollary 3.3) for every critical combination of the number of pinned nodes 𝑙_𝑢 and the control strength 𝜀_𝑢 for every cluster.

We see in Figures 10(b), 10(c), and 10(d) that even though the color RGB(128,0,0) represents the unacceptable coupling strengths 75, many combinations of the coupling strength, the number of pinned nodes and the control strength can still be taken to realize cluster synchronization. Furthermore, for every cluster if the number of pinned nodes 𝑙_𝑢 is determined, the coupling strength 𝑐_𝑢 decreases as the control gain 𝜀_𝑢 increases, and finally it approaches a constant. Once the control strength 𝜀_𝑢 is determined, the coupling strength 𝑐_𝑢 decreases as the number of nodes 𝑙_𝑢 increases, and finally it approaches a constant. Furthermore, if the coupling strength 𝑐_𝑢 approaches the threshold, it almost has no changes no matter how you increase the control strength 𝑙_𝑢 and the number of pinned nodes 𝑙_𝑢. In particular, a certain combination of suitable 𝑙_𝑢, 𝜀_𝑢, and 𝑐_𝑢 for every cluster exists that will minimize the synchronization cost. It is a challenging problem to find the minimal cost combination for a specific complex network.

5. Conclusion

In this paper, we investigated the cluster synchronization problem for complex networks that are nonlinearly coupled. Specifically, we achieved global cluster synchronization by applying a pinning control scheme to every individual cluster and derived sufficient conditions for the global stability of cluster synchronization. Furthermore, in view of the great disparity in the coupling strength magnitudes that exist between theoretical and real-world systems, we also rigorously proved an adaptive feedback control technique that can be used to completely cluster synchronize any real-world network. Finally, for clarity of exposition, some numerical examples were considered that illustrate the theoretical analysis and the acceptance condition was given for the number of pinned nodes and coupling and control strengths of one particular network.

Acknowledgments

This work was supported by the National Science Foundation of China under Grant no. 61070087, Guangdong Education University Industry Cooperation Projects (2009B090300355) and Shenzhen Basic Research Project (JC200903120040A, JC201006010743A), and a research fund form the Hong Kong Polytechnic University. The authors are very grateful to reviewers and the editor for their valuable comments and suggestions to improve the presentation of the paper.