Adaptive Pinning Synchronization Control of the Fractional-Order Chaos Nodes in Complex Networks

Abstract

Adaptive pinning synchronization control is studied for a class of fractional-order complex network systems which are constructed depending on small-world network algorithm. Based on the fractional-order stability theory, the suitable adaptive control scheme is designed to guarantee global asymptotic stability of all the nodes in complex network systems and the node selected algorithm is given. In numerical implementation, it is shown that the numerical solution of the fractional-order complex network systems can be obtained by applying an improved version of Adams-Bashforth-Moulton algorithm. Furthermore, simulation results are provided to confirm the validity and synchronization performance of the advocated design methodology.

1. Introduction

Fractional-order calculus can portray the physical phenomenon more accurately than integer order calculus in actual system. In recent years, the theoretical analysis and applied research of the fractional calculus have attracted many researchers’ attentions. Fractional-order calculus has become a hot research topic in dynamic system, and there are many successful applications in engineering field [1–5]. Some scholars found that the fractional-order nonlinear dynamic system also shows chaotic phenomenon, such as fractional-order Rössler system, fractional-order Liu system, and fractional-order Chen system [6–13].

In nature, many complex systems can be described as complex networks. Complex networks consist of nodes and the connecting relations of nodes component. Over the past twenty years, complex networks have been widely studied in various fields such as large-scale circuit networks, power networks, computer networks, communication networks, and automatic control systems [14–16]. Complex networks have become an important research topic in these years. And many theoretical analyses of the characteristics are studied in the network [17–21].

Many synchronization control methods of the integer order nodes in complex networks are studied, such as cluster synchronization, adaptive control synchronization, pinning control synchronization, projective synchronization, robust impulsive synchronization [22–26]. Because of the superiority of the fractional calculus in the description of physical phenomena, considering fractional-order systems as nodes in complex networks has important theoretical significance. In recent years, some synchronization control methods of fractional-order chaotic systems in complex networks are studied. The adaptive control methods are used in complex networks [27], the cluster synchronization in complex networks [28], the robust outer synchronization between two complex networks [29], and the generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes [30]. In this paper, we proposed a novel synchronization theorem for fractional-order complex networks. Based on the proposed control scheme, the adaptive feedback controllers are designed for pinning control. Theoretical analyses show that the suitable adaptive feedback gains achieved all the nodes synchronized with each other in the complex networks.

The rest of this paper is organized as follows. In Section 2, some important preliminaries are introduced for demonstrating the main results in the following sections. In Section 3, an effective adaptive pinning control scheme of the fractional-order complex networks is studied and the theoretical proof is given. In Section 4, the control scheme is applied to the fractional-order chaotic systems in complex networks, and the simulation results are given to demonstrate the theoretical analysis. Conclusions ended the paper in Section 5.

2. Model Description and Preliminaries

At present, there are several definitions of the fractional-order differential systems. Two commonly used definitions are Grünwald-Letnikov (GL) definition and Riemann-Liouville (RL) definition.

The best-known RL definition of fractional-order can be expressed as [1] d^𝑞𝑓(𝑡)d𝑡^𝑞=1Γd𝑛−𝑞d𝑡^𝑛𝑛𝑡₀𝑓(𝜏)(𝑡−𝜏)^{𝑞−𝑛+1}𝑑𝜏, (1) where 𝑛 is an integer satisfying 𝑛−1≤𝑞<𝑛 and Γ(⋅) is the Γ-function.

Considering a general fractional-order nonlinear dynamical system as follows: 𝐷^𝑞𝑋=𝑓(𝑋)or𝐷^𝑞𝑋=𝐴𝑋, (2) where 𝑋∈𝑅^𝑛(∈𝑁), 𝐴∈𝑅^𝑛×𝑛, 0<𝑞≤1.

Lemma 1. For a given autonomous linear system of fractional-order system (2) with 𝑥(0)=𝑥₀, where 𝑥(𝑡)∈𝑅^𝑛 is the state vector, from paper [31], we know the following. (1)The system is asymptotically stable if and only if |arg(𝜆_𝑖(𝐴))|>𝛼𝜋/2, 𝑖=1,2,…,𝑛, where arg(𝜆_𝑖(𝐴)) denotes the argument of the eigenvalues 𝜆_𝑖 of 𝐴.(2)The system is stable if and only if either it is asymptotically stable or those critical eigenvalues which satisfying |arg(𝜆_𝑖(𝐴))|=𝛼𝜋/2 have geometric multiplicity one.

Lemma 2 (see [32, 33]). For the nonlinear fractional-order system (2) with the order as 0<𝑞≤1, if there exists a real symmetric positive definite matrix 𝑃 satisfying 𝐽(𝑡)=𝑋^𝑇(𝑡)𝑃𝐷^𝑞𝑋(𝑡)≤0, where 𝑋(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)), then system (2) is asymptotically locally stable.
Consider that the generic dynamical complex networks model consists of 𝑁 coupled identical nodes, and each node is an 𝑛-dimensional dynamical system. The fractional-order network model is described as follows [34, 35]: d^𝑞𝑥_𝑖(𝑡)d𝑡^𝑞=𝑓𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑖=1,2,…,𝑁, (3) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑁(𝑡))^𝑇∈𝑅^𝑛 represent the 𝑖th node of the 𝑛-dimensional dynamical system. Γ is the inner-coupling term which links the coupled variables in networks. The constant 𝑐>0 is the coupling strength. 𝐺=(𝑔_𝑖𝑗)_𝑁×𝑁 is the coupling configuration matrix of any of the two nodes, in which 𝑔_𝑖𝑗 is defined as follows. If there is a connection between the 𝑖th node and the 𝑗th node, (𝑗≠𝑖), then 𝑔_𝑖𝑗=𝑔_𝑗𝑖=1; otherwise, 𝑔_𝑖𝑗=𝑔_𝑗𝑖=0(𝑗≠𝑖), and the diagonal elements of matrix 𝐺 are defined by 𝑔_𝑖𝑖=−^𝑁_{𝑗=1𝑗≠𝑖}𝑔_𝑖𝑗=−^𝑁_{𝑗=1𝑗≠𝑖}𝑔_𝑗𝑖,𝑖=1,2,…,𝑁. (4)

Definition 3. If 𝑥₁(𝑡)=𝑥₂(𝑡)=⋯=𝑥_𝑁(𝑡)→𝑠(𝑡), 𝑡→∞. Consider lim_𝑡→∞‖𝑥_𝑖(𝑡)−𝑠(𝑡)‖=0(1≤𝑖≤𝑁), the complex network is achieved synchronization.
Where 𝑠(𝑡)∈𝑅^𝑛 is a solution of an isolated node, satisfying 𝐷^𝑞𝑠(𝑡)=𝑓[𝑠(𝑡)], (5)
𝑠(𝑡) can be an equilibrium point, a periodic orbit, or even a chaotic attractor.
The error vector is defined as 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡),(𝑖=1,2,…,𝑁). (6)

Assumption 4 (see [36]). Suppose that there exists a nonnegative constant 𝛼, Γ is a symmetric positive semidefinite matrix, and arbitrary 𝑥,𝑦∈𝑅^𝑛, such that 𝑥−𝑦^𝑇𝑓(𝑥)−𝑓𝑦≤𝑥−𝑦^𝑇𝛼Γ𝑥−𝑦.∀𝑥,𝑦∈𝑅^𝑛. (7)

Lemma 5 (see [37]). The linear matrix inequality (LMI) 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22}<0 is equivalent to any one of the two following conditions:(1)𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0, (2)𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0, where 𝑆₁₁=𝑆^𝑇₁₁ and 𝑆₂₂=𝑆^𝑇₂₂.

3. Adaptive Pinning Synchronization

In this section, a novel adaptive pinning control scheme is proposed for the synchronization in complex networks, and this method is proved based on the fractional-order stability theory. The controlled fractional-order complex network model can be described in the following equations: d^𝑞𝑥_𝑖(𝑡)d𝑡^𝑞=𝑓𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑢_𝑖(𝑡)𝑓𝑥,𝑖=1,2,…,𝑙,_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑖=𝑙+1,𝑙+2,…,𝑁. (8)

The adaptive pinning controllers and the updating laws are defined as 𝑢_𝑖(𝑡)=−𝑑_𝑖(𝑡)Γ𝑒_𝑖(𝑡)d,𝑖=1,2,…,𝑙,_𝑞𝑑_𝑖(𝑡)d𝑡^𝑞=𝑘_𝑖𝑒^𝑇_𝑖(𝑡)Γ𝑒_𝑖(𝑡)𝑢,𝑖=1,2,…,𝑙,_𝑖(𝑡)=0,𝑖=𝑙+1,𝑙+2,…,𝑁, (9) where 𝑘_𝑖>0 is an arbitrary constant, the complex network realized pinning synchronization based on the above adaptive controllers.

Based on the above equations, we can get the error equations as follows: d^𝑞𝑒_𝑖(𝑡)d𝑡^𝑞=𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑒_𝑗(𝑡)−𝑑_𝑖(𝑡)Γ𝑒_𝑖(𝑡)𝑓𝑥𝑖=1,2,…,𝑙,_𝑖(𝑡)−𝑓(𝑠(𝑡))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑒_𝑗(𝑡),𝑖=𝑙+1,𝑙+2,…,𝑁. (10)

Theorem 6. The complex networks (8) are realized adaptive pinning global synchronizations under the controllers (9), which is satisfied in 𝜆(𝐺)_𝑁−𝑙<−𝛼/𝑐, and 𝑑^∗>𝜆_max(𝑆₁₁−𝑆₁₂𝑆⁻¹_𝑁−𝑙𝑆^𝑇₁₂), where 𝐼_𝑁=diag(1,…^𝑙,1,0,…^𝑁−𝑙,0) and 𝑑^∗=min_{1≤𝑖≤𝑙}{𝑑^∗_𝑖}.

Proof. The error system (10) can realize asymptotic stability at the state 𝑥(𝑡)=𝑠(𝑡), denote 𝑑_𝑖(𝑡)=𝑑_𝑖(𝑡)−𝑑^∗, and  𝑑^∗ is a positive constant which need to be determined.
Further denote 𝑋(𝑡)=(𝐸(𝑡),𝑑(𝑡))^𝑇, where 𝐸(𝑡)=𝑒₁(𝑡),𝑒₂(𝑡),…,𝑒_𝑁(𝑡)^𝑇,𝑒_𝑖=𝑒_𝑖1(𝑡),𝑒_𝑖2(𝑡),…,𝑒_𝑖𝑁(𝑡)^𝑇,𝑑(𝑡)=𝑑₁(𝑡),𝑑₂(𝑡),…,𝑑_𝑁(𝑡)^𝑇. (11)
Choose the real symmetric positive definite matrix 𝑃 as 𝑃=diag^𝑁11,…,1,𝑘₁,1𝑘₂1,…,𝑘_𝑁. (12)
We obtain that 𝐽(𝑡)=𝑋^𝑇(𝑡)𝑃𝐷^𝑞𝑋(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝐷^𝑞𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑑_𝑖1𝑘_𝑖𝐷^𝑞𝑑_𝑖(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝐷^𝑞𝑒_𝑖(𝑡)+^𝑁_𝑖=11𝑘_𝑖𝑑_𝑖−𝑑^∗_𝑖𝑘_𝑖𝑒^𝑇_𝑖(𝑡)Γ𝑒_𝑖(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑥_𝑗(𝑡)−𝑠(𝑡)−𝑑_𝑖Γ𝑒_𝑖(𝑡)+^𝑙_𝑖=1𝑑_𝑖−𝑑^∗_𝑖𝑒^𝑇_𝑖(𝑡)Γ𝑒_𝑖(𝑡)≤𝛼^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)Γ𝑒_𝑖(𝑡)+𝑐^𝑁𝑁_𝑖=1𝑗=1𝑒^𝑇_𝑖(𝑡)𝑔_𝑖𝑗Γ𝑒_𝑗(𝑡)−𝑑^∗_𝑖^𝑙_𝑖=1𝑒^𝑇_𝑖(𝑡)Γ𝑒_𝑖(𝑡)=𝑒^𝑇(𝑡)𝛼𝐼_𝑁−𝐷_𝑁+𝑐𝐺⊗Γ𝑒(𝑡)=𝑒^𝑇(𝑡)(𝑄⊗Γ)𝑒(𝑡), (13) where ⊗ is Kronecker product, 𝑄=𝛼𝐼_𝑁+𝑐𝐺−𝐷_𝑁, and 𝐷_𝑁=diag(1,…^𝑙,1,0,…^𝑁−𝑙,0). Let 𝑆=𝛼𝐼_𝑁+𝑐𝐺; that is, 𝑄=𝑆−𝐷_𝑁. By Lemma 5, 𝑆−𝐷_𝑁=^{𝑆11−𝐷𝑙𝑆12}_{𝑆𝑇12𝑆𝑁−𝑙}, where 𝐷_𝑙=diag(𝑑^∗₁,…,𝑑^∗_𝑙). Consider 𝜆(𝑆_𝑁−𝑙)=𝜆(𝛼𝐼_𝑁+𝑐𝐺)_𝑁−𝑙≤𝛼+𝑐𝜆(𝐺)_𝑁−𝑙, and from Theorem 6, 𝛼+𝑐𝜆(𝐺)_𝑁−𝑙<0, 𝜆(𝐺)_𝑁−𝑙<−𝛼/𝑐, we can get 𝑆_𝑁−𝑙<0. By Lemma 5, if 𝑆<0, that is, (𝑆₁₁−𝐷_𝑙)−𝑆₁₂𝑆⁻¹_𝑁−𝑙𝑆^𝑇₁₂<0, 𝐷_𝑙>𝑆₁₁−𝑆₁₂𝑆⁻¹_𝑁−𝑙𝑆^𝑇₁₂, which indicates that 𝑑^∗>𝜆_max(𝑆₁₁−𝑆₁₂𝑆⁻¹_𝑁−𝑙𝑆^𝑇₁₂), where 𝑑^∗=min_{1≤𝑖≤𝑙}{𝑑^∗_𝑖}. There exists a suitable 𝛼 to satisfy Assumption 4 and 𝑑^∗ is large enough to make the 𝑄<0, and Γ is a positive semidefinite matrix; it is easy to see that 𝑄⊗Γ≤0. According to the fractional-order stability theory, the system is asymptotically stable. Then the proof is completed.

Proposition 7. If the matrix 𝛼𝐼_𝑁+𝑐𝐺 has 𝑚 nonnegative eigenvalues, and if 𝛼𝐼_𝑁+𝑐𝐺−𝐷_𝑁<0, then the number of nodes to be selected for control cannot be less than 𝑚.

Proof. By the inequality 𝑄=𝛼𝐼_𝑁+𝑐𝐺−𝐷_𝑁<0, we can get 𝑄_𝑖𝑖<0, 𝑄_𝑖𝑖=𝛼+𝑐𝑔_𝑖𝑖−𝑑_𝑖<0, and the degree of the networks Deg_𝑖=−𝑔_𝑖𝑖, so 𝛼−𝑐Deg_𝑖−𝑑_𝑖<0, Deg_𝑖>(𝛼−𝑑_𝑖)/𝑐(𝑖=1,…,𝑙), Deg_𝑖>𝛼/𝑐(𝑖=𝑙+1,…,𝑁).

For the above theoretical proof, we can obtain that if the coupling strength 𝑐 is large enough, selecting the arbitrary 𝑙 nodes is satisfied. But when the coupling strength 𝑐 is small, we need to select the small node degree nodes to add the controllers.

4. Simulation Results

To demonstrate the effectiveness of the proposed approach, the fractional-order Liu chaotic system as node in the complex network systems is applied to construct the network model with 50 nodes based on the NW small-world network algorithm. The controlled nodes in the complex networks can be written as follows: d^𝑞𝑥_𝑖1(𝑡)d𝑡^𝑞=10𝑥_𝑖2(𝑡)−𝑥_𝑖1(𝑡)+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗𝑥_𝑗1(𝑡)−𝑑_𝑖(𝑡)Γ𝑥_𝑖1(𝑡)(𝑡),d−𝑠_𝑞𝑥_𝑖2(𝑡)d𝑡^𝑞=40𝑥_𝑖1(𝑡)−𝑥_𝑖1(𝑡)𝑥_𝑖3(𝑡)+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗𝑥_𝑗2(𝑡)−𝑑_𝑖(𝑡)Γ𝑥_𝑖2(𝑡)(𝑡),d−𝑠_𝑞𝑥_𝑖3(𝑡)d𝑡^𝑞=−2.5𝑥_𝑖3(𝑡)+4𝑥_𝑖1(𝑡)𝑥_𝑖1(𝑡)+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗𝑥_𝑗3(𝑡)−𝑑_𝑖(𝑡)Γ𝑥_𝑖3(𝑡)−𝑠(𝑡). (14)

In order to achieve the synchronization control of the above system, the Adams-Bashforth-Moulton algorithm is applied to the system (14). Consider the following differential equations: ^𝐶₀𝐷^𝛼_𝑡𝑦_𝑖(𝑡)=𝑓𝑡,𝑦_𝑖(𝑡)+𝑐^𝑁_{𝑗=1𝑗≠𝑖}𝑔_𝑖𝑗Γ𝑦_𝑗(𝑡)𝑦,0≤𝑡≤𝑇,_𝑖(𝑘)(0)=𝑦^(𝑘)_𝑖0,𝑘=0,1,…,[𝛼]−1,𝑖=1,2,…,𝑁. (15)

It is equivalent to the Volterra integral equation as follows [1, 38]: 𝑦_𝑖(𝑡)=^[𝛼]−1_𝑘=0𝑡^𝑘𝑦𝑘!^(𝑘)𝑖0+1Γ(𝛼)×^𝑡₀(𝑡−𝑤)^𝛼−1𝑓𝑤,𝑦_𝑖(𝑤)+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑦_𝑗(𝑤)𝑑𝑤. (16)

Here, ℎ=𝑇/𝑁, 𝑡_𝑛=𝑛ℎ, 𝑛=0,1,…,𝑁∈𝑍, and (16) can be discretized into [9] 𝑦_𝑖ℎ𝑡_𝑛+1=^[𝛼]−1_𝑘=0𝑡^𝑘_𝑛+1𝑦𝑘!^(𝑘)𝑖0+ℎ^𝛼Γ()×𝑓𝑡𝛼+2_𝑛+1,𝑦^∗_𝑖ℎ𝑡_𝑛+1+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑦_𝑗ℎ𝑡_𝑛+1+ℎ^𝛼Γ(𝛼+2)^𝑛_𝑙=0𝑎_𝑙,𝑛+1𝑓𝑡_𝑙,𝑦_𝑖ℎ𝑡_𝑙+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑦_𝑗ℎ𝑡_𝑙,𝑖=1,2,…,𝑁, (17) where 𝑎_𝑙,𝑛+1=𝑛^𝛼+1−(𝑛−𝛼)(𝑛+1)^𝛼()𝑙=0𝑛−𝑙+2^𝛼+1+(𝑛−𝑙)^𝛼+1−2(𝑛−𝑙+1)^𝛼+1𝑦1≤𝑙≤𝑛1𝑙=𝑛+1,_∗𝑖ℎ𝑡_𝑛+1=^[𝛼]−1_𝑘=0𝑡^𝑘_𝑛+1𝑦𝑘!^(𝑘)𝑖0+1Γ(𝛼)×^𝑛_𝑙=0𝑏_𝑙,𝑛+1𝑓𝑡_𝑙,𝑦_𝑖ℎ𝑡_𝑙+𝑐^𝑁_𝑗=1𝑔_𝑖𝑗Γ𝑦_𝑗ℎ𝑡_𝑙,𝑏_𝑙,𝑛+1=ℎ^𝛼𝛼(𝑛+1−𝑙)^𝛼−(𝑛−𝑙)^𝛼. (18)

The error equation is max_{𝑗=0,1,…,𝑁}|𝑦(𝑡_𝑗)−𝑦_ℎ(𝑡_𝑗)|=𝑂(ℎ^𝑝), where 𝑝=min(2,1+𝛼).

Where 𝑔_𝑖𝑗 is the coupling configuration matrix which is generated by the NW small-world algorithm, 𝑐 is the coupling strength, and the inner-coupling term Γ=^10000100_00100001. When the fractional-order 0.83≤𝑞≤1, the 𝑖th node of the fractional-order Liu chaotic system (14) is chaotic [10]. Figure 1 shows the chaotic attractor of the fractional-order Liu system with the order 𝑞=0.9.

(a) 𝑥-𝑦 phase diagram

(b) 𝑦-𝑧 phase diagram

(c) 𝑥-𝑧 phase diagram

Figure 1 

2D strange attractors of the fractional-order Liu chaotic system.

In the following, we choose 𝛼=135, and coupling strength 𝑐=12. Select the initial value 𝑥_𝑖1=20×rand, 𝑥_𝑖2=20×rand, and 𝑥_𝑖3=20×rand, where rand∈(0,1) is a random number, and 𝑑^∗=2. When 𝑙=20, 𝜆(𝐺)_𝑁−𝑙=−11.9644, which is satisfied in 𝜆(𝐺)_𝑁−𝑙<−𝛼/𝑐=−11.25. That is, we need to pin 20 nodes for achieving synchronization in the networks.

The synchronization errors are defined as (10), and the total synchronization error is defined as 𝐸(𝑡)=(1/𝑁)∑^𝑁_𝑖=1(𝑒²_𝑖1+𝑒²_𝑖2+𝑒²_𝑖3). Control-effort trajectory of error evolutions is shown in Figures 2 and 3. These figures show that the nodes of the networks are globally asymptotically stable under the pinning adaptive controllers. The evolutions of the pinning feedback gains with (9) are illustrated in Figure 4.

Figure 2 

Synchronization errors of 𝑒𝑖1, 𝑒𝑖2, 𝑒𝑖3, 1≤𝑖≤50.

Figure 3 

Time evolution of 𝐸(𝑡).

Figure 4 

Adaptive feedback gains.

5. Conclusions

In this paper, a novel adaptive pinning control scheme is proposed to deal with chaos synchronization for a class of fractional-order complex network systems by employing fractional-order stability theory. Based on this scheme, the new adaptive pinning controllers are designed to realize the synchronization in complex networks and the node selected algorithm is given. In the simulation, the fractional-order Liu chaotic system as nodes for constructing network model and the Adams-Bashforth-Moulton algorithm are applied to the fractional differential equations of Liu chaotic system in the networks. The suitable adaptive feedback control gain adding to the pinned nodes for achieving all the nodes synchronized with each other in the complex networks.

Conflict of Interests

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

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (nos. 51177117 and 51307130), the Creative Research Groups Fund of the National Natural Science Foundation of China (no. 51221005), and the Research Fund for the Doctoral Program of Higher Education of China (no. 20100201110023).