Pinning Two Nonlinearly Coupled Complex Networks with an Asymmetrical Coupling Matrix

Abstract

This paper addresses the hybrid synchronization problem in two nonlinearly coupled complex networks with asymmetrical coupling matrices under pinning control schemes. The hybrid synchronization of two complex networks is the outer antisynchronization between the driving network and the response network while the inner complete synchronization in the driving network and the response network. We will show that only a small number of pinning feedback controllers acting on some nodes are effective for synchronization control of the mentioned dynamical networks. Based on Lyapunov Stability Theory, some simple criteria for hybrid synchronization are derived for such dynamical networks by pinning control strategy. Numerical examples are provided to illustrate the effectiveness of our theoretical results.

1. Introduction

Complex networks become more and more important because they abound both in nature and in the artificial networks (easy examples include biological ecosystems, internet connections, the World Wide Web, and various social and neural networks [1, 2]). Complex networks also provide models for many large-scale and complicated systems occurring in nature and society that would otherwise be extremely difficult to study. Indeed, complex networks have been extensively studied over the past two decades in many fields of science and engineering because of their general ability to represent virtually all natural and man-made structures. The focus of most of these previous works, however, have been on the structural properties of the dynamical networks [1–4] with little regard for their collective behavior which is where some interesting phenomena such as synchronization, spatiotemporal chaos, autowaves, and spiral waves [5, 6] occur. Synchronization, in particular, is the time-correlated behavior between the nodes of a dynamical network and it is a collective behavior of coupled nonlinear systems that promises immense potential engineering applications [7–12].

Many synchronization patterns have been recorded in the literature such as complete synchronization [7–10], 𝜇-synchronization [11], cluster synchronization [13, 14], generalized synchronization [15], lag synchronization [16, 17], and phase synchronization [18] and it is known that they can all be encompassed by a unified definition [19]. One particular type of generalized synchronization that occurs commonly in symmetrical oscillators is antisynchronization (AS). Specifically, AS is characterized by the vanishing of the sum of the relevant variables and is a phenomenon that has both been experimentally and numerically observed in coupled chaotic systems. AS has been observed, for example, in piecewise linearly coupled Chua’s circuits [20] and in coupled laser systems [21] and it is an important phenomenon in chaotic systems that has great application potential. Using antisynchronization in lasers, for instance, one can generate not only drop-outs of the intensity with ordinary low frequency fluctuations but also short pulses of high intensity that provides a new way of generating pulses of special shapes. Communications security and secrecy can similarly be enhanced during transmission by continuously transforming the digital signals between the synchronized and antisynchronized modes. Indeed, antisynchronization has been fairly well studied and many results exist about the phenomenon. An adaptive control method, for example, was proposed in [22] for antisynchronizing an uncertain Chua’s chaotic system and sufficient conditions for both the synchronization and antisynchronization of the Colpitts oscillators were obtained using active control in [23]. In another attempt, a nonlinear control technique was used to antisynchronize two distinct chaotic systems in [24] and several sufficient conditions were obtained for the attainment of AS and adaptive AS in linearly coupled systems in [25]. The antisynchronization of a class of delayed chaotic neural networks was also investigated in [26, 27] using the Halanany Inequality and Lyapunov Stability Theory.

Recently, there is much interest in the coexistence of antisynchronization and complete synchronization (called hybrid synchronization) in chaotic systems [28–31] although much of these studies have been concerned with coupled chaotic systems rather than with coupled networks. The only investigation of the hybrid synchronization of two-coupled complex networks, to date, was that carried out by Sun et al. in [32] in which linearly coupled complex networks (i.e., the coupling function being linear) were considered. The fact remains, however, that many state variables of a network could at times be totally unobservable with ℎ(𝑥_𝑖(𝑡)) (which is a nonlinear function of the state [32]) being the only known quantity.

Since it is literally impossible to add controllers to all nodes, local feedback injections are applied to only some of the nodes (this is known as pinning control). It has been revealed that, in the process of controlling various networks, pinning control serves as a simple and effective strategy for stabilization and synchronization. Pinning state feedback controllers have recently been used for the synchronization of complex dynamical networks by controlling a small fraction of nodes [33–35]. The pinning strategy has been shown to be effective for synchronization of networks. In this paper, we investigate the hybrid synchronization of nonlinearly coupled complex networks that are described by ordinary differential equations (ODE) with coupling functions that are nonlinear and whose coupling matrix is asymmetrical. Some simple criteria are derived for the hybrid synchronization of such dynamical networks by using the pinning control method and the Lyapunov stability theory. The results of this paper are a nontrivial extension of those in [32], where the coupling function is linear and the controllers were added to all nodes for getting hybrid synchronization.

This paper is organized as follows. In Section 2, we construct the hybrid synchronization of nonlinearly coupled complex networks with differential drive and response network local dynamics. In Section 3, we give some necessary definitions, lemmas, and hypotheses. In Section 4, we derive some simple criteria for the hybrid synchronization of such dynamical networks by pinning control using Lyapunov stability theory. In Section 5, we present some numerical simulations that validate the theoretical results. This paper concludes in Section 6.

2. Model Description

In [32], the authors used linear feedback or adaptive linear feedback methods to realize the hybrid synchronization of two-coupled networks. The driving network is 𝑥̇^𝑖(𝑡)=𝑓𝑥^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑥^𝑗(𝑡),𝑖=1,2,…,𝑁, (1) and the response network is 𝑦̇^𝑖(𝑡)=𝑓𝑦^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑦^𝑗(𝑡)−𝑑_𝑖Γ𝑥^𝑖+𝑦^𝑖,𝑖=1,2,…,𝑁. (2) For the coupling system (1), we let the inner coupled matrix Γ to be the identity matrix. Then the driving network of the nonlinearly coupled differential equation is the following: 𝑥̇^𝑖(𝑡)𝑓𝑥^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗ℎ𝑥^𝑖(𝑡),𝑖=1,2,…,𝑁. (3) In the following, the hybrid synchronization of the nonlinearly coupled complex network model (3) is investigated. To realize the hybrid synchronization, some pinning controllers will be added to parts of the response network. Without loss of generality, rearrange the order of the nodes in the response network and let the first 𝑙 nodes be controlled. Then the pinning controlled response network can be described by 𝑦̇^𝑖(𝑡)=𝑓𝑦^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗ℎ𝑦^𝑗(𝑡)−𝑐𝑑_𝑖𝑥^𝑖(𝑡)+𝑦^𝑖(𝑡𝑦̇),𝑖=1,2,…,𝑙,_𝑖(𝑡)=𝑓𝑦^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗ℎ𝑦^𝑗(𝑡),𝑖=𝑙+1,𝑙+2,…,𝑁, (4) where 𝑁>1 is the network size, 𝑥^𝑖(𝑡)=(𝑥^𝑖₁(𝑡),𝑥^𝑖₂(𝑡),…, 𝑥^𝑖_𝑛(𝑡))^𝑇∈𝑅^𝑛, 𝑦^𝑖(𝑡)=(𝑦^𝑖₁(𝑡),𝑦^𝑖₂(𝑡),…,𝑦^𝑖_𝑛(𝑡))^𝑇∈𝑅^𝑛 are the state variables of the 𝑖-node in the driving network and the response network, respectively, 𝑓∶𝑅^𝑛×[0,+∞)→𝑅^𝑛 is a continuous function that describes the local dynamics of the driving networks and the response network, respectively, 𝑐 is the coupling strength, and 𝑘_𝑖(𝑖=1,2,…,𝑙) are control gains. 𝐴=[𝑎_𝑖𝑗]∈𝑅^𝑛×𝑛 denotes the coupling configuration of the network and 𝑎_𝑖𝑗 is defined as follows. If nodes 𝑖 and 𝑗 are linked by an edge then 𝑎_𝑖𝑗>0, otherwise 𝑎_𝑖𝑗=0. This ensures that the network is connected, so 𝐴 is irreducible although 𝐴 is not necessary symmetric. The nonlinear coupling function is ℎ(𝑥^𝑖(𝑡))=(𝑔(𝑥^𝑖₁(𝑡)),𝑔(𝑥^𝑖₂(𝑡)),…,𝑔(𝑥^𝑖_𝑛(𝑡)))^𝑇, 𝑖=1,2,…,𝑁.

3. Model Description

In this section, we write down some definitions, notations, and lemmas that will be used throughout this paper.

Definition 1 (see [32]). If lim_𝑡→∞𝑥^𝑖(𝑡)−𝑥^𝑗(𝑡)=0,lim_𝑡→∞𝑦^𝑖(𝑡)−𝑦^𝑗(𝑡)=0,lim_𝑡→∞𝑥^𝑖(𝑡)+𝑦^𝑖(𝑡)=0,𝑖,𝑗=1,2,…,𝑁, (5) then the driving network (3) and the response network (4) are said to achieve hybrid synchronization.

Definition 2 (see [36]). A nonlinear coupling function 𝑔(⋅)∶𝑅→𝑅 is said to belong to the acceptable nonlinear coupling function class, denoted by ℎ∈NCF(𝛾,𝛽), if there exist two nonnegative scalers 𝛾 and 𝛽 such that 𝑔(𝑥)−𝛾𝑥 satisfies the following Lipschitz condition: 𝑔𝑥₁−𝑔𝑥₂−𝛾𝑥₁−𝑥₂≤𝛽𝑥₁−𝑥₂ (6) Holds, for any 𝑥₁,𝑥₂∈𝑅.

Remark 3. Let 𝑟(𝑥)=𝑔(𝑥)−𝛾𝑥. Then the above condition (6) can be changed as 𝑟𝑥₁−𝑟𝑥₂≤𝛽𝑥₁−𝑥₂ (7) and if 𝑔(⋅) is an odd function, then we have 𝑟𝑥₁+𝑟𝑥₂≤𝛽𝑥₁+𝑥₂. (8)

Definition 4 (see [37]). 𝐴=[𝑎_𝑖𝑗]∈𝑅^𝑁×𝑁 is said to belong to class 𝐴₁, denoted by 𝐴∈𝐴₁, if(1)𝑎_𝑖𝑗≥0(𝑖≠𝑗), and 𝑎_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗, for 𝑖=1,2,…,𝑁, (2)𝐴 is irreducible. If 𝐴∈𝐴₁ is symmetrical, then we say that 𝐴 belongs to 𝐴₂, denoted by 𝐴∈𝐴₂.

Lemma 5 (see [36]). Assume that 𝐴∈𝐴₁. Then (1)rank(𝐴)=𝑁−1, that is, 0 is an eigenvalue of 𝐴 of multiplicity 1, and all nonzero eigenvalues of 𝐴 have negative real parts; (2)𝟏=(1,1,…,1)^𝑇 is the right eigenvector of 𝐴 corresponding to the eigenvalue 0 of multiplicity 1, that is, 𝐴⋅𝟏=0; (3)let 𝜉=(𝜉¹,𝜉²,…,𝜉^𝑁)^𝑇 be the left eigenvector of 𝐴 corresponding to the eigenvalue 0, that is, 𝜉^𝑇𝐴=0. Then, 𝜉^𝑖>0, 𝑖=1,2,…,𝑁, and its multiplicity is 1. In the following, we will always assume that ∑^𝑁_𝑖=1𝜉^𝑖=1.

Definition 6 (see [36]). Denote Ξ=diag{𝜉¹,𝜉²,…,𝜉^𝑁}, 𝐼_𝑁=diag{1,1,…,1}, and define 𝑈≡𝑢_𝑖𝑗=Ξ−𝜉𝜉^𝑇,𝑄=𝐼_𝑁−1𝑁𝟏⋅𝟏^𝑇. (9) Obviously, 𝑈,𝑄∈𝐴₂. Moreover, it is easy to check that if 𝐴∈𝐴₁ then 𝐴𝑄=𝐴.

Lemma 7 (see [37]). If 𝐴=(𝑎_𝑖𝑗)∈𝐴₂, then for any two vectors 𝑢=[𝑢₁,𝑢₂,…,𝑢_𝑛]^𝑇 and 𝑣=[𝑣₁,𝑣₂,…,𝑣_𝑛]^𝑇, we have 𝑢^𝑇𝐴𝑣=^𝑛𝑛_𝑖=1𝑗=1𝑢_𝑖𝑎_𝑖𝑗𝑣_𝑗=−_𝑗>𝑖𝑎_𝑖𝑗𝑢_𝑖−𝑢_𝑗𝑣_𝑖−𝑣_𝑗. (10)
Hypothesis. Let 𝑃=diag{𝑝₁,𝑝₂,…,𝑝_𝑛},𝑝_𝑖>0,𝑖=1,2,…,𝑛,𝑓(𝑥,𝑡)∈𝐶[𝑅^𝑛×[0,∞),𝑅^𝑛], satisfies 𝑥−𝑦^𝑇𝑃𝑓(𝑥,𝑡)−𝑓𝑦,𝑡−Δ𝑥−𝑦≤−𝛼𝑥−𝑦^𝑇𝑥−𝑦, (11) for some 𝛼>0, all 𝑥,𝑦∈𝑅^𝑛 and 𝑡>0 where Δ=diag{𝛿₁,𝛿₂,…,𝛿_𝑛}.

Remark 8. When 𝑓(𝑥,𝑡) is odd with regard to 𝑥, then we have 𝑥+𝑦^𝑇𝑃𝑓(𝑥,𝑡)+𝑓𝑦,𝑡−Δ𝑥+𝑦≤−𝛼𝑥+𝑦^𝑇𝑥+𝑦, (12) for some 𝛼>0, all 𝑥,𝑦∈𝑅^𝑛, and 𝑡>0.

4. Main Result

In this part, we investigate the hybrid synchronization of a class of nonlinearly coupled complex dynamical networks and obtain some criteria for hybrid synchronization.

Theorem 9. Suppose Hypothesis holds, 𝐴=(𝑎_𝑖𝑗)∈𝐴₁, 𝑔(⋅)∈𝑁𝐶𝐹(𝛾,𝛽), and 𝑔 is an odd function, 𝑓(𝑥,𝑡) is odd with respect to 𝑥. If there exist positive scalers 𝜃_𝑘, 𝜗_𝑘(𝑘=1,2,…,𝑛) such that the following inequalities 𝛿_𝑘𝑈+𝑐𝛾(Ξ𝐴)^𝑠+𝑐𝜃_𝑘2Ξ𝐴𝐴^𝑇Ξ+𝑐𝛽²2𝜃_𝑘𝑄≤0,𝑐𝛾𝐵^𝑠+𝑐2𝜗_𝑘𝐵𝐵^𝑇+𝛽²𝜗_𝑘𝐼_𝑁+𝜗_𝑘𝐼_𝑁𝑄≤0,𝑘=1,2,…,𝑛 (13) hold, where Ξ, 𝑈, 𝑄 are defined as in Definition 6, 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑙,0,…,0}∈𝑅^𝑁×𝑁 and 𝐵=𝐴−𝐷=(𝑏_𝑖𝑗)∈𝑅^𝑁×𝑁 with 𝑏_𝑖𝑖=𝑎_𝑖𝑖−𝑑_𝑖 (𝑖=1,2,…,𝑙), otherwise 𝑏_𝑖𝑗=𝑎_𝑖𝑗, then the nonlinearly coupled driving network (3) can achieve hybrid synchronization with the response network (4).

Proof. Denote 𝑋(𝑡)=𝑥¹(𝑡)^𝑇,𝑥²(𝑡)^𝑇,…,𝑥^𝑁(𝑡)^𝑇𝑇,𝐹(𝑋(𝑡))=𝑓𝑥¹(𝑡),𝑡^𝑇,𝑓𝑥²(𝑡),𝑡^𝑇𝑓𝑥,…,_𝑁(𝑡),𝑡^𝑇𝑇,𝐻(𝑋(𝑡))=ℎ𝑥¹(𝑡)^𝑇,ℎ𝑥²(𝑡)^𝑇,…,ℎ𝑥^𝑁(𝑡)^𝑇𝑇. (14) Then the driving network (3) can be written in a compact form as 𝑋(𝑡)=𝐹(𝑋(𝑡))+𝑐𝐴⊗𝐼_𝑛𝐻(𝑋(𝑡)). (15) In order to achieve the hybrid synchronization of two nonlinearly coupled complex networks with different local dynamical systems, we first choose the synchronization state of the drive network to be 𝑠^𝜉(𝑡)=∑^𝑁_𝑖=1𝜉^𝑖𝑥^𝑖(𝑡) so that the drive network can be synchronized with 𝑠^𝜉(𝑡). At the same time, we antisynchronize the sates 𝑥^𝑖(𝑡) of the drive network and the state 𝑦^𝑖(𝑡) of the response network using pinning control so that (5) is satisfied, that is, the drive and response networks are hybrid synchronized.
Let 𝑒^𝑖(𝑡)=𝑥^𝑖(𝑡)+𝑦^𝑖(𝑡). Then the antisynchronization error systems between the driving network (3) and the response network (4) can be written as 𝑑𝑒^𝑖(𝑡)𝑥𝑑𝑡=𝑓^𝑖(𝑡),𝑡+𝑓𝑦^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗ℎ𝑥^𝑗(𝑡)+ℎ𝑦^𝑗(𝑡)−𝑐𝑑_𝑖ℎ𝑥^𝑖(𝑡)+ℎ𝑦^𝑖(𝑡),𝑖=1,2,…,𝑙,𝑑𝑒^𝑖(𝑡)𝑥𝑑𝑡=𝑓^𝑖(𝑡),𝑡+𝑓𝑦^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗ℎ𝑥^𝑗(𝑡)+ℎ𝑦^𝑗(𝑡),𝑖=𝑙+1,𝑙+2,…,𝑁. (16) Construct a Lyapunov function candidate as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡), (17) where 𝑉₁(𝑡)=12^𝑁_𝑖=1𝜉^𝑖𝑥^𝑖(𝑡)−𝑥^𝜉(𝑡)^𝑇𝑃𝑥^𝑖(𝑡)−𝑥^𝜉(𝑡)=12𝑋(𝑡)^𝑇()𝑋(𝑡),𝑉𝑈⊗𝑃₂(𝑡)=12^𝑁_𝑖=1𝑒^𝑖(𝑡)^𝑇𝑃𝑒^𝑖(𝑡). (18)
Differentiating the function 𝑉₁(𝑡) along the drive network (15), we have 𝑉̇₁(𝑡)=𝑋(𝑡)^𝑇(𝑈⊗𝑃)𝐹(𝑋(𝑡))+𝑐𝐴⊗𝐼_𝑛𝐻(𝑋(𝑡(𝑡)))=𝑋^𝑇(𝐹(𝑋(𝑡−()𝑋(𝑡(𝑡)𝑈⊗𝑃)())𝐼⊗Δ))+𝑋^𝑇()×)𝑋(𝑡)𝑈⊗𝑃(𝐼⊗Δ+𝑐𝐴⊗𝐼_𝑛𝐻(𝑋(𝑡))=𝑉₁₁(𝑡)+𝑉₁₂(𝑡). (19)
Because of Hypothesis (11) and Lemma 7, we have 𝑉₁₁(𝑡)=𝑋(𝑡)^𝑇(𝑈⊗𝑃)(𝐹(𝑋(𝑡))−(𝐼⊗Δ)𝑋(𝑡))=−^𝑁_𝑖>𝑗𝑢_𝑖𝑗𝑥^𝑖(𝑡)−𝑥^𝑗(𝑡)^𝑇×𝑃𝑓𝑥^𝑖(𝑡),𝑡−𝑓𝑥^𝑗(𝑡)𝑥,𝑡−Δ^𝑖(𝑡)−𝑥^𝑗(𝑡)≤𝛼^𝑁_𝑖>𝑗𝑢_𝑖𝑗𝑥^𝑖(𝑡)−𝑥^𝑗(𝑡)^𝑇𝑥^𝑖(𝑡)−𝑠^𝑗(𝑡)=−𝛼𝑋(𝑡)^𝑇(𝑡)𝛼𝑈⊗𝐼𝑋≤−max_𝑘𝑝_𝑘𝑋(𝑡)^𝑇(𝑈⊗𝑃)𝑋(𝑡)=−2𝛼max_𝑘𝑝_𝑘𝑉₁(𝑡). (20)
Denote 𝑥_𝑘(𝑡)=(𝑥¹_𝑘(𝑡),𝑥²_𝑘(𝑡),…,𝑥^𝑁_𝑘(𝑡))^𝑇, ℎ_𝑘(𝑥_𝑘(𝑡))=(𝑔(𝑥¹_𝑘(𝑡)),𝑔(𝑥²_𝑘(𝑡)),…,𝑔(𝑥^𝑁_𝑘(𝑡)))^𝑇 for 𝑘=1,2,…,𝑛 and rewrite 𝑉₁₂(𝑡) as follows 𝑉₁₂(𝑡)=𝑋(𝑡)^𝑇(𝑈⊗𝑃Δ)𝑋(𝑡)+𝑐𝑋(𝑡)^𝑇()𝐻(𝑋(𝑡=𝑈𝐴⊗𝑃))_𝑛𝑘=1𝑝_𝑘𝛿_𝑘𝑥_𝑘(𝑡)^𝑇𝑈𝑥_𝑘(𝑡)+𝑐^𝑛_𝑘=1𝑝_𝑘𝑥_𝑘(𝑡)^𝑇Ξ𝐴ℎ_𝑘𝑥_𝑘(𝑡)=^𝑛_𝑘=1𝑝_𝑘𝑥_𝑘(𝑡)^𝑇𝛿_𝑘𝑈+𝑐𝛾Ξ𝐴𝑥_𝑘(𝑡)+𝑐^𝑛_𝑘=1𝑝_𝑘𝑥_𝑘(𝑡)^𝑇Ξ𝐴ℎ_𝑘𝑥_𝑘(𝑡)−𝛾𝑥_𝑘(𝑡). (21) Since 𝑈𝐴=Ξ𝐴, 𝐴∈𝐴₁, and from the property of 𝑄 in Definition 6 gives 𝑥_𝑘(𝑡)^𝑇Ξ𝐴ℎ_𝑘𝑥_𝑘(𝑡)−𝛾𝑥_𝑘(𝑡=𝑥_𝑘)(𝑡)^𝑇Ξ𝐴𝑄ℎ_𝑘𝑥_𝑘(𝑡)−𝛾𝑥_𝑘(𝑡)≤12𝜃_𝑘𝑥_𝑘(𝑡)^𝑇Ξ𝐴𝐴^𝑇Ξ𝑥_𝑘(𝑡)+1𝜃_𝑘ℎ_𝑘𝑥_𝑘(𝑡)−𝛾𝑥_𝑘(𝑡)^𝑇×𝑄ℎ_𝑘𝑥_𝑘(𝑡)−𝛾𝑥_𝑘(𝑡). (22)
By Lemma 7 and 𝑔∈NCF(𝛾,𝛽), we have ℎ_𝑘𝑥_𝑘(𝑡)−𝛾𝑥_𝑘(𝑡)^𝑇𝑄ℎ_𝑘𝑥_𝑘(𝑡)−𝛾𝑥_𝑘(𝑡)=−_𝑖>𝑖𝑄_𝑖𝑗𝑔𝑥^𝑗_𝑘(𝑡)−𝛾𝑥^𝑗_𝑘(𝑡)−𝑔𝑥^𝑖_𝑘(𝑡)+𝛾𝑥^𝑖_𝑘(𝑡)²≤−𝛽²_𝑖>𝑗𝑄_𝑖𝑗𝑥^𝑗_𝑘(𝑡)−𝑥^𝑖_𝑘(𝑡)²=𝛽²𝑥_𝑘(𝑡)^𝑇𝑄𝑥_𝑘(𝑡). (23)
Combining inequalities (22) and (23), we obtain 𝑐^𝑛_𝑘=1𝑝_𝑘𝑥_𝑘(𝑡)^𝑇Ξ𝐴ℎ_𝑘𝑥_𝑘(𝑡)−𝛾𝑥_𝑘(𝑡≤𝑐)2^𝑛_𝑘=1𝑝_𝑘𝑥_𝑘(𝑡)^𝑇𝜃_𝑘Ξ𝐴𝐴^𝑇𝛽Ξ+²𝜃_𝑘𝑄𝑥_𝑘(𝑡). (24) Substituting inequalities (24) into (21), we get 𝑉₁₂(𝑡)≤^𝑛_𝑘=1𝑝_𝑘𝑥_𝑘(𝑡)^𝑇×𝛿_𝑘𝑐𝑈+𝑐𝛾Ξ𝐴+2𝜃_𝑘Ξ𝐴𝐴^𝑇𝑐Ξ+2𝛽²𝜃_𝑘𝑄×𝑥_𝑘(𝑡)≤0, (25) which means that 𝑑𝑉₁(𝑡)𝑑𝑡≤−2𝛼max_𝑘𝑝_𝑘𝑉₁(𝑡)≜−𝜙𝑉₁(𝑡),𝜙=−2𝛼max_𝑘𝑝_𝑘<0. (26) Thus, 𝑉₁(𝑡)=𝑜(𝑒^−𝜙𝑡). From the definition of 𝑉₁(𝑡), we have (𝑥^𝑖(𝑡)−𝑥^𝜉(𝑡))^𝑇(𝑥^𝑖(𝑡)−𝑥^𝜉(𝑡))=𝑜(𝑒^−𝜙𝑡) for 𝑖=1,2,…,𝑁, which means 𝑥^𝑖(𝑡)−𝑥^𝜉(𝑡) converges to zero exponentially. Therefore, 𝑥^𝑖(𝑡) can be synchronized with 𝑥^𝜉(𝑡) globally and exponentially.
Evaluating the time derivative of 𝑉₂(𝑡) along the trajectory of (16), we get 𝑑𝑉₂(𝑡)=𝑑𝑡^𝑁_𝑖=1𝑒^𝑖(𝑡)^𝑇𝑃𝑓𝑥^𝑖(𝑡),𝑡+𝑓𝑦^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗ℎ𝑥^𝑖(𝑡)+ℎ𝑦^𝑖(𝑡)−𝑐𝑑_𝑖ℎ𝑥^𝑗(𝑡)+ℎ𝑦^𝑗(𝑡)=^𝑁_𝑖=1𝑒^𝑖(𝑡)^𝑇𝑃𝑓𝑥^𝑖(𝑡),𝑡+𝑓𝑦^𝑖(𝑡)𝑥,𝑡−Δ^𝑖(𝑡)+𝑦^𝑖(𝑡)+^𝑁_𝑖=1𝑒^𝑖(𝑡)^𝑇𝑃Δ𝑒^𝑖(𝑡)+𝑐^𝑁_𝑖=1𝑒^𝑖(𝑡)^𝑇𝑃^𝑁_𝑗=1𝑏_𝑖𝑗ℎ𝑥^𝑗(𝑡)+ℎ𝑦^𝑗(𝑡)=𝑉₂₁(𝑡)+𝑉₂₂(𝑡)+𝑉₂₃(𝑡). (27)
By Hypothesis and because 𝑓 is odd respect to 𝑥, we obtain 𝑉₂₁(𝑡)=^𝑁_𝑖=1𝑒^𝑖(𝑡)^𝑇𝑃𝑓𝑥^𝑖(𝑡),𝑡+𝑓𝑦^𝑖(𝑡)𝑥,𝑡−Δ^𝑖(𝑡)+𝑦^𝑖(𝑡)≤−𝛼^𝑁_𝑖=1𝑒^𝑖(𝑡)^𝑇𝑒^𝑖(𝑡). (28)
Let 𝑒_𝑘̃(𝑡)=𝑥_𝑘(𝑡)+𝑦_𝑘(𝑡). Since 𝑔∈NCF(𝛾,𝛽), ℎ_𝑘(𝑥_𝑘(𝑡)) can be as ℎ_𝑘(𝑥_𝑘(𝑡))=𝛾𝑥_𝑘(𝑡)+𝑟_𝑘̃(𝑥_𝑘(𝑡)) where 𝑟_𝑘̃(𝑥_𝑘(𝑡))=(𝑟(𝑥¹_𝑘(𝑡)),𝑟(𝑥²_𝑘(𝑡)),…,𝑟(𝑥^𝑁_𝑘(𝑡)))^𝑇, 𝑘=1,2,…,𝑛, we have 𝑉₂₂(𝑡)=𝑐^𝑁_𝑖=1𝑒^𝑖(𝑡)^𝑇𝑃^𝑁_𝑗=1𝑏_𝑖𝑗ℎ𝑥^𝑗(𝑡)+ℎ𝑦^𝑗(𝑡)=𝑐^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝐵ℎ_𝑘𝑥_𝑘(𝑡)+ℎ_𝑘𝑦_𝑘(𝑡)=𝑐𝛾^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝐵^𝑠𝑒_𝑘̃(𝑡)+𝑐^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝐵𝑟_𝑘̃𝑥_𝑘(𝑡)+𝑟_𝑘̃𝑦_𝑘(𝑡). (29) Since 𝑔 is odd and there exists 𝜗_𝑘>0 for 1≤𝑘≤𝑛, we have 𝑒_𝑘̃(𝑡)^𝑇𝐵𝑟_𝑘̃𝑥_𝑘(𝑡)+𝑟_𝑘̃𝑦_𝑘(𝑡≤1)2𝜗_𝑘𝑒_𝑘̃(𝑡)^𝑇𝐵𝐵^𝑇𝑒_𝑘̃(𝑡)+1𝜗_𝑘𝑟_𝑘̃𝑥_𝑘(𝑡)+𝑟_𝑘̃𝑦_𝑘(𝑡)^𝑇×𝑟_𝑘̃𝑥_𝑘(𝑡)+𝑟_𝑘̃𝑦_𝑘(𝑡)=12𝜗_𝑘𝑒_𝑘̃(𝑡)^𝑇𝐵𝐵^𝑇𝑒_𝑘̃(𝑡)+1𝜗_𝑘^𝑁_𝑠=1𝑟𝑥^𝑠_𝑘(𝑡)+𝑟𝑦^𝑠_𝑘(𝑡)²≤12𝜗_𝑘𝑒_𝑘̃(𝑡)^𝑇𝐵𝐵^𝑇𝑒_𝑘̃(𝑡)+𝛽²𝜗_𝑘^𝑁_𝑠=1𝑥^𝑠_𝑘(𝑡)+𝑦^𝑠_𝑘(𝑡)²=12𝑒_𝑘̃(𝑡)^𝑇𝜗_𝑘𝐵𝐵^𝑇+𝛽²𝜗_𝑘𝐼_𝑁𝑒_𝑘̃(𝑡). (30) Substituting inequality (30) into (29), we get 𝑉₂₂(𝑡)≤^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝑐𝛾𝐵^𝑠+𝑐2𝜗_𝑘𝐵𝐵^𝑇+𝛽²𝜗_𝑘𝐼_𝑁𝑒_𝑘̃(𝑡). (31) On the other hand, 𝑉₂₃(𝑡) can be directly written as 𝑉₂₃(𝑡)=^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝛿_𝑘𝑒_𝑘̃(𝑡). (32)
Finally, substituting (28), (31), and (32) into (27), we get 𝑑𝑉₂(𝑡)𝑑𝑡≤−𝛼^𝑁_𝑖=1𝑒^𝑖(𝑡)^𝑇𝑒^𝑖(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇×𝑐𝛾𝐵^𝑠+𝑐2𝜗_𝑘𝐵𝐵^𝑇+𝛽²𝜗_𝑘𝐼_𝑁+𝛿_𝑘𝐼_𝑁𝑒_𝑘̃(𝑡). (33) It now implies that 𝑑𝑉₂(𝑡)𝑑𝑡≤−𝛼^𝑁_𝑖=1𝑒^𝑖(𝑡)^𝑇𝑒^𝑖(𝑡), (34) and 𝑉₂(𝑡)=𝑜(𝑒^−𝛼𝑡) follows. Thus, 𝑒^𝑖(𝑡)→0 (𝑡→∞) for 𝑖=1,2,…,𝑁. This implies that lim_𝑡→+∞𝑥^𝑖(𝑡)+𝑦^𝑖(𝑡)=0,𝑖=1,2,…,𝑁. (35)
The proof of the theorem is complete.

From the proof, one can see that the difficulty in investigating the hybrid synchronization with an asymmetrical coupling matrix 𝐴 lies in how to deal with the terms 𝑥_𝑘(𝑡)^𝑇Ξ𝐴ℎ_𝑘(𝑥_𝑘(𝑡)) of (22) and 𝑒_𝑘̃(𝑡)^𝑇𝐵(ℎ_𝑘(𝑥_𝑘(𝑡))+ℎ_𝑘(𝑦_𝑘(𝑡))) of (31), where Ξ𝐴 and 𝐵 are asymmetrical. If 𝐴 is symmetric, then Ξ𝐴 and 𝐵 are symmetric. In the case, by Lemma 7, we have 𝑐^𝑛_𝑘=1𝑝_𝑘𝑥_𝑘(𝑡)^𝑇𝐴ℎ_𝑘𝑥_𝑘(𝑡)=𝑐^𝑛_𝑘=1𝑝_{𝑘𝑖>𝑗}𝑎_𝑖𝑗𝑥^𝑖_𝑘−𝑥^𝑗_𝑘𝑔𝑥^𝑖_𝑘−𝑔𝑥^𝑗_𝑘⩽−𝑐𝛾^𝑛_𝑘=1𝑝_{𝑘𝑖>𝑗}𝑎_𝑖𝑗𝑥^𝑖_𝑘−𝑥^𝑗_𝑘²=𝑐𝛾^𝑛_𝑘=1𝑝_𝑘𝑥_𝑘(𝑡)^𝑇𝐴𝑥_𝑘(𝑡), (36) where 𝛾 is defined as 𝛾=inf_{𝑤1≠𝑤2}(𝑔(𝑤₁)−𝑔(𝑤₂))/(𝑤₁−𝑤₂)>0, for 𝑤₁,𝑤₂∈𝑅.

Therefore 𝑉₁₂(𝑡)≤^𝑛_𝑘=1𝑝_𝑘𝑥_𝑘(𝑡)^𝑇𝛿_𝑘𝑐𝑈+𝑁𝛾𝐴𝑥_𝑘(𝑡), (37) on the other hand, let 𝑒^𝑖_𝑘̃(𝑡)=𝑥^𝑖_𝑘(𝑡)+𝑦^𝑖_𝑘(𝑡), from (29), we get 𝑉₂₂(𝑡)=𝑐^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝐵ℎ𝑥_𝑘(𝑡)+ℎ𝑦_𝑘(𝑡)=𝑐^𝑛_𝑘=1𝑝_{𝑘𝑖>𝑗}𝑏_𝑖𝑗𝑒^𝑖_𝑘̃(𝑡)−𝑒^𝑗_𝑘̃(𝑡)×𝑔𝑥^𝑖_𝑘(𝑡)+𝑔𝑦^𝑖_𝑘(𝑡)−𝑔𝑥^𝑗_𝑘(𝑡)−𝑔𝑦^𝑗_𝑘(𝑡)=𝑐𝛾^𝑛_𝑘=1𝑝_{𝑘𝑖>𝑗}𝑏_𝑖𝑗𝑒^𝑖_𝑘̃(𝑡)−𝑒^𝑗_𝑘̃(𝑡)²=𝑐𝛾^𝑛_𝑘=1𝑝_𝑘𝑒_𝑘̃(𝑡)^𝑇𝐵𝑒_𝑘̃(𝑡). (38) Notice (36), (37), and (38), we obtain the following corollaries.

Corollary 10. Suppose 𝑓(⋅) satisfy Hypothesis. 𝐴=(𝑎_𝑖𝑗)∈𝐴₂, 𝑔(⋅)∈𝑁𝐶𝐹(𝛾,𝛽), and 𝑔 is odd function, 𝑓(𝑥,𝑡) is odd with regard to 𝑥, if the following inequalities 𝛿_𝑘𝑐𝑈+𝑁𝛾𝐴≤0,𝑐𝛾𝐵+𝛿_𝑘𝐼_𝑁≤0,𝑘=1,2,…,𝑛 (39) hold, then the nonlinearly coupled driving network (3) can hybrid synchronized to the response network (4).
If 𝑔∈𝑁𝐶𝐹(𝛾,0), then the driving network (3) becomes linear-coupling systems 𝑥̇^𝑖(𝑡)=𝑓𝑥^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑥^𝑗(𝑡),𝑖=1,2,…,𝑁, (40) respectively, the pinning controlled response network (4) can be rewritten as 𝑦̇^𝑖(𝑡)=𝑓𝑦^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑦^𝑗(𝑡)−𝑐𝑑_𝑖𝑥^𝑖(𝑡)+𝑦^𝑖(𝑡),𝑦̇𝑖=1,2,…,𝑙,_𝑖(𝑡)=𝑓𝑦^𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑦^𝑗(𝑡),𝑖=𝑙+1,𝑙+2,…,𝑁. (41) As a direct consequence of Theorem 9, we obtain the following result.

Corollary 11. Suppose Hypothesis holds. 𝐴=(𝑎_𝑖𝑗)∈𝐴₁, 𝑔(⋅)∈𝑁𝐶𝐹(𝛾,0), and 𝑔 is odd function, 𝑓(𝑥,𝑡) is odd with regard to 𝑥, if the following inequalities 𝛿_𝑘𝑈+𝑐𝛾(Ξ𝐴)^𝑠≤0,𝑐𝛾𝐵^𝑠+𝛿_𝑘𝐼_𝑁≤0 (42) hold, then the linearly coupled driving network (40) can hybrid synchronized to the response network (41).
The case has been discussed in [30]. Therefore, one can see the main result of the paper as a nontrivial extension of the Theorem 1 of [30].

Remark 12. By using the Lyapunov method combined with some other technique, the hybrid synchronization criterion of the pinning-controlled dynamical networks has been obtained. It means that the outer antisynchronization between system (3) and system (4) while the inner synchronization in them. Our result displays that pinning control is effective for the hybrid synchronization of complex dynamical network. Pinning impulsive control strategy has been used effectively for the synchronization of stochastic dynamical networks with nonlinear coupling in [33, 35]. Our pinning control strategy for the hybrid synchronization of two nonlinearly coupled complex networks in this paper differs from them.

5. Numerical Simulation

In this section, we given numerical simulation to verify the theorem given in the previous section. In order to verify our results, we consider the driving complex network [32] as 𝑥̇^𝑖(𝑡)=𝑓𝑥^𝑖(𝑡),𝑡+𝑐⁵_𝑗=1𝑎_𝑖𝑗ℎ𝑥^𝑗(𝑡),𝑖=1,2,…,5 (43) and the response networks [32] as 𝑦̇^𝑖(𝑡)=𝑓𝑦^𝑖(𝑡),𝑡+⁵_𝑗=1𝑎_𝑖𝑗ℎ𝑦^𝑗(𝑡)−𝑐𝑑_𝑖𝑥^𝑖(𝑡)+𝑦^𝑖(𝑡)𝑦̇,𝑖=1,2,_𝑖(𝑡)=𝑓𝑦^𝑖(𝑡),𝑡+⁵_𝑗=1𝑎_𝑖𝑗ℎ𝑦^𝑗(𝑡),𝑖=3,4,5, (44) where 𝑓𝑥^𝑖(𝑡),𝑡=−𝑆𝑥^𝑖(𝑡)+𝑇𝑤𝑥^𝑖(𝑡),𝑆=,3.2−100−11−1014.870,𝑇=5.900000000 (45) and 𝑤(𝑥)=(|𝑥+1|−|𝑥−1|)/2.

Let 𝑃=𝐼₃, Δ=𝛿𝐼₃, if there exists a positive-definite diagonal matrix Σ=diag{𝜎₁,…,𝜎_𝑛}, such that 2𝑃(𝑆+Δ)−Σ−𝑃𝑇−𝑇^𝑇𝑃Σ>0, (46) then 𝑓(𝑥^𝑖(𝑡),𝑡) satisfies the condition (11) [38] in Hypothesis. Using Matlab LMI Control Toolbox, we get 𝛿=9.0637.

Example 13. As for the asymmetrical coupling matrix 𝐴, we use 𝐴=−0.7940.7940000−0.840.31100.5290.1660−0.166000.60200.263−0.86500000.654−0.654, (47) whose left eigenvector corresponding to eigenvalue 0 is 𝜉=(0.3157,0.2984,0.8483,0.1825,0.2414)^𝑇. It is clear that 𝜉^𝑇𝐿=0. Let the nonlinear function 𝑔(𝑥(𝑡))=5𝑥(𝑡)+sin(𝑥(𝑡)), so 𝑔(⋅)∈NCF(5,1) (𝛾=5, 𝛽=1).
The initial values are chosen as 𝑥¹(0)=(−0.4946,−4.1618,−2.7102)^𝑇,𝑥²(0)=(4.1334,−3.4762,3.2582)^𝑇,𝑥³(0)=(0.3834,4.9613,−4.2182)^𝑇,𝑥⁴(0)=(−0.5732,−3.9335,4.619)^𝑇,𝑥⁵(0)=(−4.9537,2.7491,3.173)^𝑇,𝑦¹(0)=(−3.7009,0.6882,−0.3061)^𝑇,𝑦²(0)=(−4.881,−1.6288,−3.3782)^𝑇,𝑦³(0)=(2.9428,−1.8878,0.2853)^𝑇,𝑦⁴(0)=(−3.3435,1.0198,−2.3703)^𝑇,𝑦⁵(0)=(1.5408,1.8921,2.4815)^𝑇, (48) and the errors between nodes are defined by 𝐸_𝑥=_𝑖>𝑗𝑥^𝑖(𝑡)−𝑥^𝑗(𝑡,𝐸_𝑦)=_𝑖>𝑗𝑦^𝑖(𝑡)−𝑦^𝑗(𝑡,𝐸_𝑥𝑦)=⁵⁵_𝑖=1𝑗=1𝑥^𝑖(𝑡)+𝑦^𝑗(𝑡), (49) as measures of hybrid synchronization.
Let 𝑐=11.868, 𝜃_𝑘=2.309, 𝜗_𝑘=0.965, (𝑘=1,2,3), and 𝑑₁=𝑑₂=1.76, 𝑑₃=𝑑₄=𝑑₅=0, the calculations show that eigenvalues of matrix 𝛿_𝑘𝑈+𝑐𝛾(Ξ𝐴)^𝑠+(c𝜃_𝑘/2)Ξ𝐴𝐴^𝑇Ξ+(𝑐𝛽²/2𝜃_𝑘)𝑄 are −16.3119, −0.1990, −7.8756, −3.8328, and −5.5047, and 𝑐𝛾𝐵^𝑠+(𝑐/2)(𝜗_𝑘𝐵𝐵^𝑇+(𝛽²/𝜗_𝑘)𝐼_𝑁)+𝛿_𝑘𝐼_𝑁 are −0.0823, −21.76, −45.804, −103.63, and −77.347.
The trajectories evolution of the pinning control gains are shown in Figure 1. The evolution of synchronization error of the driver and the response system in Figures 2(a) and 2(b), respectively. The evolution of synchronization error of between the driver and the response system Figure 2(c).

(a)

(b)

(c)

Figure 1 

The trajectories evolution of the state variable, (a) 𝑥𝑖1(𝑡), 𝑦𝑖1(𝑡), (b) 𝑥𝑖2(𝑡), 𝑦𝑖2(𝑡), and (c) 𝑥𝑖3(𝑡), 𝑦𝑖3(𝑡)(𝑖=1,2,…,5).

(a)

(b)

(c)

Figure 2 

(a) The synchronization error evolution of the driver networks 𝐸𝑥(𝑡); (b) The synchronization error evolution of the response networks 𝐸𝑦(𝑡); (c) The anti-synchronization error evolution between the driver networks and the response networks 𝐸𝑥𝑦(𝑡) with the pinning control.

Example 14. As for the symmetrical coupling matrix 𝐴, we use 𝐴=−2.4070.7920.95900.6560.792−0.7920000.9590−2.5720.9340.679000.934−0.93400.65600.6790−1.335. (50) Let 𝑐=9.182, 𝑑₁=𝑑₁=1.76, and 𝑑₃=𝑑₄=𝑑₅=0, the calculations show that eigenvalues of matrix 𝛿_𝑘𝑈+(𝑐/𝑁)𝛾𝐴 are −30.4559, −18.9805, −15.2688, −9.3311, and −0.1760, and 𝑐𝛾𝐵+𝛿_𝑘𝐼_𝑁 are: −59.0165, −36.2877, −28.4531, −14.4174, and −0.2446.
The initial values are chosen as above. The trajectories evolution of the pinning control gains are shown in Figure 3. The evolution of synchronization error of the driver and the response system in Figures 4(a) and 4(b), respectively. The evolution of synchronization error of between the driver and the response system in Figure 4(c).

(a)

(b)

(c)

Figure 3 

The trajectories evolution of the state variable, (a) 𝑥𝑖1(𝑡), 𝑦𝑖1(𝑡), (b) 𝑥𝑖2(𝑡), 𝑦𝑖2(𝑡), and (c) 𝑥𝑖3(𝑡), 𝑦𝑖3(𝑡)(𝑖=1,2,…,5).

(a)

(b)

(c)

Figure 4 

(a) The synchronization error evolution of the driver networks 𝐸𝑥(𝑡). (b) The synchronization error evolution of the response networks 𝐸𝑦(𝑡). (c) The antisynchronization error evolution between the driver networks and the response networks 𝐸𝑥𝑦(𝑡) with the pinning control.

6. Conclusions and Future Directions

In this paper, we considered the hybrid synchronization of two nonlinearly coupled complex dynamical networks with asymmetrical coupling matrices under the pinning control scheme. By placing a small number of feedback controllers on some nodes, we obtained some criteria for the hybrid synchronization of such dynamical networks based on Lyapunov Stability Theory. It is shown that under certain conditions, two nonlinearly coupled complex networks can achieve an intriguing hybrid synchronization: the outer antisynchronization between the driving network and the response network and the inner complete synchronization between the driving network and the response network, respectively. Numerical examples were also provided to demonstrate the effectiveness of the theoretical result.

Because the random phenomenon appears frequently in dynamical complex network, in our further efforts, this feature should be taken into account in order to solve problems more practically. More precisely, our future works may be extended to the consensus problem of the multiagent systems with stochastic disturbance, the synchronization issue of the dynamical complex networks with stochastic disturbance, and so on. On the other side, control methods would not be limited to the feedback control any more. For example, in real world, the states of nodes in the networks often suffer from instantaneous perturbations or abrupt changes at certain instants, such as switching phenomena, frequency change, or sudden noise. In order to investigate the synchronization matters for this situation, suitable pinning impulsive controllers could be applied to deal with the impulsive-coupled dynamical networks. Moreover, couplings or communications between nodes in this paper are considered to be continuous. However, there are other modes of the information exchanging among the nodes, such as impulsive communication and intermittent communication. The studies on these problems will be also interesting and meaningful in the further research work.

Acknowledgments

This work was supported by the National Science Foundation of China under Grant no. 61273220, the Guangdong Education University Industry Cooperation Project (2009B090300355), and the Shenzhen Basic Research Project (JC201006010743A, JCYJ20120613105730482). The authors are very grateful to the reviewers and editors for their valuable comments on this paper.