Synchronization of General Complex Networks with Hybrid Couplings and Unknown Perturbations

Abstract

The issue of synchronization for a class of hybrid coupled complex networks with mixed delays (discrete delays and distributed delays) and unknown nonstochastic external perturbations is studied. The perturbations do not disappear even after all the dynamical nodes have reached synchronization. To overcome the bad effects of such perturbations, a simple but all-powerful robust adaptive controller is designed to synchronize the complex networks even without knowing a priori the functions and bounds of the perturbations. Based on Lyapunov stability theory, integral inequality Barbalat lemma, and Schur Complement lemma, rigorous proofs are given for synchronization of the complex networks. Numerical simulations verify the effectiveness of the new robust adaptive controller.

1. Introduction

Over the past decade, complex networks have attracted much attention from authors of many disciplines since the pioneer works of Watts and Strogatz [1, 2]. In fact, many phenomena in nature and our daily life can be explained by using complex networks, such as the Internet, World Wide Web, social networks, and neural networks. A complex network can be considered as a graph which consists of a set of nodes and edges connecting these nodes [3].

In recent years, chaos synchronization [3–7] has been intensively studied due to its important applications in many different areas, such as secure communication, biological systems, and information science [8–11]. Particularly, the synchronization of all the dynamical nodes in complex networks has become a hot research topic [3], and several results have been appeared in the literature. The authors of [12] studied the synchronization in complex networks with switching topology. In [13], Wu and Jiao investigated the synchronization in complex dynamical networks with nonsymmetric coupling. They showed that the synchronizability of a dynamical network with nonsymmetric coupling is not always characterized by its second-largest eigenvalue, even though all the eigenvalues of the nonsymmetric coupling matrix are real. Liu and Chen [14] gave some criteria for the global synchronization of complex networks in virtual of the left eigenvector corresponding to the zero eigenvalue of the coupling matrix. For a given network with identical node dynamics, the authors of [15] showed that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix. Some synchronization criteria were given in [16–19] for coupled neural networks with or without delayed couplings. In [20], the robust impulsive synchronization of coupled delayed neural networks with uncertainties is considered; several new criteria are obtained to guarantee the robust synchronization via impulses.

Complex networks have the properties of robustness and fragility. A complex network can synchronize itself when parameter mismatch is within some limit. If parameter mismatch exceeds this limit, networks cannot realize synchronization themselves. Thus the controlled synchronization of coupled networks is believed to be a rather significant topic in both theoretical research and practical applications [21–29]. Some effective control scheme has been proposed, for instance, state feedback control with constant control gains, impulsive control, intermittent control, and adaptive control. Adaptive control method receives particular attention of researchers in recently rears. In [3], the authors studied synchronization in complex networks by using distributed adaptive control scheme. By designing a simple adaptive controller, authors of [23] investigated the locally and globally adaptive synchronization of an uncertain complex dynamical network. Authors in [24] investigated synchronization of neural networks with time-varying delays and distributed delays via adaptive control method. By using the adaptive feedback control scheme, Chen and Zhou [25] studied synchronization of complex nondelayed networks and Cao et al. [26] investigated the complete synchronization in an array of linearly stochastically coupled identical networks with delays. By using adaptive pinning control method, Zhou et al. [27] studied local and global synchronization of complex networks without delays, authors of [28, 29] considered the global synchronization of the complex networks with nondelayed and delayed couplings and the authors of [30] investigated lag synchronization of complex networks via state feedback pinning strategy. Outer synchronization of complex delayed networks with uncertain parameters was considered by using adaptive coupling in [31]. However, models in the previous references are special; that is, each of them does not consider general complex networks in which every dynamical node has mixed delays (discrete delay and distributed delay), and the complex networks have nondelayed, discrete-delayed, and distributed-delayed couplings.

Complex networks are always affected by some unknown external perturbations due to environmental causes and human causes. White noises brought by some random fluctuations in the course of transmission and other probabilities causes have received extensive attention in the literatures [21, 24, 32–35]. However, not all the external perturbations are white noise, and some of them may be nonlinear and nonstochastic perturbations. When complex networks are disturbed by nonlinear and nonstochastic perturbations, the states of the nodes will be changed dramatically, which will affect the stability and synchronization of the complex networks. Due to the fragility of complex networks, if some important nodes are perturbed by such external perturbations, whole states of the network will be affected or even the network cannot operate normally. Hence, how to realize synchronization of all nodes for complex networks with uncertain nonlinear nonstochastic external perturbations is an urgent practical problem to be solved. Obviously, the controllers for stability and synchronization of stochastic perturbations are not applicable to the case of nonlinear nonstochastic perturbations, especially when the functions and bounds of the perturbations are unknown. Therefore, to enhance antiperturbations capability and to realize synchronization of complex networks, more effective controller should be designed.

Motivated by the previous analysis, in this paper, a class of more general complex networks is proposed. The new model has nondelayed, discrete-delayed, and distributed-delayed couplings, and every dynamical node has mixed delays. Unknown nonstochastic external perturbations to the complex networks are also considered. Then we study the global complete synchronization of the proposed model. A new simple but robust adaptive controller is designed to overcome the effects of such perturbations and synchronize the complex networks even without knowing the exact functions and bounds of the perturbations. Moreover, the adaptive controller can also synchronize coupled systems with stochastic perturbations since it includes existing adaptive controller as special case. Two cases are considered: all nodes or partial nodes are perturbed. All nodes should be controlled for the former case. Pinning control scheme can also be used for the latter case. Based on Lyapunov stability theory, integral inequality, Barbalat lemma, and Schur Complement lemma, rigorous proofs are given for synchronization of the complex networks with unknown perturbations of the previous two cases. It should be noted that our new adaptive controllers can also prevent external perturbations. Therefore, the new adaptive controllers are better than those in [23–29]. Numerical simulations verify the effectiveness of our theoretical results.

Notations. In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. 𝐼_𝑁 denotes the identity matrix of 𝑁 dimension. The Euclidean norm in ℝ^𝑛 is denoted as ‖⋅‖; accordingly, for vector 𝑥∈ℝ^𝑛, ‖𝑥‖=𝑥^𝑇𝑥, where 𝑇 denotes transposition. 𝐴=(𝑎_𝑖𝑗)_𝑚×𝑚 denotes a matrix of 𝑚 dimension, ‖𝐴‖=𝜆_max(𝐴^𝑇𝐴), and 𝐴^𝑠=(1/2)(𝐴+𝐴^𝑇). 𝐴>0 or 𝐴<0 denotes that the matrix 𝐴 is symmetric and positive or negative definite matrix. 𝜆_min(𝐴^𝑠) is the minimum eigenvalues of the symmetric matrices 𝐴^𝑠, and 𝐴_𝑙 denotes the matrix of the first 𝑙 row-column pairs of 𝐴. 𝐴^𝑐_𝑙 denotes the minor matrix of matrix 𝐴 by removing all the first 𝑙 row-column elements of 𝐴.

The rest of this paper is organized as follows. In Section 2, a class of general complex networks with mixed delays and external perturbations is proposed. Some necessary assumptions and lemmas are also given in this section. In Section 3, synchronization of the complex networks with all nodes perturbed is studied. Synchronization with only partial nodes perturbed is considered in Section 4. Then, in Section 5, numerical simulations are given to show the effectiveness of our results. Finally, in Section 6, conclusions are given.

2. Preliminaries

The general complex networks consisting of 𝑁 identical nodes with external perturbations and mixed-delay couplings are described as 𝑥̇_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)+𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑓𝑥_𝑖(𝑠)𝑑𝑠+𝐼(𝑡)+𝛼^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑥_𝑗(𝑡)+𝛽^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑥_𝑗(𝑡−𝜏(𝑡))+𝛾^𝑁_𝑗=1𝑤_𝑖𝑗Λ^𝑡_{(𝑡)𝑡−𝜃}𝑥_𝑗(𝑠)𝑑𝑠+𝜎_𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)),^𝑡_{(𝑡)𝑡−𝜃}𝑥_𝑖(𝑠)𝑑𝑠+𝑅_𝑖,𝑖=0,1,…,𝑁, (1) where 𝑥_𝑖(𝑡)=[𝑥_𝑖1(𝑡),…,𝑥_𝑖𝑛(𝑡)]^𝑇∈ℝ^𝑛 represents the state vector of the 𝑖th node of the network at time 𝑡, and 𝐶,𝐴,𝐵,𝐷 are matrices with proper dimension. 𝑓(⋅) is a continuous vector function. 𝐼(𝑡) is the external input vector. 𝑅_𝑖∈ℝ^𝑛 is the control input. 𝜏(𝑡)>0, 𝜃(𝑡)>0 are time-varying discrete delay and distributed delay, respectively. Constants 𝛼>0, 𝛽>0, 𝛾>0 are coupling strengths of the whole network corresponding to nondelay, discrete delay, and distributed delay, respectively. Φ,Υ,Λ∈ℝ^𝑛×𝑛 are inner coupling matrices of the networks, which describe the individual coupling between two subsystems. Matrices 𝑈=(𝑢_𝑖𝑗)_𝑁×𝑁, 𝑉=(𝑣_𝑖𝑗)_𝑁×𝑁, 𝑊=(𝑤_𝑖𝑗)_𝑁×𝑁 are outer couplings of the whole networks satisfying the following diffusive conditions: 𝑢_𝑖𝑗≥0𝑖≠𝑗,𝑢_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑢_𝑖𝑗,𝑣_𝑖𝑗≥0𝑖≠𝑗,𝑣_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑢_𝑖𝑗,𝑤_𝑖𝑗≥0𝑖≠𝑗,𝑤_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑤_𝑖𝑗, (2) where 𝑖,𝑗=1,2,…,𝑁. Vector 𝜎_𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)),∫^𝑡_{𝑡−𝜃(𝑡)}𝑥_𝑖(𝑠)𝑑𝑠)∈ℝ^𝑛 describes the unknown perturbation to 𝑖th node of the complex networks. In this paper, we always assume that 𝜏̇≤ℎ_𝜏<1 and 𝜃̇≤ℎ_𝜃<1. 𝜃(𝑡) is bounded and we denote 𝜃_min>0 the minimum of 𝜃(𝑡) and 𝜃_max the maximum of 𝜃(𝑡).

We assume that (1) has a unique continuous solution for any initial condition in the following form: 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),−𝜚≤𝑠≤0,𝑖=0,1,2,…,𝑁, (3) where 𝜚=max{𝜏_max,𝜎_max} and 𝜏_max is the maximum of 𝜏(𝑡).

For convenience of writing, in the sequel, we denote 𝜎_𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)),∫^𝑡_{𝑡−𝜃(𝑡)}𝑥_𝑗(𝑠)𝑑𝑠) with 𝜎_𝑖(𝑡).

The system of an isolate node without external perturbation is described as 𝑧̇(𝑡)=𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓(𝑧(𝑡−𝜏(𝑡)))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑓(𝑧(𝑠))𝑑𝑠+𝐼(𝑡), (4)

and 𝑧(𝑡) can be any desired state: equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit.

Remark 1. The nonstochastic perturbations 𝜎_𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)),∫^𝑡_{𝑡−𝜃(𝑡)}𝑥_𝑖(𝑠)𝑑𝑠) are different from stochastic ones in the literature [21, 24, 32–35]. The distinct feature of the such stochastic perturbations is that the stochastic perturbations disappear when the synchronization goal is realized. However, perturbations of this paper still exist even when complete synchronization has been achieved. Therefore, the controllers in most of existing papers including those in [21, 24, 32–35] are invalid for perturbations of this paper.

When system (4) is perturbed, then (4) turns to the following system: 𝑧̇‌(𝑡)=𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓(𝑧(𝑡−𝜏(𝑡)))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑓(𝑧(𝑠))𝑑𝑠+𝐼(𝑡)+𝜎_𝑖𝑡,𝑧(𝑡),𝑧(𝑡−𝜏(𝑡)),^𝑡_{(𝑡)𝑡−𝜃}𝑧(𝑠)𝑑𝑠. (5)

Generally, the state of a system will be changed when the system is perturbed. We assume that the state of system (5) remains to be any one of the previous three states but not necessarily the original one.

The following assumptions are needed in this paper:(H₁)𝑓(0)≡0, and there exists positive constant ℎ such that 𝑓(𝑢)−𝑓(𝑣)≤ℎ‖𝑢−𝑣‖,forany𝑢,𝑣∈ℝ^𝑛, (6)(H₂)𝜎_𝑖𝑘(𝑡,0,0,0)≡0, and there exist positive constants 𝑀_𝑖𝑘 such that |𝜎_𝑖𝑘(𝑡,𝑢,𝑣,𝑤)|≤𝑀_𝑖𝑘 for any bounded 𝑢,𝑣,𝑤∈ℝ^𝑛,𝑖=1,2,…,𝑁,𝑘=1,2,…,𝑛.

Remark 2. Note that (4) unifies many well-known chaotic systems with or without delays, such as Chua system, Lorenz system, Rössler system, Chen system, and chaotic neural networks with mixed delays [12–29]. Hence, results of this paper are general.

Remark 3. Condition (H₂) is very mild. We do not impose the usual conditions such as Lipschitz condition, differentiability on the external perturbation functions. It can be discontinuous or even impulsive functions. If the state of (5) is a equilibrium point or a nontrivial periodic orbit, the condition (H₂) can be easily satisfied. If the state of (5) is a chaotic orbit, the condition (H₂) can also be satisfied. Since chaotic system has strange attractors, there exists a bounded region containing all attractors of it such that every orbit of the system never leaves them. Anyway, condition (H₂) can be satisfied for equilibrium point, a nontrivial periodic orbit, and a chaotic orbit. Moreover, we will subsequently prove that the complex networks (1) can be synchronized even without knowing the exact values of ℎ and 𝑀_𝑖𝑘, 𝑖=1,2,…,𝑁,𝑘=1,2,…,𝑛.

The aim of this paper is to synchronize all the states of complex networks (1) to the following manifold: 𝑥₁(𝑡)=𝑥₂(𝑡)=⋯=𝑥_𝑁(𝑡)=𝑧(𝑡), (7) where 𝑧(𝑡) is immune to external perturbations.

Lemma 4 ((Schur Complement) see [36]). The linear matrix inequality (LMI) 𝑆𝑆=¹¹𝑆₁₂𝑆^𝑇₁₂𝑆₂₂<0 (8) is equivalent to any one of the following two conditions:(L₁)𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0, (L₂)𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂S^𝑇₁₂<0, where 𝑆₁₁=𝑆^𝑇₁₁, 𝑆₂₂=𝑆^𝑇₂₂.

Lemma 5 (see [37]). For any constant matrix 𝐷∈ℝ^𝑛×𝑛, 𝐷^𝑇=𝐷>0, scalar 𝜎>0, and vector function 𝜔∶[0,𝜎]→ℝ^𝑛, one has 𝜎^𝜎₀𝜔^𝑇(𝑠)𝐷𝜔(𝑠)d𝑠≥^𝜎₀𝜔(𝑠)d𝑠^𝑇𝐷^𝜎₀𝜔(𝑠)d𝑠 (9) provided that the integrals are all well defined.

Lemma 6 ((Barbalat lemma) see [38]). If 𝑓(𝑡)∶ℝ→ℝ⁺ is a uniformly continuous function for 𝑡≥0 and if the limit of the integral lim_𝑡→∞^𝑡₀𝑓(𝑠)d𝑠 (10) exists and is finite, then lim_𝑡→∞𝑓(𝑡)=0.

3. Synchronization with All the Nodes Perturbed

In this section, we consider the case when all the nodes are perturbed. To realize synchronization goal (7), we have to introduce an isolate node (4).

Let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑧(𝑡). Subtracting (4) from (1), we get the following error dynamical system: 𝑒_𝑖̇(𝑡)=𝐶𝑒_𝑖(𝑡)+𝐴𝑔𝑒_𝑖(𝑡)+𝐵𝑔𝑒_𝑖(𝑡−𝜏(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑔𝑒_𝑖(𝑠)𝑑𝑠+𝛼^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑒_𝑗(𝑡)+𝛽^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑒_𝑗(𝑡−𝜏(𝑡))+𝜎_𝑖(𝑡)+𝑅_𝑖+𝛾^𝑁_𝑗=1𝑤_𝑖𝑗Λ^𝑡_{(𝑡)𝑡−𝜃}𝑒_𝑗(𝑠)𝑑𝑠, (11) where 𝑔(𝑒_𝑖)=𝑓(𝑥_𝑖(𝑡))−𝑓(𝑧(𝑡)),𝑖=1,2,…,𝑁.

From (H₁) and (H₂) we know that (11) admits a trivial solution 𝑒_𝑖(0)≡0,𝑖=1,2,…,𝑁. Obviously, to reach the goal (7), we have only to prove that system (11) is asymptotically stable at the origin.

Theorem 7. Under the assumption conditions (H₁) and (H₂), the networks (1) are synchronized with the following adaptive controllers: 𝑅_𝑖=−𝛼𝜀_𝑖𝑒_𝑖(𝑡)−𝜔𝛽_𝑖sign𝑒_𝑖(𝑡),𝜀_𝑖̇=𝑝_𝑖𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡),𝛽̇_𝑖=𝜉_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡), (12) where 𝜔>1, 𝑝_𝑖>0, and 𝜉_𝑖>0 are arbitrary constants, respectively, 𝑖=1,2,…,𝑁.

Proof. Define the Lyapunov function as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (13) where 𝑉₁(𝑡)=12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+12^𝑁_𝑖=1𝛼𝜀_𝑖−𝑘_𝑖²𝑝_𝑖+12^𝑁_𝑖=1𝑀_𝑖−𝛽_𝑖²𝜉_𝑖,𝑉₂(𝑡)=^𝑡_{(𝑡)𝑡−𝜏}𝜂^𝑇(𝑠)(𝑠)𝑉𝑄𝜂𝑑𝑠,₃(𝑡)=^𝑡_{(𝑡)𝑡−𝜃}^𝑡_𝜇𝜂^𝑇(𝑠)𝐺𝜂(𝑠)𝑑𝑠𝑑𝜇, (14)
𝜂(𝑡)=(‖𝑒₁(𝑡)‖,‖𝑒₂(𝑡)‖,…,‖𝑒_𝑁(𝑡)‖)^𝑇, 𝑀_𝑖=max_{1≤𝑘≤𝑛}{𝑀_𝑖𝑘}, 𝑘_𝑖,𝑖=1,2,…,𝑁, are constants, 𝑄 and 𝐺 are symmetric positive definite matrices, and 𝑘_𝑖, 𝑄, and 𝐺 are to be determined.
Differentiating 𝑉₁(𝑡) along the solution of (11) and from (H₁) and (H₂), we obtain 𝑉̇₁(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)+𝛼^𝑁_𝑖=1𝜀_𝑖−𝑘_𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)−^𝑁_𝑖=1𝑀_𝑖−𝛽_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡≤_𝑁𝑖=1)‖𝐶‖+‖𝐴‖ℎ−𝛼𝑘_𝑖𝑒_𝑖(𝑡)²+‖𝐵‖ℎ𝑒_𝑖(𝑡)𝑒_𝑖((𝑡+‖𝐷‖ℎ𝑒𝑡−𝜏))_𝑖(𝑡)^𝑡_{𝑡−𝜃(𝑡)}𝑒_𝑖(𝑠)𝑑𝑠+𝛼^𝑁_{𝑗=1,𝑗≠𝑖}𝑢_𝑖𝑗‖Φ‖𝑒_𝑖(𝑡)𝑒_𝑗(𝑡)+𝛼𝜆_minΦ^𝑠𝑢_𝑖𝑖𝑒_𝑖(𝑡)²+𝛽^𝑁_𝑗=1𝑣_𝑖𝑗‖Υ‖𝑒_𝑖(𝑡)𝑒_𝑗(𝑡−𝜏(𝑡))+𝛾^𝑁_𝑗=1𝑤_𝑖𝑗‖Λ‖𝑒_𝑖(𝑡)^𝑡_{(𝑡)𝑡−𝜃}𝑒_𝑗(𝑠)𝑑𝑠=𝜂^𝑇(𝑡)(‖𝐶‖+‖𝐴‖ℎ)𝐼_𝑁+𝛼‖Φ‖𝑈^𝑠−𝐾𝜂(𝑡)+𝜂^𝑇(𝑡)‖𝐵‖ℎ𝐼_𝑁+𝛽‖Υ‖|𝑉|𝜂(𝑡−𝜏(𝑡))+𝜂^𝑇(𝑡)‖𝐷‖ℎ𝐼_𝑁+𝛾‖Λ‖|𝑊|^𝑡_{(𝑡)𝑡−𝜃}𝜂(𝑠)𝑑𝑠≤𝜂^𝑇(𝑡)(‖𝐶‖+‖𝐴‖ℎ+1)𝐼_𝑁+𝛼‖Φ‖𝑈^𝑠−𝐾𝜂(𝑡)+𝜂^𝑇(𝑡−𝜏(𝑡))𝐵^𝑇((𝑡+𝐵𝜂𝑡−𝜏))_{𝑡(𝑡)𝑡−𝜃}𝜂(𝑠)𝑑𝑠^𝑇𝐷^𝑇𝐷^𝑡_{(𝑡)𝑡−𝜃}𝜂(𝑠)𝑑𝑠, (15) where 𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑁), 𝑈=(𝑢_𝑖𝑗)_𝑁×𝑁, 𝑢_𝑖𝑗=𝑢_𝑖𝑗, 𝑖≠𝑗,  𝑢_𝑖𝑖=(𝜆_min(Φ^𝑠)/‖Φ‖)𝑢_𝑖𝑖, 𝐵=‖𝐵‖ℎ𝐼_𝑁+𝛽‖Υ‖|𝑉|, 𝐷=‖𝐷‖ℎ𝐼_𝑁+𝛾‖Λ‖|𝑊|, |𝑉|=(|𝑣_𝑖𝑗|)_𝑁×𝑁, |𝑊|=(|𝑤_𝑖𝑗|)_𝑁×𝑁, and we have used the following deduction: ^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝜎_𝑖(𝑡)−𝜔^𝑛_𝑘=1𝛽_𝑖𝑒_𝑖𝑘(𝑡)−^𝑛_𝑘=1𝑀_𝑖−𝛽_𝑖𝑒_𝑖𝑘(𝑡≤_{𝑁𝑛𝑖=1𝑘=1})𝑒_𝑖𝑘(𝑡)𝑀_𝑖𝑘−𝑀_𝑖𝑒_𝑖𝑘−(𝜔−1)𝛽_𝑖𝑒_𝑖𝑘≤−^𝑁𝑛_𝑖=1𝑘=1(𝜔−1)𝛽_𝑖𝑒_𝑖𝑘≤0. (16)
Differentiating 𝑉₂(𝑡), we get 𝑉̇₂(𝑡)=𝜂^𝑇(𝑡)𝑄𝜂(𝑡)−(1−𝜏̇(𝑡))𝜂^𝑇(𝑡−𝜏(𝑡))𝑄𝜂(𝑡−𝜏(𝑡))≤𝜂^𝑇(𝑡)𝑄𝜂(𝑡)−1−ℎ_𝜏𝜂^𝑇(𝑡−𝜏(𝑡))𝑄𝜂(𝑡−𝜏(𝑡)). (17)
Differentiating 𝑉₃(𝑡) from Lemma 5 we have 𝑉̇₃(𝑡)=𝜃(𝑡)𝜂^𝑇(𝑡)𝐺𝜂(𝑡)−1−𝜃̇(𝑡)^𝑡_{(𝑡)𝑡−𝜃}𝜂^𝑇(𝑠)𝐺𝜂(𝑠)𝑑𝑠≤𝜃_max𝜂^𝑇(𝑡)(𝑡)−𝐺𝜂1−ℎ_𝜃𝜃_min^𝑡_{(𝑡)𝑡−𝜃}𝜂(𝑠)𝑑𝑠^𝑇𝐺^𝑡_{(𝑡)𝑡−𝜃}𝜂(𝑠)𝑑𝑠. (18)
Take 𝑄=(1/(1−ℎ_𝜏))𝐵^𝑇𝐵, 𝐺=(𝜃_min/(1−ℎ_𝜃))𝐷^𝑇𝐷. From the definition of 𝑉(𝑡) we reach the following inequality: 𝑉̇(𝑡)≤𝛼𝜂^𝑇(𝑡)1𝛼(‖𝐶‖+‖𝐴‖ℎ+1)𝐼_𝑁+‖Φ‖𝑈^𝑠+1𝛼1−ℎ_𝜏𝐵^𝑇𝐵+𝜃_min𝜃_max𝛼1−ℎ_𝜃𝐷^𝑇𝐷−𝐾𝜂(𝑡). (19)
Let 𝑘_𝑖=𝜆_max((1/𝛼)(‖𝐶‖+‖𝐴‖ℎ+1)𝐼_𝑁+‖Φ‖𝑈^𝑠+(1/𝛼(1−ℎ_𝜏))𝐵^𝑇𝐵+(𝜃_min𝜃_max/(𝛼(1−ℎ_𝜃)))𝐷^𝑇𝐷)+1, where 𝜆_max((1/𝛼)(‖𝐶‖+‖𝐴‖ℎ+1)𝐼_𝑁+‖Φ‖𝑈^𝑠+(1/𝛼(1−ℎ_𝜏))𝐵^𝑇𝐵+(𝜃_min𝜃_max/𝛼(1−ℎ_𝜃))𝐷^𝑇𝐷) denotes the maximum eigenvalue of (1/𝛼)(‖𝐶‖+‖𝐴‖ℎ+1)𝐼_𝑁+‖Φ‖𝑈^𝑠+(1/𝛼(1−ℎ_𝜏))𝐵^𝑇𝐵+(𝜃_min𝜃_max/𝛼(1−ℎ_𝜃))𝐷^𝑇𝐷. Then, from the previous inequality, we get 𝑉̇(𝑡)≤−𝛼𝜂^𝑇(𝑡)𝜂(𝑡). (20)
Integrating both sides of the previous equation from 0 to 𝑡 yields 𝑉(0)≥𝑉(𝑡)+𝛼^𝑁_𝑖=1^𝑡₀𝑒_𝑖(𝑠)²𝑑𝑠≥𝛼^𝑁_𝑖=1^𝑡₀𝑒_𝑖(𝑠)²𝑑𝑠. (21)
Therefore, 𝛼lim^𝑁_{𝑡→∞𝑖=1}^𝑡₀𝑒_𝑖(𝑠)²𝑑𝑠≤𝑉(0). (22)
In view of Lemma 6 and the previous inequality, one can easily get lim^𝑁_{𝑡→∞𝑖=1}𝑒_𝑖(𝑡)²=0, (23)
which in turn means lim_𝑡→∞𝑒_𝑖(𝑡)=0,𝑖=1,2,…,𝑁. (24)
This completes the proof.

4. Synchronization with Partial Nodes Perturbed

Usually, only partial nodes of complex networks are perturbed. If some important nodes are perturbed, then the entire network will not work correctly. Theoretically speaking, nodes with larger degree (undirected networks) or larger outdegree (directed networks) are more vulnerable to perturbation [39], since the states of these nodes have more effect on networks than those with smaller degree (undirected networks) or outdegree (directed networks). On the other hand, the real-world complex networks normally have a large number of nodes; it is usually impractical and impossible to control a complex networks by adding the controllers to all nodes. Therefore, from both practical point of view and the view of reducing control cost, we can use the scheme of pinning control [27–29, 40–42] to prevent external perturbations and synchronize complex networks.

In this section, we assume that matrix 𝑈 is irreducible in the sense that there is no isolate cluster in the network and there are 𝑙₁ nodes affected by external perturbations.

Without loss of generality, rearrange the order of the nodes in the network, and take the first 𝑙(𝑙≥𝑙₁) nodes to be controlled. Thus, the pinning controlled network can be described as 𝑥̇_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)+𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑓𝑥_𝑖(𝑠)𝑑𝑠+𝛼^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑥_𝑗(𝑡)+𝐼(𝑡)+𝛽^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑥_𝑗(𝑡−𝜏(𝑡))+𝛾^𝑁_𝑗=1𝑤_𝑖𝑗Λ^𝑡_{(𝑡)𝑡−𝜃}𝑥_𝑗(𝑠)𝑑𝑠+𝜎_𝑖(𝑡)+𝑅_𝑖,𝑖=1,2,…,𝑙₁,𝑥̇_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)+𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑓𝑥_𝑖(𝑠)𝑑𝑠+𝛼^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑥_𝑗(𝑡)+𝐼(𝑡)+𝛽^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑥_𝑗(𝑡−𝜏(𝑡))+𝛾^𝑁_𝑗=1𝑤_𝑖𝑗Λ^𝑡_{(𝑡)𝑡−𝜃}𝑥_𝑗(𝑠)𝑑𝑠+𝑅_𝑖,𝑖=𝑙₁+1,𝑙₁𝑥̇+2,…,𝑙,_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)+𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑓𝑥_𝑖(𝑠)𝑑𝑠+𝛼^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑥_𝑗(𝑡)+𝐼(𝑡)+𝛽^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑥_𝑗(𝑡−𝜏(𝑡))+𝛾^𝑁_𝑗=1𝑤_𝑖𝑗Λ^𝑡_{(𝑡)𝑡−𝜃}𝑥_𝑗(𝑠)𝑑𝑠,𝑖=𝑙+1,𝑙+2,…,𝑁, (25) where 𝑅_𝑖,𝑖=1,2,…,𝑙, are control inputs.

Let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑧(𝑡). Subtracting (4) from (25) we obtain the following error dynamical system: 𝑒_𝑖̇(𝑡)=𝐶𝑒_𝑖(𝑡)+𝐴𝑔𝑒_𝑖(𝑡)+𝐵𝑔𝑒_𝑖(𝑡−𝜏(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑔𝑒_𝑖(𝑠)𝑑𝑠+𝛼^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑒_𝑗(𝑡)+𝛽^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑒_𝑗(𝑡−𝜏(𝑡))+𝛾^𝑁_𝑗=1𝑤_𝑖𝑗Λ^𝑡_{(𝑡)𝑡−𝜃}𝑒_𝑗(𝑠)𝑑𝑠+𝜎_𝑖(𝑡)+𝑅_𝑖,𝑖=1,2,…,𝑙₁,𝑒_𝑖̇(𝑡)=𝐶𝑒_𝑖(𝑡)+𝐴𝑔𝑒_𝑖(𝑡)+𝐵𝑔𝑒_𝑖(𝑡−𝜏(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑔𝑒_𝑖(𝑠)𝑑𝑠+𝛼^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑒_𝑗(𝑡)+𝛽^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑒_𝑗(𝑡−𝜏(𝑡))+𝛾^𝑁_𝑗=1𝑤_𝑖𝑗Λ^𝑡_{(𝑡)𝑡−𝜃}𝑒_𝑗(𝑠)𝑑𝑠+𝑅_𝑖,𝑖=𝑙₁+1,𝑙₁𝑒̇+2,…,𝑙,_𝑖(𝑡)=𝐶𝑒_𝑖(𝑡)+𝐴𝑔𝑒_𝑖(𝑡)+𝐵𝑔𝑒_𝑖(𝑡−𝜏(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑔𝑒_𝑖(𝑠)𝑑𝑠+𝛼^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑒_𝑗(𝑡)+𝛽^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑒_𝑗(𝑡−𝜏(𝑡))+𝛾^𝑁_𝑗=1𝑤_𝑖𝑗Λ^𝑡_{(𝑡)𝑡−𝜃}𝑒_𝑗(𝑠)𝑑𝑠,𝑖=𝑙+1,𝑙+2,…,𝑁. (26)

Similar to Theorem 7, to reach the goal (7), we have only to prove that system (26) is asymptotically stable at the origin.

Theorem 8. Suppose that matrix 𝑈 is irreducible and the assumptions (𝐻₁) and (𝐻₂) hold. If 2𝛼‖Φ‖𝑈^𝑠𝑐_𝑙+Σ𝐼_𝑁−𝑙<0, (27) then the complex networks (25) are synchronized with the adaptive pinning controllers 𝑅_𝑖=−𝛼𝜀_𝑖𝑒_𝑖(𝑡)−𝜔𝛽_𝑖sign𝑒_𝑖(𝑡),𝜀_𝑖̇=𝑝_𝑖𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡),𝛽̇_𝑖=𝜉_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡), (28) where 𝑖=1,2,…,𝑙,Σ=2(‖𝐶‖+‖𝐴‖ℎ)+((𝜃_max𝜃_min+1−ℎ_𝜃)/(1−ℎ_𝜃))‖‖𝐷‖ℎ𝐼_𝑁+𝛾‖Λ‖|𝑊|‖+((2−ℎ_𝜏)/(1−ℎ_𝜏))‖‖𝐵‖ℎ𝐼_𝑁+𝛽‖Υ‖|𝑉|‖, and the other parameters are the same as those of Theorem 7.

Proof. We define another Lyapunov function as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (29) where 𝑉₁(𝑡)=12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+12^𝑙_𝑖=1𝛼𝜀_𝑖−𝑘_𝑖²𝑝_𝑖+12^𝑙1_𝑖=1𝑀_𝑖−𝛽_𝑖²𝜉_𝑖, (30)
𝑘_𝑖,𝑖=1,2,…,𝑙, are constants to be determined, and 𝑉₂(𝑡) and 𝑉₃(𝑡) are defined as those in the proof of Theorem 7.
In view of (H₁) and (H₂), differentiating 𝑉₁(𝑡) along the solution of (26) yields 𝑉̇‌₁(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)+𝛼^𝑙_𝑖=1𝜀_𝑖−𝑘_𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)−^𝑙1_𝑖=1𝑀_𝑖−𝛽_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡≤_𝑁𝑖=1)(‖𝐶‖+‖𝐴‖ℎ)𝑒_𝑖(𝑡)²+‖𝐵‖ℎ𝑒_𝑖(𝑡)𝑒_𝑖((𝑡+‖𝐷‖ℎ𝑒𝑡−𝜏))_𝑖(𝑡)^𝑡_{(𝑡)𝑡−𝜃}𝑒_𝑖(𝑠)𝑑𝑠−𝛼^𝑙_𝑖=1𝑘_𝑖𝑒_𝑖(𝑡)²+𝛼^𝑁_𝑖=1𝜆^Φ𝑠_min𝑢_𝑖𝑖𝑒_𝑖(𝑡)²+𝛼^𝑁𝑁_{𝑖=1𝑗=1,𝑗≠𝑖}𝑢_𝑖𝑗‖Φ‖𝑒_𝑖(𝑡)𝑒_𝑗(𝑡)+𝛽^𝑁𝑁_𝑖=1𝑗=1𝑣_𝑖𝑗‖Υ‖𝑒_𝑖(𝑡)𝑒_𝑗(𝑡−𝜏(𝑡))+𝛾^𝑁𝑁_𝑖=1𝑗=1𝑤_𝑖𝑗‖Λ‖𝑒_𝑖(𝑡)^𝑡_{𝑡−𝜃(𝑡)}𝑒_𝑗(𝑠)𝑑𝑠=𝜂^𝑇(𝑡)(‖𝐶‖+‖𝐴‖ℎ)𝐼_𝑁+𝛼‖Φ‖𝑈^𝑠−𝐾𝜂(𝑡)+𝜂^𝑇(𝑡)‖𝐵‖ℎ𝐼_𝑁+𝛽‖Υ‖|𝑉|𝜂(𝑡−𝜏(𝑡))+𝜂^𝑇(𝑡)‖𝐷‖ℎ𝐼_𝑁+𝛾‖Λ‖|𝑊|^𝑡_{(𝑡)𝑡−𝜃}𝜂(𝑠)𝑑𝑠, (31) where 𝐾=diag(𝑘₁,…,𝑘_𝑙,0,…,0_𝑁−𝑙), and the following deduction is used: ^𝑙1_𝑖=1𝑒^𝑇_𝑖(𝑡)𝜎_𝑖(𝑡)−𝜔^𝑙1𝑛_𝑖=1𝑘=1𝛽_𝑖𝑒_𝑖𝑘(𝑡−_{𝑙1𝑛𝑖=1𝑘=1})𝑀_𝑖−𝛽_𝑖𝑒_𝑖𝑘(𝑡)−𝜔^𝑙_𝑖=𝑙1^𝑛_+1𝑘=1𝛽_𝑖𝑒_𝑖𝑘(𝑡≤_{𝑙1𝑛𝑖=1𝑘=1})𝑒_𝑖𝑘(𝑡)𝑀_𝑖𝑘−𝑀_𝑖𝑒_𝑖𝑘−(𝜔−1)𝛽_𝑖𝑒_𝑖𝑘−𝜔^𝑙_𝑖=𝑙1^𝑛_+1𝑘=1𝛽_𝑖𝑒_𝑖𝑘(𝑡)≤−^𝑙1𝑛_𝑖=1𝑘=1(𝜔−1)𝛽_𝑖𝑒_𝑖𝑘−𝜔^𝑙_𝑖=𝑙1^𝑛_+1𝑘=1𝛽_𝑖𝑒_𝑖𝑘(𝑡)≤0. (32)
Combining (31) with (17) and (25), we have 𝑉̇‌(𝑡)≤12𝜁^𝑇(𝑡)𝚷𝜁(𝑡), (33) where 𝜁(𝑡)=(𝜂^𝑇(𝑡),𝜂^𝑇(𝑡−𝜏(𝑡)),(∫^𝑡_{𝑡−𝜃(𝑡)}𝜂(𝑠)𝑑𝑠)^𝑇)^𝑇 and Π𝚷=¹¹Π₁₂Π₁₃Π^𝑇₁₂−21−ℎ_𝜏Π𝑄0_𝑇1320−1−ℎ_𝜃𝜃_min𝐺 (34)
with Π₁₁=2((‖𝐶‖+‖𝐴‖ℎ)𝐼_𝑁+𝛼(‖Φ‖𝑈^𝑠−𝐾)+𝑄+𝜃_max𝐺), Π₁₂=‖𝐵‖ℎ𝐼_𝑁+𝛽‖Υ‖|𝑉|, Π₁₃=‖𝐷‖ℎ𝐼_𝑁+𝛾‖Λ‖|𝑊|.
According to Lemma 4, 𝚷<0 is equivalent to Δ=2(‖𝐶‖+‖𝐴‖ℎ)𝐼_𝑁+𝛼‖Φ‖𝑈^𝑠−𝐾+𝑄+𝜃_max𝐺+121−ℎ_𝜏‖𝐵‖ℎ𝐼_𝑁+𝛽‖Υ‖|𝑉|𝑄⁻¹×‖𝐵‖ℎ𝐼_𝑁+𝛽‖Υ‖|𝑉|^𝑇+𝜃_min21−ℎ_𝜃‖𝐷‖ℎ𝐼_𝑁+𝛾‖Λ‖|𝑊|𝐺⁻¹×‖𝐷‖ℎ𝐼_𝑁+𝛾‖Λ‖|𝑊|^𝑇<0. (35)
Let 𝑄=(1/2(1−ℎ_𝜏))‖‖𝐵‖ℎ𝐼_𝑁+𝛽‖Υ‖|𝑉|‖𝐼_𝑁, 𝐺=(𝜃_min/2(1−ℎ_𝜃))‖‖𝐷‖ℎ𝐼_𝑁+𝛾‖Λ‖|𝑊|‖𝐼_𝑁. We have Δ≤2(‖𝐶‖+‖𝐴‖ℎ)𝐼_𝑁+2𝛼‖Φ‖𝑈^𝑠+𝜃−𝐾_max𝜃_min+1−ℎ_𝜃1−ℎ_𝜃‖𝐷‖ℎ𝐼_𝑁+𝛾‖Λ‖|𝑊|𝐼_𝑁+2−ℎ_𝜏1−ℎ_𝜏‖𝐵‖ℎ𝐼_𝑁+𝛽‖Υ‖|𝑉|𝐼_𝑁−1=Δ₁₁2𝛼‖Φ‖𝑈^𝑠_∗2𝛼‖Φ‖𝑈^𝑠𝑇_∗2𝛼‖Φ‖𝑈^𝑠𝑐_𝑙+Σ𝐼_𝑁−𝑙, (36) where Δ₁₁=2𝛼‖Φ‖(𝑈^𝑠)_𝑙−2𝛼𝐾_𝑙+Σ𝐼_𝑙, 2𝛼‖Φ‖(𝑈^𝑠)_∗ is matrix with appropriate dimension.
Since 2𝛼‖Φ‖(𝑈^𝑠)^𝑐_𝑙+Σ𝐼_𝑁−𝑙<0 and there exist positive constants 𝑘₁,𝑘₂,…,𝑘_𝑙 such that 2𝛼‖Φ‖𝑈^𝑠_𝑙−2𝛼𝐾_𝑙+Σ𝐼_𝑙−(2𝛼‖Φ‖)²𝑈^𝑠_∗×2𝛼‖Φ‖𝑈^𝑠𝑐_𝑙+Σ𝐼_𝑁−𝑙⁻¹𝑈^𝑠𝑇_∗<0, (37)
again, from Lemma 4 we obtain Δ<0. Hence, 𝚷<0. Denote 𝜆_min to be the minimum eigenvalue of −𝚷; then 𝑉̇‌(𝑡)≤−𝜆^𝑁_min𝑖=1𝑒_𝑖(𝑡)²≤0. (38)
Integrating both sides of the previous equation from 0 to 𝑡 yields 𝑉(0)≥𝑉(𝑡)+𝜆^𝑁_min𝑖=1^𝑡₀𝑒_𝑖(𝑠)²𝑑𝑠≥𝜆^𝑁_min𝑖=1^𝑡₀𝑒_𝑖(𝑠)²𝑑𝑠. (39)
Therefore, lim_𝑡→∞𝜆^𝑁_min𝑖=1^𝑡₀𝑒_𝑖(𝑠)²𝑑𝑠≤𝑉(0). (40)
By Lemma 6 we obtain lim_𝑡→∞𝜆^𝑁_min𝑖=1𝑒_𝑖(𝑡)²=0, (41)
which in turn means lim_𝑡→∞𝑒_𝑖(𝑡)=0,𝑖=1,2,…,𝑁. (42)
This completes the proof.

When there is no external perturbation, that is, 𝜎_𝑖(𝑡)=0,𝑖=1,2,…,𝑁, one can easily get the following corollaries from Theorems 7 and 8, respectively. We omit their proofs here.

Corollary 9. Suppose that 𝜎_𝑖(𝑡)=0,𝑖=1,2,…,𝑁, and the assumption condition (𝐻₁) holds. Then complex networks (1) are synchronized with the adaptive controllers (12). Moreover, the scalar 𝜔 can be relaxed to any positive constant.

Corollary 10. Suppose that matrix 𝑈 is irreducible and the assumption (𝐻₁) holds. The complex networks (25) are synchronized with the adaptive pinning controllers (28), if (27) holds. Moreover, the scalar 𝜔 can be relaxed to any positive constant.

Remark 11. From the inequalities (16) and (32) one can see that the designed adaptive controllers (12) and (28) are very useful. They can overcome the bad effects of the uncertain nonlinear perturbations without knowing the exact functions and bounds of the perturbations as long as the perturbed systems are chaotic. Especially, when there are only partial nodes perturbed (the first 𝑙₁ nodes in the system (25)), the designed controllers still are effective to stabilize the error system by adding them to nodes with and without such perturbations, (see the inequality (32)). Obviously, in the case of no perturbation, the parameter 𝜔 can also be taken as 0. When 𝜔=0, the controllers (12) and (28) turn out to be the usual adaptive controller, which is extensively utilized to synchronize coupled systems with or without stochastic perturbations [8, 23–34, 40–42]. However, the controllers in [8, 23–34, 40–42] cannot synchronize coupled systems with nonstochastic perturbations. Therefore, the designed controllers can deal with both stochastic and nonstochastic perturbations to the systems, and hence they have better robustness than usual adaptive controllers.

Remark 12. Model (1) can be extended to the following more general complex networks: 𝑥̇_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)+𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑓_𝜏𝑥_𝑖(𝑡−𝜏(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑓_𝜃𝑥_𝑖(𝑠)𝑑𝑠+𝐼(𝑡)+𝛼^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑥_𝑗(𝑡)+𝛽^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑥_𝑗(𝑡−𝜏(𝑡))+𝛾^𝑁_𝑗=1𝑤_𝑖𝑗Λ^𝑡_{(𝑡)𝑡−𝜃}𝑥_𝑗(𝑠)𝑑𝑠+𝜎_𝑖(𝑡)+𝑅_𝑖,𝑖=0,1,…,𝑁. (43)
Moreover, we can also consider stochastic perturbations [21] and Markovian jump [43, 44] in (43) to get more general results. For simplicity, we omit the corresponding results and only consider model (1).

5. Numerical Examples

In this section, we provide two examples to illustrate the general model and the advantage of the new adaptive controller.

Example 13. The Lorenz system is described as 𝑧̇(𝑡)=𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡)), (44) where 𝐶=−1010028−10008/3,𝐴=000010001, (45)
𝑓(𝑧(𝑡))=(0,−𝑧₁(𝑡)𝑧₃(𝑡),𝑧₁(𝑡)𝑧₂(𝑡))^𝑇. When initial values are taken as 𝑧₁(0)=0.8, 𝑧₂(0)=2, 𝑧₃(0)=2.5, chaotic trajectory of (44) can be seen in Figure 1.
The following three perturbed Lorenz systems are chaotic: 𝑧̇‌(𝑡)=𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡))+𝜎_𝑖(𝑡),𝑖=1,2,3, (46) where 𝜎₁(𝑡)=(0.1𝑧²₁(𝑡),0.2𝑧₂(𝑡),0.2𝑧₃(𝑡))^𝑇, 𝜎₂(𝑡)=(0.1𝑧₁(𝑡),0.05𝑧²₂(𝑡),sin𝑧₃(𝑡))^𝑇, 𝜎₃(𝑡)=(0.1𝑧₁(𝑡),cos𝑧₂(𝑡),sin𝑧₃(𝑡))^𝑇. Chaotic trajectories of the three perturbed Lorenz systems are showed in Figure 2 with the same initial values 𝑧₁(0)=0.8, 𝑧₂(0)=2, 𝑧₃(0)=2.5.
Now consider the following complex networks with each node as the previous perturbed Lorenz system: 𝑥̇_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)+𝐴𝑓𝑥_𝑖(𝑡)+𝛼^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑥_𝑗(𝑡)+𝜎_𝑖(𝑡)+𝑅_𝑖,𝑖=0,1,3, (47) where 𝛼=0.5,Φ=𝐼₃ and 𝑈=−1101−1011−2. (48)
Obviously, conditions (H₁) and (H₂) are satisfied. According to Theorem 7, the complex networks (47) can be synchronized with adaptive controllers (12).
The initial conditions of the numerical simulations are as follows: 𝜔=4, step=0.0005, 𝑥₁(0)=(−2,−1,0)^𝑇, 𝑥₂(0)=(1,2,3)^𝑇, 𝑥₃(0)=(4,5,6)^𝑇, 𝜀_𝑖(0)=1, 𝛽_𝑖(0)=2, 𝑝_𝑖=𝜉_𝑖=0.5, 𝑖=1,2,3. Figure 3 describes the synchronization errors 𝑒_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡)−𝑧_𝑗(𝑡), 𝑖,𝑗=1,2,3. Figure 4 shows the adaptive feedback gains. Numerical simulations verify the effectiveness of Theorem 7.

Figure 1 

Chaotic trajectory of (44).

(a)

(b)

(c)

Figure 2 

Chaotic trajectories of (46) with 𝜎1(𝑡) (a), 𝜎2(𝑡) (b), 𝜎3(𝑡) (c).

(a)

(b)

(c)

Figure 3 

Synchronization errors of (47): (a) 𝑒𝑖1(𝑡), (b) 𝑒𝑖2(𝑡), (c) 𝑒𝑖3(𝑡),𝑖=1,2,3.

(a)

(b)

Figure 4 

The adaptive control gains of (47): (a) the adaptive gains 𝜀𝑖,1≤𝑖≤3, (b) the adaptive gains 𝛽𝑖,1≤𝑖≤3.

Example 14. Consider the following chaotic neural networks with mixed delays: 𝑧̇(𝑡)=𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓(𝑧(𝑡−𝜏(𝑡)))+𝐷^𝑡_{(𝑡)𝑡−𝜎}𝑓(𝑧(𝑠))𝑑𝑠+𝐼(𝑡), (49) where 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡))^𝑇,𝜏(𝑡)=1,𝜎(𝑡)=0.3,𝑓(𝑧(𝑡))=(tanh(𝑧₁(𝑡),tanh(𝑧₂(𝑡))^𝑇, ,02.𝐶=−1.200−1,𝐴=3−0.385𝐼=,𝐵=−1.40.10.3−8,𝐷=−1.20.1−2.8−1 (50)
In the case that the initial condition is chosen as 𝑧₁(𝑡)=0.4, 𝑧₂(𝑡)=0.6, for  all 𝑡∈[−1,0], the chaotic attractor can be seen in Figure 5.
The perturbed system of (49) is 𝑧̇‌(𝑡)=𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡))+𝐵𝑓(𝑧(𝑡−𝜏(𝑡)))+𝐷^𝑡_{𝑡−𝜎(𝑡)}𝑓(𝑧(𝑠))𝑑𝑠+𝐼(𝑡)+𝜎(𝑡), (51) where 𝜎(𝑡)=(0.2𝑧₁(𝑡−1),0.2∫^𝑡_𝑡−0.3𝑧₂(𝑠)𝑑𝑠)^𝑇. The chaotic attractor of (51) can be seen in Figure 6 with 𝑧₁(𝑡)=0.4, 𝑧₂(𝑡)=0.6, for  all 𝑡∈[−1,0].
Now consider the following complex networks with each node as the previous neural networks with mixed delays (49), while the second node is disturbed with the previous 𝜎(𝑡). 𝑥̇_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)+𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))+𝐷^𝑡_{(𝑡)𝑡−𝜃}𝑓𝑥_𝑖(𝑠)𝑑𝑠+𝐼(𝑡)+𝛼¹⁰_𝑗=1𝑢_𝑖𝑗Φ𝑥_𝑗(𝑡)+𝛽¹⁰_𝑗=1𝑣_𝑖𝑗Υ𝑥_𝑗(𝑡−𝜏(𝑡))+𝛾¹⁰_𝑗=1𝑤_𝑖𝑗Λ^𝑡_{(𝑡)𝑡−𝜃}𝑥_𝑗(𝑠)𝑑𝑠+𝜎_𝑖(𝑡)+𝑅_𝑖,𝑖=0,1,…,10, (52) where 𝛼=3, 𝛽=𝛾=1, 𝜎₂(𝑡)=𝜎(𝑡), else 𝜎_𝑖(𝑡)=0.
Figure 7 depicts the WS Small-World networks [2] corresponding to nondelay (a), discrete delay, and distributed delay (b). The corresponding Laplacian matrices are shown as following: 𝑈=−41100000111−51100001111−41100000011−41100000011−41100000011−41100000011−41100000011−41111000011−40110000010−3,𝑉=𝑊=−21000100001−21000000001−20000010000−11000000001−30100110000−10000000010−21000000001−21000100001−31000010001−2. (53)(54)
Take the first two nodes (corresponding to matrix 𝑈) to be controlled. According Theorem 8, the complex networks (52) can be synchronized with adaptive controllers (27).
The initial conditions of the numerical simulations are as follows: 𝜔=2, step=0.0005, 𝑥_𝑖(0)=(−11+2𝑖,−10+2𝑖)^𝑇,𝑖=1,2,…,10.𝜀_𝑖(0)=𝛽_𝑖(0)=𝑝_𝑖=𝜉_𝑖=1,𝑖=1,2. Figure 8 describes the synchronization errors 𝑒_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡)−𝑧_𝑗(𝑡),𝑖=1,2,…,10,𝑗=1,2. Figure 9 depicts the adaptive feedback gains. Numerical simulations verify the effectiveness of Theorem 8.

Figure 5 

Chaotic trajectory of (49).

Figure 6 

Chaotic trajectory of (51).

(a)

(b)

Figure 7 

WS Small-Worlds with 10 nodes. In (a) each node connects 4 nodes, and the rewire probability is 0.2; in (b) each node connects 2 nodes, and the rewire probability is 0.4.

(a)

(b)

Figure 8 

Synchronization errors of (52): (a) 𝑒𝑖1, (b) 𝑒𝑖2,1≤𝑖≤10.

(a)

(b)

Figure 9 

The adaptive pinning control gains of (52): (a) the adaptive pinning control gains 𝜀𝑖,𝑖=1,2. (b) The adaptive pinning control gains 𝛽𝑖,𝑖=1,2.

6. Conclusions

External perturbations to networks are unavoidable in practice. On the other hand, many chaotic models have discrete delay and distributed delay. Therefore, in this paper, we introduced a class of hybrid coupled complex networks with mixed delays and unknown nonstochastic external perturbations. A simple robust adaptive controller is designed to synchronize the complex networks even without knowing a priori the bounds and the exact functions of the perturbations. It should be emphasized that we do not assume that the coupling matrix is symmetric or diagonal. The controller can enhance robustness and reduce fragility of complex networks; hence, it has great practical significance. Moreover, we also verify the effectiveness of the theoretical results by numerical simulations.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China (NSFC) under Grants 61263020 and 11101053, the Scientific Research Fund of Yunnan Province under Grant 2010ZC150, the Scientific Research Fund of Chongqing Normal University under Grants 940115 and 12XLB031, the Key Project of Chinese Ministry of Education under Grant 211118, and the Excellent Youth Foundation of Educational Committee of Hunan Provincial under Grant 10B002.