Synchronization of Coupled Networks with Uncertainties

Abstract

Asymptotic synchronization for a class of coupled networks with nondelayed and delayed couplings is investigated. A distinct feature of the network is that all the dynamical nodes are affected by uncertain nonlinear nonidentical perturbations. In order to synchronize the network onto a given isolate trajectory, a novel adaptive controller is designed to overcome the effects of the nonidentical uncertain nonlinear perturbations. The designed controller has better robustness than classical adaptive controller, since it can realize the synchronization goal whether the nodes have these perturbations or not. Based on the Lyapunov stability theory and the Barbalat lemma, sufficient conditions guaranteeing the asymptotic synchronization of the coupled network are derived. Two examples with numerical simulations are given to illustrate the effectiveness of the theoretical results. Simulations also demonstrate that our adaptive controller has better robustness than existing ones.

1. Introduction

A coupled network or complex network is a set of interconnected nodes, where each node is a dynamical system. In fact, many natural and man-made systems, such as the Internet networks [1], biological networks [2], epidemic spreading networks [3], collaborative networks [4], and social networks [5], can be described by coupled networks. Since the seminal work of Wu and Chua [6], much attention has been payed to the dynamical behaviors in an array of coupled networks [7–9]. A coupled network can exhibit complicated dynamical behaviors or collective behaviors which may be absolutely different from those of a single node. Hence, investigating the dynamical behaviors of coupled networks is of great importance.

One of the important collective behaviors of coupled networks is synchronization. Actually, many natural phenomena can be well explained by the synchronization of coupled networks. Moreover, synchronization has potential applications in many fields such as secure communication [10, 11] and information processing [12]. Therefore, in recent years, an increasing number of researchers have been devoted to the investigation of synchronization in an array of coupled networks [6, 7, 9, 10, 13–22]. It is well known that sometimes the coupled networks may not be synchronized when a controller is not added into the infrastructure of individual nodes [7]. Thus the controlled synchronization of coupled networks is believed to be a rather significant topic in both theoretical research and practical applications. Consequently, many control methods have been proposed to realize chaos synchronization of coupled networks, for instance, intermittent control [7], state feedback control [9], and adaptive control [23]. Particularly, adaptive control technique is an effective method due to its good robustness.

In the literature, there were many results concerning synchronization of coupled networks via adaptive control [14–16, 19–22]. Cao et al. [14] investigated the complete synchronization in an array of linearly stochastically coupled identical neural networks with delays. In [15], Chen and Zhou studied the synchronization of coupled nondelayed networks under the adaptive control. By designing a simple adaptive controller, authors of [20] investigated the locally and globally adaptive synchronization of an uncertain coupled dynamical network. Recently, authors of [21] considered the local and global synchronization of coupled networks without delays via adaptive pinning control. Authors of [16, 19, 22] investigated the global synchronization of the coupled networks with nondelayed and delayed couplings by utilizing adaptive control technique.

It should be noted that most of the above results are valid only for the coupled networks without perturbation. However, in practice, chaotic systems are inevitably subject to many types of uncertain or even nonlinear perturbations, such as unknown parameters, exogenous disturbances, and artificial factors [23]. For instance, it is reported that the famous Lorenz system is derived from partial differential equations after a series of approximations [24]. Therefore, there usually exist modelling errors between the deterministic model and practical system. One can also find that dynamical nodes are difficult to be identical all the time since the parameters of dynamical nodes may be variant due to some environmental changes [25]. Another example is the social network [5], in which a person is a node and the relationship between persons is the edge. A person’s mood may change according to the evolution of the time and environments and hence many uncertainties may interfere with the individual’s behaviors. Furthermore, in the process of signal transmitting, the states of subsystems in coupled networks are unavoidably subject to some uncertain perturbations. Thus, synchronization of coupled chaotic systems with uncertain perturbations is essential and useful in both theoretical research and practical applications. Although the usual adaptive technique used in [14–17, 19–22] has good robustness when synchronizing coupled networks without perturbation, it is not always effective to synchronize coupled networks with various types of uncertain nonlinear perturbations. Simulations of this paper show that the adaptive technique used in [14–17, 19–22] cannot effectively synchronize coupled dynamical systems with some uncertain perturbations onto a given trajectory. Note that the uncertain perturbations of this paper are not stochastic as those in [7, 9, 19]. A distinct feature of the uncertain perturbations in this paper is that their effects cannot disappear even after the synchronization has been realized.

On the other hand, time delays usually exist in spreading due to the finite speeds of transmission as well as traffic congestions. Therefore, it is significant to investigate the synchronization in an array of coupled dynamical systems with delayed couplings. In [14, 16, 22], synchronization in an array of coupled dynamical systems with delayed couplings was studied. However, to the best of our knowledge, result on synchronization in an array of coupled systems with nondelayed and delayed couplings and uncertain nonlinear perturbations is seldom.

Motivated by the above analysis, in this paper, a general model of coupled networks with nondelayed and delayed couplings as well as uncertain nonlinear perturbations is proposed. Our model is applicable to most of known chaotic systems with or without uncertainties. The coupling configuration matrices are not assumed to be symmetric. Global asymptotic synchronization of the proposed model is studied. A new simple but robust adaptive controller is designed to overcome the effects of uncertainties and nonlinear external perturbations. The designed controller can synchronize the considered network to a given isolate trajectory. Moreover, our controller includes controllers used in [14–17, 19–22] as a special case. The designed adaptive controller can also synchronize the coupled dynamical network without any uncertainty and perturbation. Based on the Lyapunov stability theory and the Barbalat lemma, sufficient conditions are obtained to guarantee the realization of the synchronization of the coupled network. Numerical simulations verify the effectiveness of the theoretical results. Simulations also demonstrate that our adaptive controller has better robustness than those used in [14–17, 19–22].

The rest of this paper is organized as follows. In Section 2, a class of general coupled networks with uncertainties is proposed. Some necessary assumptions and lemmas are also given in this section. In Section 3, synchronization of the coupled networks with or without uncertainties is studied. Then, in Section 4, numerical simulations are given to show the effectiveness of our results. Section 5 concludes the investigation and expresses the acknowledgements. Future research field is also discussed in this section.

Notations. In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. 𝐼_𝑁 stands for the identity matrix of 𝑁-dimension. ℝ^𝑛 is the Euclidean space of 𝑛-dimension and ℝ⁺ is the set of nonnegative real numbers. For vector 𝑥∈ℝ^𝑛, the Euclidean norm is ‖𝑥‖=𝑥^𝑇𝑥, where 𝑇 denotes transposition. 𝐴=(𝑎_𝑖𝑗)_𝑚×𝑚 denotes a matrix of 𝑚-dimension, ‖𝐴‖=𝜆_max(𝐴^𝑇𝐴), 𝐴^𝑠=(1/2)(𝐴+𝐴^𝑇). 𝐴>0 or 𝐴<0 means that the matrix 𝐴 is symmetric and positive or negative definite matrix. 𝜆_min(𝐴) and 𝜆_max(𝐴) are the minimum and maximum eigenvalues of the matrix 𝐴, respectively.

2. Model Formulation and Preliminaries

The general coupled network consisting of 𝑁 identical nodes with nondelayed and delayed couplings as well as uncertain nonlinear perturbations is described as follows: 𝑥̇_𝑖(𝑡)=𝑓₁𝑥_𝑖(𝑡)+𝑓₂𝑥_𝑖((𝑡+𝑡−𝜏))_𝑁𝑗=1𝑢_𝑖𝑗Φ𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑥_𝑗(𝑡−𝜏(𝑡))+𝜎_𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))+𝑅_𝑖,𝑖=1,2,…,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 𝑓₁(⋅) and 𝑓₂(⋅) are continuous vector functions. 𝑅_𝑖∈ℝ^𝑛 is the control input. 𝜏(𝑡)>0 is time-varying delay and Φ, Υ∈ℝ^𝑛×𝑛 are inner coupling matrices of the network, which describe the individual coupling between two subsystems. Matrices 𝑈=(𝑢_𝑖𝑗)_𝑁×𝑁 and 𝑉=(𝑣_𝑖𝑗)_𝑁×𝑁 are outer couplings of the whole network satisfying the following diffusive conditions: 𝑢_𝑖𝑗≥0𝑖≠𝑗,𝑢_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑢_𝑖𝑗𝑣,𝑖,𝑗=1,2,…,𝑁,_𝑖𝑗≥0𝑖≠𝑗,𝑣_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑣_𝑖𝑗,𝑖,𝑗=1,2,…,𝑁. (2) Vector 𝜎_𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))=(𝜎_𝑖1(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))),…,𝜎_𝑖𝑛(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))))^𝑇∈ℝ^𝑛 describes the uncertain nonlinear perturbation of 𝑖th node of the coupled systems at time 𝑡. In this paper, we always assume that the derivative of 𝜏(𝑡) satisfies 𝜏̇(𝑡)≤ℎ_𝜏<1, where ℎ_𝜏 is constant.

We assume that (1) has a unique continuous solution for any initial condition in the following form: 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),−𝜏≤𝑠≤0,𝑖=1,2,…,𝑁, (3) where 𝜏 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))) as 𝜎_𝑖(𝑡).

The aim of this paper is to synchronize all the states of the coupled network (1) to the following manifold: 𝑥₁(𝑡)=𝑥₂(𝑡)=⋯=𝑥_𝑁(𝑡)=𝑧(𝑡), (4) where 𝑧(𝑡) is the state of an isolate node system without perturbation, which is described as follows: 𝑧(̇𝑡)=𝑓₁(𝑧(𝑡))+𝑓₂(𝑧(𝑡−𝜏(𝑡))), (5) and 𝑧(𝑡) can be any desired state: equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit.

The following assumptions are needed in this paper.(H₁)𝑓₁(0)=𝑓₂(0)≡0 and there exist positive constants ℎ₁ and ℎ₂ such that 𝑓₁(𝑢)−𝑓₁(𝑣)≤ℎ₁‖‖,𝑓𝑢−𝑣₂(𝑢)−𝑓₂(𝑣)≤ℎ₂‖𝑢−𝑣‖,forany𝑢,𝑣∈ℝ^𝑛. (6)(H₂)Solutions of all the nodes in network (1) without control are bounded; that is, there exists a positive constant 𝐷 such that ‖𝑥_𝑖(𝑡)‖≤𝐷, 𝑖=1,2,…,𝑁.(H₃)𝜎_𝑖𝑘(𝑡,0,0)≡0, and for any positive constants 𝐷 and 𝐷^′ such that ‖𝑢‖≤𝐷 and ‖𝑣‖≤𝐷^′, 𝑢,𝑣∈ℝ^𝑛, there exists positive constant 𝑀_𝑖𝑘 such that |𝜎_𝑖𝑘(𝑡,𝑢,𝑣)|≤𝑀_𝑖𝑘, 𝑖=1,2,…,𝑁,𝑘=1,2,…,𝑛.

Remark 1. It follows from (H₁) that system (5) unifies many well-known chaotic systems with or without delays such as Chua system [26], Lorenz system [27], Rössler system [28], Chen system [29], delayed chaotic neural networks [14], and delayed Chua system [30, 31]. A similar condition to (H₁) is also used in [32–34] for neural networks. Hence, results of this paper are general.

Remark 2. Conditions (H₂) and (H₃) are reasonable. We do not impose the usual condition such as Lipschitz condition or differentiability on the uncertain perturbation function 𝜎_𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))). It can be discontinuous or even impulsive function. If the state of each node system in (1) is an equilibrium point or a nontrivial periodic orbit, the condition (H₂) can be easily satisfied. If the state of each node in system (1) 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. For such three cases, all the solutions of (1) are bounded and (H₃) is satisfied. Moreover, we shall subsequently prove that, with suitable robust adaptive controller, the coupled network (1) can be synchronized to the trajectory of (5) even without knowing the exact values of ℎ₁, ℎ₂, and 𝑀_𝑖𝑘, 𝑖=1,2,…,𝑁,𝑘=1,2,…,𝑛. However, examples at the end of this paper show that the usual adaptive controller used in [14–17, 19–22] is not sufficient to achieve this goal.

Lemma 3 ([35, Barbalat lemma]). 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 (7) exists and is finite, then lim_𝑡→∞𝑓(𝑡)=0.

3. Synchronization with Uncertain Nonlinear Perturbations

In this section, an adaptive controller is designed such that the controlled network (1) can be asymptotically synchronized onto the isolate system (5). Corresponding results are also given for the network in the two cases: only partial dynamical nodes are perturbed and none of the dynamical nodes is perturbed. The advantages of the designed adaptive controller over those of existing ones are discussed in Remark 7.

Let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑧(𝑡). Subtracting (5) from (1) yields the following error dynamical system: 𝑒_𝑖̇(𝑡)=𝑔₁𝑒_𝑖(𝑡)+𝑔₂𝑒_𝑖((𝑡+𝑡−𝜏))_𝑁𝑗=1𝑢_𝑖𝑗Φ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑒_𝑗(𝑡−𝜏(𝑡))+𝜎_𝑖(𝑡)+𝑅_𝑖,𝑖=1,2,…,𝑁, (8) where 𝑔₁(𝑒_𝑖(𝑡))=𝑓₁(𝑥_𝑖(𝑡))−𝑓₁(𝑧(𝑡)), 𝑔₂(𝑒_𝑖(𝑡))=𝑓₂(𝑥_𝑖(𝑡))−𝑓₂(𝑧(𝑡)).

From (H₁) and (H₃) we know that (8) admits a trivial solution 𝑒_𝑖(0)≡0, 𝑖=1,2,…,𝑁. Obviously, to reach goal (4), we only need to prove that the trivial solution of system (8) is asymptotically stable. Theorem 4 is our main result.

Theorem 4. Under the assumption conditions (𝐻₁)--(𝐻₃), the trivial solution of the error system (8) is asymptotically stable with the following adaptive controllers: 𝑅_𝑖=−𝜀_𝑖(𝑡)𝑒_𝑖(𝑡)−𝜔𝛽_𝑖(𝑡)sign𝑒_𝑖(𝑡)𝜀̇,𝑖=1,2,…,𝑁,_𝑖(𝑡)=𝑝_𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)𝛽̇,𝑖=1,2,…,𝑁,_𝑖(𝑡)=𝜉_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡),𝑖=1,2,…,𝑁, (9) where sign(𝑒_𝑖(𝑡))=(sign(𝑒_𝑖1(𝑡)),…,sign(𝑒_in(𝑡)))^𝑇, 𝜔>1, 𝑝_𝑖>0, and 𝜉_𝑖>0 are arbitrary constants, respectively.

Proof. It follows from (H₂) that there exists positive constant 𝐷 such that ‖𝑥_𝑖(𝑡)‖≤𝐷 for 𝑡∈[−𝜏,+∞)(𝑖=1,2,…,𝑁). In view of (H₃), there exists positive constant 𝑀_𝑖𝑘 such that |𝜎_𝑖𝑘(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))|≤𝑀_𝑖𝑘(𝑖=1,2,…,𝑁,𝑘=1,2,…,𝑛). Take 𝑀_𝑖=max{𝑀_𝑖𝑘,𝑘=1,2,…,𝑛}(𝑖=1,2,…,𝑁) and define the Lyapunov function as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡), (10) where 𝑉₁(𝑡)=12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+12^𝑁_𝑖=1𝜀_𝑖(𝑡)−𝑘_𝑖²𝑝_𝑖+12^𝑁_𝑖=1𝑀_𝑖−𝛽_𝑖(𝑡)²𝜉_𝑖,𝑉₂(𝑡)=^𝑡_{𝑡−𝜏(𝑡)}𝜂^𝑇(𝑠)𝑄𝜂(𝑠)d𝑠, (11) in which 𝜂(𝑡)=(‖𝑒₁(𝑡)‖,‖𝑒₂(𝑡)‖,…,‖𝑒_𝑁(𝑡)‖)^𝑇, 𝑄 is symmetric positive definite matrix, 𝑘_𝑖 is positive constant, and 𝑘_𝑖 and 𝑄 are to be determined.
Differentiating 𝑉₁(𝑡) along the solution of (8), one obtains from (H₁) and (H₂) that 𝑉̇₁(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖̇(𝑡)+^𝑁_𝑖=1𝜀_𝑖(𝑡)−𝑘_𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)−^𝑁_𝑖=1𝑀_𝑖−𝛽_𝑖(𝑡)^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡=_𝑁𝑖=1)𝑒^𝑇_𝑖(𝑡)𝑔₁𝑒_𝑖(𝑡)+𝑔₂𝑒_𝑖((𝑡+𝑡−𝜏))_𝑁𝑗=1𝑢_𝑖𝑗Φ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑒_𝑗((𝑡+𝑡−𝜏))_𝑁𝑖=1𝑒^𝑇_𝑖(𝑡)𝜎_𝑖(𝑡)−^𝑁_𝑖=1𝑘_𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)−𝜔^𝑁𝑛_𝑖=1𝑘=1𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡)−^𝑁𝑛_𝑖=1𝑘=1𝑀_𝑖−𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡≤_𝑁𝑖=1)ℎ₁𝑒_𝑖(𝑡)²+^𝑁_𝑖=1ℎ₂𝑒_𝑖(𝑡)𝑒_𝑖((𝑡−𝑡−𝜏))_𝑁𝑖=1𝑘_𝑖𝑒_𝑖(𝑡)²+^𝑁_𝑖=1𝜆_minΦ^𝑠𝑢_𝑖𝑖𝑒_𝑖(𝑡)²+^𝑁𝑁_{𝑖=1𝑗=1,𝑗≠𝑖}𝑢_𝑖𝑗‖Φ‖𝑒_𝑖(𝑡)𝑒_𝑗(𝑡+_{𝑁𝑁𝑖=1𝑗=1})𝑣_𝑖𝑗‖Υ‖𝑒_𝑖(𝑡)𝑒_𝑗(𝑡−𝜏(𝑡))=𝜂^𝑇(𝑡)ℎ₁𝐼_𝑁+‖Φ‖𝑈^𝑠−𝐾𝜂(𝑡)+𝜂^𝑇(𝑡)ℎ₂𝐼_𝑁+‖Υ‖|𝑉|𝜂(𝑡−𝜏(𝑡))≤𝜂^𝑇(𝑡)ℎ₁+12𝐼_𝑁+‖Φ‖𝑈^𝑠−𝐾𝜂(𝑡)+𝜂^𝑇(𝑡−𝜏(𝑡))𝑃^𝑇𝑃𝜂(𝑡−𝜏(𝑡)), (12) 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, 𝑃=ℎ₂𝐼_𝑁+‖Υ‖|𝑉|, |𝑉|=(|𝑣_𝑖𝑗|)_𝑁×𝑁, and we have used the following deduction: ^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝜎_𝑖(𝑡)−𝜔^𝑁𝑛_𝑖=1𝑘=1𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡−_{𝑁𝑛𝑖=1𝑘=1})𝑀_𝑖−𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡≤_{𝑁𝑛𝑖=1𝑘=1})𝑒_𝑖𝑘(𝑡)𝑀_𝑖𝑘−𝑀_𝑖𝑒_𝑖𝑘(𝑡)−(𝜔−1)𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡)≤−^𝑁𝑛_𝑖=1𝑘=1(𝜔−1)𝛽_𝑖(𝑡)𝑒_𝑖𝑘≤0. (13)
Differentiating 𝑉₂(𝑡), one has 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)). (14)
Taking 𝑄=(1/(1−ℎ_𝜏))𝑃^𝑇𝑃, one gets from the definition of 𝑉(𝑡) that 𝑉̇(𝑡)≤𝜂^𝑇(𝑡)ℎ₁+12𝐼_𝑁+‖Φ‖𝑈^𝑠+11−ℎ_𝜏𝑃^𝑇𝑃−𝐾𝜂(𝑡). (15) Take 𝑘_𝑖=𝜆_max[(ℎ₁+1/2)𝐼_𝑁+‖Φ‖𝑈^𝑠+(1/(1−ℎ_𝜏))𝑃^𝑇𝑃]+1. Then, one can easily derive from the above inequality that 𝑉̇(𝑡)≤−𝜂^𝑇(𝑡)𝜂(𝑡). (16)
Integrating both sides of the above equation from 0 to 𝑡 yields 𝑉(0)≥𝑉(𝑡)+^𝑁_𝑖=1^𝑡₀𝑒_𝑖(𝑠)²d𝑠≥^𝑁_𝑖=1^𝑡₀𝑒_𝑖(𝑠)²d𝑠. (17) Therefore, lim^𝑁_{𝑡→∞𝑖=1}^𝑡₀𝑒_𝑖(𝑠)²d𝑠≤𝑉(0). (18)
It is obvious that 𝑒_𝑖(𝑡)(𝑖=1,2,…,𝑁) is uniformly continuous for 𝑡≥0; hence ∑^𝑁_𝑖=1‖𝑒_𝑖(𝑡)‖² is uniformly continuous for 𝑡≥0 according to the continuity of norm. Moreover, the above inequality means that lim_𝑡→∞∑^𝑁_𝑖=1∫^𝑡₀‖𝑒_𝑖(𝑠)‖²d𝑠 exists and is finite. In view of Lemma 3, one can easily get lim^𝑁_{𝑡→∞𝑖=1}𝑒_𝑖(𝑡)²=0, (19) which in turn means lim_𝑡→∞𝑒_𝑖(𝑡)=0,𝑖=1,2,…,𝑁. (20) Therefore, the trivial solution of the error system (8) is asymptotically stable. This completes the proof.

If some nodes are effected by uncertain perturbations while others are not perturbed, the trivial solution of the error system (8) can also be asymptotically stabilized by the adaptive controllers (9). Without loss of generality, rearrange the order of the nodes in the network, and let the first 𝑙 nodes have uncertainties; that is, 𝜎_𝑖(𝑡)≠0,𝑖=1,2,…,𝑙, 𝜎_𝑖(𝑡)≡0,𝑖=𝑙+1,𝑙+2,…,𝑁. We have Theorem 5.

Theorem 5. Suppose 𝜎_𝑖(𝑡)≠0,𝑖=1,2,…,𝑙, 𝜎_𝑖(𝑡)≡0,𝑖=𝑙+1,𝑙+2,…,𝑁. Then, under the assumption conditions (𝐻₁)--(𝐻₃), the trivial solution of the error system (8) is asymptotically stable with the adaptive controllers (9).

Proof. Define the Lyapunov function as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡), (21) where 𝑉₁(𝑡)=12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+12^𝑁_𝑖=1𝜀_𝑖(𝑡)−𝑘_𝑖²𝑝_𝑖+12^𝑙_𝑖=1𝑀_𝑖−𝛽_𝑖(𝑡)²𝜉_𝑖, (22)𝑉₂(𝑡) is defined as that in the proof of Theorem 4.
Differentiating 𝑉(𝑡) along the solution of (8) and noting that the following inequality holds ^𝑙_𝑖=1𝑒^𝑇_𝑖(𝑡)𝜎_𝑖(𝑡)−𝜔^𝑙𝑛_𝑖=1𝑘=1𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡−_{𝑙𝑛𝑖=1𝑘=1})𝑀_𝑖−𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡)−𝜔^𝑁𝑛_{𝑖=𝑙+1𝑘=1}𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡≤_{𝑙𝑛𝑖=1𝑘=1})𝑒_𝑖𝑘(𝑡)𝑀_𝑖𝑘−𝑀_𝑖𝑒_𝑖𝑘(𝑡)−(𝜔−1)𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡)−𝜔^𝑁𝑛_{𝑖=𝑙+1𝑘=1}𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡)≤−^𝑙𝑛_𝑖=1𝑘=1(𝜔−1)𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡)−𝜔^𝑁𝑛_{𝑖=𝑙+1𝑘=1}𝛽_𝑖(𝑡)𝑒_𝑖𝑘(𝑡)≤0, (23) by the same procedure of the proof of Theorem 4, one can easily finish the proof. This completes the proof.

A special case of Theorem 5 is that none of the nodes is perturbed. In this case, the designed adaptive controller (9) can obviously realize the synchronization goal. Moreover, we have more general result than the adaptive controllers [14–16, 19–22], which is presented in Corollary 6 and Remark 7.

Corollary 6. Suppose 𝜎_𝑖(𝑡)≡0,𝑖=1,2,…,𝑁, and the assumption condition (𝐻₁) holds. Then the trivial solution of the error system (8) is asymptotically stable with the adaptive controller (9). Moreover, the scalar 𝜔 can be relaxed to any nonnegative constant.

Proof. Define the Lyapunov function as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡), (24) where 𝑉₂(𝑡) is defined as that in the proof of Theorem 4: 𝑉₁(𝑡)=12^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+12^𝑁_𝑖=1𝜀_𝑖(𝑡)−𝑘_𝑖²𝑝_𝑖. (25)
Differentiating 𝑉(𝑡) along the solution of (8) and noting that −𝜔∑^𝑁_𝑖=1∑^𝑛_𝑘=1𝛽_𝑖(𝑡)|𝑒_𝑖𝑘(𝑡)|≤0, by the same procedure of the proof of Theorem 4, one can easily finish the proof. This completes the proof.

Remark 7. The new adaptive controller of this paper has better property of robustness than that used in [14–16, 19–22], which is 𝑅_𝑖=−𝜀_𝑖(𝑡)𝑒_𝑖(𝑡)𝜀̇,𝑖=1,2,…,𝑁,_𝑖(𝑡)=𝑝_𝑖𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡),𝑖=1,2,…,𝑁. (26) The adaptive controller (26) is a special case of (9) when 𝜔=0. Although controller (26) has robustness, it is not sufficient to synchronize networks with nonlinear perturbations. From Corollary 6, one sees that both adaptive controller (9) and (26) can synchronize a coupled network without any external perturbation to a given isolate system. Since uncertain perturbations to coupled networks are unavoidable in practice, our adaptive controller (9) is more practical than (26). Numerical simulations of this paper verify that new adaptive controller (9) has better robustness than (26).

Remark 8. Note that controller (9) are discontinuous and the phenomenon of chattering will appear [36, 37]. In order to eliminate the chattering, controller (9) can be modified as 𝑅_𝑖=−𝜀_𝑖(𝑡)𝑒_𝑖(𝑡)−𝜔𝛽_𝑖(𝑡)𝑆_𝑖𝜀̇,𝑖=1,2,…,𝑁,_𝑖(𝑡)=𝑝_𝑖𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)𝛽̇,𝑖=1,2,…,𝑁,_𝑖(𝑡)=𝜉_𝑖^𝑛_𝑘=1𝑒_𝑖𝑘(𝑡),𝑖=1,2,…,𝑁, (27) where 𝑆_𝑖=((𝑒_𝑖1(𝑡)/(|𝑒_𝑖1(𝑡)|+𝜁_𝑖)),…,(𝑒_𝑖𝑛(𝑡)/(|𝑒_𝑖𝑛(𝑡)|+𝜁_𝑖)))^𝑇, 𝜁_𝑖,𝑖=1,2,…,𝑁, are sufficiently small positive constants and 𝜔>1,   𝑝_𝑖>0, and 𝜉_𝑖>0, are arbitrary constants, respectively.

Remark 9. If only some nodes of (1) are effected by uncertain nonlinear perturbations and matrix 𝑈 is irreducible, we can also consider adaptive pinning control scheme (see [16, 21, 22]) with the new adaptive controller to synchronize coupled networks (1). Moreover, we can also consider stochastic perturbations [14] and the Markovian jump (see [18, 38]) in (1) to get more general results. This is our next research topic.

4. Numerical Examples

In this section, we first consider a network composed of three Lorenz systems with uncertain perturbations. Then one WS small-world model with partially perturbed nodes, in which the node systems are delayed neural networks, is given. Numerical simulations demonstrating better robustness of the designed adaptive controller than existing one are also given.

Example 1. The Lorenz system is described as [23, 27] 𝑧(̇𝑡)=𝑓₁(𝑧(𝑡))=𝐶𝑧(𝑡)+𝑓(𝑧(𝑡)), (28) where 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡))^𝑇, 𝑓(𝑧(𝑡))=(0,−𝑧₁(𝑡)𝑧₃(𝑡),𝑧₁(𝑡)𝑧₂(𝑡))^𝑇, and 8𝐶=−1010028−1000−3. (29) When initial values are taken as 𝑧₁(0)=0.8, 𝑧₂(0)=2, and 𝑧₃(0)=2.5, chaotic trajectory of (28) can be seen in Figure 1.
Now consider the following controlled network with nondelayed coupling, where each node system is the above Lorenz system with uncertainties: 𝑥̇_𝑖(𝑡)=𝑓₁𝑥_𝑖(𝑡)+𝛼³_𝑗=1𝑢_𝑖𝑗Φ𝑥_𝑗(𝑡)+𝜎_𝑖(𝑡)+𝑅_𝑖,𝑖=1,2,3, (30) where 𝛼=0.5,Φ=𝐼₃, 𝜎₁(𝑡)=[0.1𝑥²₁₁(𝑡),0.2𝑥₁₂(𝑡),0.2𝑥₁₃(𝑡)]^𝑇, 𝜎₂(𝑡)=[0.1𝑥₂₁(𝑡),0.05𝑥²₂₂(𝑡),sin𝑥₂₃(𝑡)]^𝑇, 𝜎₃(𝑡)=[0.1𝑥₃₁(𝑡),(sin𝑥₃₂(𝑡))²,sin𝑥₃₃(𝑡)]^𝑇, and 𝑈=−1101−1011−2. (31) Numerical simulations show that the states of (30) without control are bounded; see Figure 2. Therefore, (H₂) is satisfied.
It is obvious that 𝑓₁(0)=0. Let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑧(𝑡)=(𝑒_𝑖1(𝑡),𝑒_𝑖2(𝑡),𝑒_𝑖3(𝑡))^𝑇; then 𝑓₁𝑥_𝑖(𝑡)−𝑓₁(𝑧(𝑡=))𝐶𝑒_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡)(𝑧(𝑡≤‖𝐶𝑒−𝑓))‖_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡)−𝑓(𝑧(𝑡)). (32) On the other hand, one can easily get 𝑓𝑥_𝑖(𝑡)−𝑓(𝑧(𝑡))^𝑇𝑓𝑥_𝑖(𝑡)(𝑧(𝑡=𝑥−𝑓))_𝑖1𝑒_𝑖3+𝑧₃𝑒_𝑖1²+𝑥_𝑖1𝑒_𝑖2+𝑧₂𝑒_𝑖1²≤𝑎𝑒²_𝑖1(𝑡)+𝑏𝑒²_𝑖2(𝑡)+𝑐𝑒²_𝑖3(𝑡), (33) where 𝑎=𝑧²₃(𝑡)+𝑧²₂(𝑡)+|𝑥_𝑖1(𝑡)𝑧₃(𝑡)|+|𝑥_𝑖1(𝑡)𝑧₂(𝑡)|, 𝑏=𝑥²_𝑖1(𝑡)+|𝑥_𝑖1(𝑡)𝑧₂(𝑡)|, and 𝑐=𝑥²_𝑖1(𝑡)+|𝑥_𝑖1(𝑡)𝑧₃(𝑡)|. From Figures 1 and 2, it is obvious that 𝑥_𝑖𝑗(𝑡),𝑖,𝑗=1,2,3, and 𝑧_𝑗(𝑡),𝑗=1,2,3, are all bounded; hence 𝑎,𝑏, and 𝑐 are all bounded. Take 𝑑=max{𝑎,𝑏,𝑐}; then 𝑓𝑥_𝑖(𝑡)−𝑓(𝑧(𝑡))≤𝑑^1/2𝑒_𝑖(𝑡). (34) One gets from (32) and (34) that 𝑓𝑥_𝑖(𝑡)−𝑓(𝑧(𝑡))≤‖𝐶‖+𝑑^1/2𝑒_𝑖(𝑡). (35) Take ℎ₁=‖𝐶‖+𝑑^1/2; then (H₁) is satisfied. Since 𝑥_𝑖𝑗(𝑡),𝑖,𝑗=1,2,3, are bounded, every component of 𝜎_𝑖(𝑡),𝑖=1,2,3, is bounded, and (H₃) is satisfied. According to Theorem 4, the coupled network (30) can be asymptotically synchronized onto (28) with adaptive controller (9).
In the simulations, the forward Euler method is used to simulate in Matlab (math works). The initial conditions of the numerical simulations are as follows: step =0.0005, 𝑥₁(0)=(−2,−1,0)^𝑇, 𝑥₂(0)=(1,2,3)^𝑇, 𝑥₃(0)=(4,5,6)^𝑇, 𝜔=4, 𝜀_𝑖(0)=1, 𝛽_𝑖(0)=1, 𝑝_𝑖=𝜉_𝑖=0.05,𝑖=1,2,3. Figure 3 describes the synchronization errors 𝑒_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡)−𝑧_𝑗(𝑡)(𝑖,𝑗=1,2,3) between (30) and (28) by using controllers (9), from which one can see that the synchronization errors quickly turn to zero as time goes. Figure 4 presents the time response of the feedback gain parameters 𝜀_𝑖(𝑡) and 𝛽_𝑖(𝑡),1≤𝑖≤3, which reach constants eventually. Numerical simulations show that when all nodes are perturbed by uncertainties, the new adaptive controller (9) can synchronize the coupled networks onto a given trajectory, which verify the effectiveness of Theorem 4.
In order to show the better robustness and advantage of our new controller, we now control (30) with the adaptive controller (26), where all the values of parameters are the same as those above. Figure 5 describes the trajectories of synchronization errors between (30) and (28) by using adaptive controller (26), from which one can see that 𝑒_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡)−𝑧_𝑗(𝑡)(𝑖,𝑗=1,2,3) do not turn to zero as time goes. Hence, controllers (26) cannot synchronize (30) onto the trajectory of (28).

Figure 1 

Chaotic trajectory of system (28).

(a) 𝑥𝑖1(𝑡),1≤𝑖≤3

(b) 𝑥𝑖2(𝑡),1≤𝑖≤3

(c) 𝑥𝑖3(𝑡),1≤𝑖≤3

Figure 2 

State trajectories of (30) without control: (a) 𝑥𝑖1(𝑡),1≤𝑖≤3; (b) 𝑥𝑖2(𝑡),1≤𝑖≤3; (c) 𝑥𝑖3(𝑡),1≤𝑖≤3.

(a) 𝑒𝑖1(𝑡),1≤𝑖≤3

(b) 𝑒𝑖2(𝑡),1≤𝑖≤3

(c) 𝑒𝑖3(𝑡),1≤𝑖≤3

Figure 3 

Synchronization errors between (30) and (28) with adaptive controllers (9).

(a) 𝜀𝑖(𝑡),1≤𝑖≤3

(b) 𝛽𝑖(𝑡),1≤𝑖≤3

Figure 4 

Feedback gain parameters 𝜀𝑖(𝑡) (a) and 𝛽𝑖(𝑡) (b) (1≤𝑖≤3) of (30) with adaptive controller (9).

(a) 𝑒𝑖1(𝑡),1≤𝑖≤3

(b) 𝑒𝑖2(𝑡),1≤𝑖≤3

(c) 𝑒𝑖3(𝑡),1≤𝑖≤3

Figure 5 

Synchronization errors between (30) and (28) by using adaptive controllers (26).

Example 2. In [39], Watts and Strogatz proposed a small-word network (WS small-world) model, which is described as follows.(1)Initialization. Starting with a regular ring lattice of 𝑁 nodes, each node is connected to 𝐾 (even number) nearest neighbors by undirected links.(2)Rewiring. Randomly rewire each link of the network with probability 𝑝 such that self-connected and duplicated links are excluded. Here, rewiring means reconnecting one end of a selected link to another randomly chosen node.
The delayed neural network is described as [14] 𝑧(̇𝑡)=𝑓₁(𝑧(𝑡))+𝑓₂(𝑧(𝑡−𝜏(𝑡))), (36) where 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡))^𝑇, 𝑓₁(𝑧(𝑡))=𝐶𝑧(𝑡)+𝐴𝑓(𝑧(𝑡)), 𝑓₂(𝑧(𝑡−𝜏(𝑡)))=𝐵𝑔(𝑧(𝑡−𝜏(𝑡))), 𝑓(𝑧(𝑡))=(tanh(𝑧₁(𝑡)),tanh(𝑧₂(𝑡)))^𝑇, 𝑔(𝑧(𝑡−𝜏(𝑡)))=(tanh(𝑧₁(𝑡−𝜏(𝑡))),tanh(𝑧₂(𝑡−𝜏(𝑡))))^𝑇, 𝜏(𝑡)=1, ,.𝐶=−100−1,𝐴=2−0.1−54.5𝐵=−1.5−0.1−0.2−4 (37) When the initial values are taken as 𝑧(𝑠)=(0.4,0.6)^𝑇, ∀𝑠∈[−1,0], system (36) is chaotic as shown in Figure 6.
It can be easily verified that 𝑓₁(0)=𝑓₂(0)=0, ‖𝐶‖=1,‖𝐴‖=6.9099, ‖𝐵‖=4.0094, and ‖𝑓(𝑥)−𝑓(𝑦)‖≤‖𝑥−𝑦‖ for any 𝑥,𝑦∈ℝ². Hence 𝑓₁(𝑥)−𝑓₁𝑦≤𝐶‖+‖𝐴,𝑓(‖‖)𝑥−𝑦=7.9099𝑥−𝑦₂(𝑥)−𝑓₂𝑦≤‖𝐵‖𝑥−𝑦=4.0094𝑥−𝑦. (38) Therefore, (H₁) is satisfied with ℎ₁=7.9099 and ℎ₂=4.0094.
Next, consider the following coupled network subject to uncertainties and control inputs: 𝑥̇_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)+𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑔𝑥_𝑖(𝑡−𝜏(𝑡))+𝑐₁^𝑁_𝑗=1𝑢_𝑖𝑗Φ𝑥_𝑗(𝑡)+𝑐₂^𝑁_𝑗=1𝑣_𝑖𝑗Υ𝑥_𝑗(𝑡−𝜏(𝑡))+𝜎_𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))+𝑅_𝑖,𝑖=1,2,…,𝑁, (39) where Φ=Υ=𝐼₂, 𝑐₁=0.05, and 𝑐₂=0.005. Suppose that the network is connected in a small-world topology. If there is a connection between nodes 𝑖 and 𝑗(𝑗≠𝑖), then 𝑢_𝑖𝑗=𝑣_𝑖𝑗=1; otherwise, 𝑢_𝑖𝑗=𝑣_𝑖𝑗=0. Taking 𝑁=30,𝐾=4, and 𝑝=0.2, we get structure of WS small world shown in Figure 7.
Suppose the uncertain perturbations of the nodes 𝑖=1,2,…,10 are 𝜎_𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))=(−0.1𝑥_𝑖1(𝑡),0.2𝑥_𝑖2(𝑡))^𝑇(1≤𝑖≤10), the uncertain perturbations of the nodes 𝑖=11,12,…,20 are 𝜎_𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))=(0.05(𝑥_𝑖1(𝑡−𝜏(𝑡)))²,0.2𝑥_𝑖2(𝑡))^𝑇(11≤𝑖≤20), and the nodes 𝑖=21,22,…,30 have no perturbation. The state trajectories 𝑥_𝑖1(𝑡) and 𝑥_𝑖2(𝑡)(1≤𝑖≤30) of the perturbed network (39) without control are shown in Figure 8, from which one can see that 𝑥_𝑖1(𝑡) and 𝑥_𝑖2(𝑡)(1≤𝑖≤30) are all bounded. Obviously, conditions (H₂) and (H₃) are also satisfied. According to Theorem 5, the coupled network (39) can be synchronized to (36) with adaptive controller (9).
Again, the forward Euler method is used to simulate in Matlab. The initial conditions of the numerical simulations are as follows: step =0.0001, the initial values of 𝑥_𝑖(𝑡)(𝑖=1,2,…,30) are arbitrary in [−2,2], 𝜔=1.5, 𝜀_𝑖(𝑡)=𝛽_𝑖(𝑡)=1,𝑡∈[−1,0], and𝑝_𝑖=𝜉_𝑖=0.5(𝑖=1,2,…,30). Figure 9 describes the synchronization errors𝑒_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡)−𝑧_𝑗(𝑡)(𝑖=1,2,…,30,𝑗=1,2,3)between (39) and (36) by using adaptive controller (9). One can see from Figure 9 that synchronization errors quickly turn to zero as time goes. Figure 10 shows the feedback gain parameters𝜀_𝑖(𝑡)and𝛽_𝑖(𝑡)(1≤𝑖≤30), which all reach constants eventually. Numerical simulations show that when only partial nodes are perturbed by uncertainties, the new adaptive controller (9) can also synchronize the coupled networks onto a given trajectory, which verify the effectiveness of Theorem 5.
However, with the same parameter values, controller (26) cannot synchronize (39) to the trajectory of (36). The synchronization errors between (39) and (36) by using adaptive controller (26) are shown in Figure 11.

Figure 6 

Trajectory of the delayed neural network (36).

Figure 7 

The structure of WS small-world network with 𝑁=30,𝐾=4, and 𝑝=0.2.

(a) 𝑥𝑖1(𝑡)(1≤𝑖≤30)

(b) 𝑥𝑖2(𝑡)(1≤𝑖≤30)

Figure 8 

State trajectories of (39) without control: (a) 𝑥𝑖1(𝑡)(1≤𝑖≤30), (b) 𝑥𝑖2(𝑡)(1≤𝑖≤30).

(a) 𝑒𝑖1(𝑡),1≤𝑖≤30

(b) 𝑒𝑖2(𝑡),1≤𝑖≤30

Figure 9 

Synchronization errors between (39) and (36) with adaptive controllers (9).

(a) Feedback gain parameters 𝜀𝑖(𝑡),1≤𝑖≤30

(b) Feedback gain parameters 𝛽𝑖(𝑡),1≤𝑖≤30

Figure 10 

Feedback gain parameters 𝜀𝑖(𝑡) (a) and 𝛽𝑖(𝑡) (b) (1≤𝑖≤30) of (39) with controller (9).

(a) 𝑒𝑖1(𝑡),1≤𝑖≤30

(b) 𝑒𝑖2(𝑡),1≤𝑖≤30

Figure 11 

Synchronization errors between (39) and (36) by using adaptive controllers (26).

Remark 10. Figure 11(a) shows that the usual adaptive controller has robustness; however, one can see from Figure 11(b) that its robustness is limited. Nevertheless, by comparing Figures 5 and 11 with Figures 3 and 9, respectively, one can easily see that the new adaptive controller (9) has better robustness than the usual one (26). Therefore, the new adaptive controller proposed in this paper is more effective and more robust than that in [14–16, 19–22] for coupled networks with uncertain perturbations.

Remark 11. Since uncertain perturbations to coupled networks in real life are unavoidable, the new adaptive controller (9) is important. Actually, external uncertain perturbations may have key effects on the dynamics of node system. For example, consider the system (36) with perturbation: 𝑧(̇𝑡)=𝑓₁(𝑧(𝑡))+𝑓₂(𝑧((𝑡((𝑡)((𝑡.𝑡−𝜏)))+𝜎𝑡,𝑧,𝑧𝑡−𝜏))) (40) Take the same initial values as those in Figure 6; that is,𝑧(𝑠)=(0.4,0.6)^𝑇,∀𝑠∈[−1,0]. Figure 12 presents the different trajectories of system (40) under different perturbations, which are completely different from those of system (36); see Figure 6. However, under these perturbations, the coupled network (39) can still be synchronized onto trajectory of (36) by the designed controller (9).

(a) Trajectory of system (40) with 𝜎(𝑡,𝑧(𝑡),𝑧(𝑡−𝜏(𝑡)))=(−0.1𝑧(𝑡),0.2𝑧(𝑡))𝑇

(b) Trajectory of system (40) with 𝜎(𝑡,𝑧(𝑡),𝑧(𝑡−𝜏(𝑡)))=(0.05(𝑧(𝑡−𝜏(𝑡)))2,0.2𝑧(𝑡))𝑇

Figure 12 

Trajectories of system (40) under different perturbations.

5. Conclusion

Uncertainties for systems are unavoidable in practice. Therefore, in this paper, we introduced a class of coupled networks with delays and uncertain nonlinear perturbations. A simple but robust adaptive controller is designed to synchronize the coupled networks onto an isolate node even without knowing priori the bounds of such perturbations. Results of this paper are also applicable to complex networks with asymmetric coupling configuration matrix. The designed controller enhances the robustness and reduces fragility of coupled networks; hence, it has great practical significance. Two types of coupled network model with uncertain perturbations, asymmetric coupled Lorenz network and WS small-world type of complex network with delayed neural network as node system, are employed to verify the effectiveness of the theoretical results. Numerical simulations also show that the designed adaptive controller is more robust and more effective in synchronizing a coupled network than the usual adaptive controller used in the references.

Obviously, it is optimal to synchronize complex networks with delays and uncertain perturbations in finite time. However, the existing controllers cannot synchronize a delayed complex network in finite time. Therefore, our next research is to design simple and effective controllers to realize finite-time synchronization of complex networks with delays and uncertain perturbations, which is challenging.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant no. 61263020, the Scientific Research Fund of Chongqing Municipal Education Commission under Grant no. KJ130613, and the Scientific Research Fund of Chongqing Normal University under Grants no. 12XLB031 and no. 940115.