The Asymptotic Synchronization Analysis for Two Kinds of Complex Dynamical Networks

Abstract

We consider a class of complex networks with both delayed and nondelayed coupling. In particular, we consider the situation for both time delay-independent and time delay-dependent complex dynamical networks and obtain sufficient conditions for their asymptotic synchronization by using the Lyapunov-Krasovskii stability theorem and the linear matrix inequality (LMI). We also present some simulation results to support the validity of the theories.

1. Introduction

A complex dynamical network is a large set of interconnected nodes that represent the individual elements of the system and their mutual relationships. Owing to their immense potential for applications to various fields, complex networks have been intensively investigated in the past decade in areas as diverse as mathematics, physics, biology, engineering, and even the social sciences [1–3]. The synchronization problem for complex networks was first posed by Saber and Murray [4, 5] who also introduced a theoretical framework for their investigation by viewing them as the adjustments of the rhythms of their interaction states [6] and many different kinds of synchronization phenomena and models have also been discovered such as complete synchronization, phase synchronization, lag synchronization, antisynchronization, impulsive synchronization, and projective synchronization.

Time delays are an important consideration for complex networks although these were usually ignored in early investigations of synchronization and control problems [6–11]. To make up for this deficiency, uniformly distributed time delays have recently been incorporated into network models [12–25] and Wang et al. [18] even considered networks with both delayed and nondelayed couplings and obtained sufficient conditions for asymptotic stability. Similarly, Wu and Lu [19] investigated the exponential synchronization of general weighted delay and nondelay coupled complex dynamical networks with different topological structures. There remains, however, much room for improvement in both the scope of the systems considered by Wang and Xu as well as in their methods of proofs.

The main contributions of this paper are two-fold. Firstly, we present a more general model for networks with both delayed and nondelayed couplings and derive criteria for their asymptotical synchronization. Secondly, we apply the Lyapunov-Krasovskii theorem and the LMIs to ensure the inevitable attainment of the required synchronization.

The rest of the paper is organized as follows. In Section 2, we present the general complex dynamical network model under consideration and state some preliminary definitions and results. In Section 3, we present the main results of this paper. In particular, we consider the situation for both time delay-independent and time delay-dependent complex dynamical networks and derive sufficient conditions for their asymptotic synchronization by using the Lyapunov-Krasovskii stability theorem and the linear matrix inequality (LMI). In Section 4, we present some numerical simulation results that verify our theoretical results. The paper concludes in Section 5.

2. Preliminaries and Model Description

In general, a linearly coupled ordinary differential equation system (LCODES) can be described as follows: 𝑑𝑥_𝑖(𝑡)𝑥𝑑𝑡=𝑓_𝑖(𝑡)+𝑐₁^𝑁_{𝑗≠𝑖,𝑗=1}𝑏_𝑖𝑗𝐴𝑥_𝑗(𝑡)+𝑐₂^𝑁_{𝑗≠𝑖,𝑗=1}𝑏^′_𝑖𝑗𝐴^′𝑥_𝑗(𝑡−𝜏). (2.1) Since 𝑥_𝑖−𝑥_𝑖=0 for all 𝑖=1,…,𝑁, we can choose any values for 𝑎_𝑖𝑖 in the above equations. Hence, letting 𝑏_𝑖𝑖=−∑^𝑁_{𝑗≠𝑖,𝑗=1}𝑏_𝑖𝑗 and 𝑏^′_𝑖𝑖=−∑^𝑁_{𝑗≠𝑖,𝑗=1}𝑏^′_𝑖𝑗, the above equations can be rewritten as follows: 𝑑𝑥_𝑖(𝑡)𝑥𝑑𝑡=𝑓_𝑖(𝑡)+𝑐₁^𝑁_𝑗=1𝑏_𝑖𝑗𝐴𝑥_𝑗(𝑡)+𝑐₂^𝑁_𝑗=1𝑏^′_𝑖𝑗𝐴^′𝑥_𝑗(𝑡−𝜏), (2.2) where 𝑁 is the number of nodes, 𝑥^𝑖(𝑡)=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑁)^𝑇∈𝑅^𝑛 are the state variables of the 𝑖th node, 𝑡∈[0,+∞) and 𝑓∶𝑅^𝑛→𝑅^𝑛 is a continuously differentiable function. The constants 𝑐₁ and 𝑐₂ (possibly distinct) are the coupling strengths, 𝑏_𝑖𝑗≥0, 𝑏^′_𝑖𝑗>0 (for 𝑖,𝑗=1,…,𝑁), 𝐴,𝐴^′∈𝑅^𝑛×𝑛 are inner-coupled matrices, 𝐵,𝐵^′∈𝑅^𝑛×𝑛 are coupled matrices with zero-sum rows with 𝑏_𝑖𝑗,𝑏^′_𝑖𝑗≥0 for 𝑖≠𝑗 that determines the topological structure of the network. We assume that 𝐵 and 𝐵^′ are symmetric and irreducible matrices so that there are no isolated nodes in the system.

If all the eigenvalues of a matrix 𝐴∈𝑅^𝑛×𝑛 are real, then we denote its 𝑖th eigenvalue by 𝜆_𝑖(𝐴) and sort them by 𝜆₁(𝐴)≤𝜆₂(𝐴)≤⋯≤𝜆_𝑛(𝐴). A symmetric real matrix 𝐴 is positive definite (semidefinite) if 𝑥^𝑇𝐴𝑥>0(≥0) for all nonzero 𝑥 and denoted by 𝐴>0(𝐴≥0). Finally, 𝐼 stands for the identity matrix and the dimensions of all vectors and matrices should be clear in the context.

Definition 2.1. A complex network with delayed and nondelayed coupling (2.2) is said to achieve asymptotic synchronization if 𝑥₁(𝑡)=𝑥₂(𝑡)=⋯=𝑥_𝑁(𝑡)=𝑠(𝑡),𝑡→+∞, (2.3) where 𝑠(𝑡) is a solution of the local dynamics of an isolated node satisfying 𝑠(𝑡̇)=𝑓(𝑠(𝑡)).

Definition 2.2. A matrix 𝐿=(𝑙_𝑖𝑗)^𝑁_𝑖,𝑗=1 is said to belong to the class 𝐴1, denoted by 𝐿∈𝐴1 if(1)𝑙_𝑖𝑗≤0, 𝑖≠𝑗, 𝑙_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑙_𝑖𝑗, 𝑖=1,2,…,𝑁,(2)L  is  irreducible.

If 𝐿∈𝐴1 is symmetrical, then we say that 𝐿 belongs to the class 𝐴2, denoted by 𝐿∈𝐴2.

Lemma 2.3 (see [26]). If 𝐿∈𝐴1, then rank(𝐿)=𝑁−1, that is, 0 is an eigenvalue of 𝐿 with multiplicity 1, and all the nonzero eigenvalues of 𝐿 have positive real part.

Lemma 2.4 (Wang and Chen [11]). If 𝐺=(𝑔_𝑖𝑗)_𝑁×𝑁 satisfies the above conditions, then there exists a unitary matrix Φ=(𝜙₁,…,𝜙_𝑁) such that 𝐺^𝑇𝜙_𝑘=𝜆_𝑘𝜙_𝑘,𝑘=1,2,…,𝑁, (2.4) where 𝜆_𝑖,𝑖=1,2,…,𝑁, are the eigenvalues of G.

Lemma 2.5 (Schur complement [22]). The linear matrix inequality (LMI) 𝑄(𝑥)𝑆(𝑥)𝑆(𝑥)^𝑇𝑅(𝑥)>0, (2.5) where 𝑄(𝑥) and 𝑅(𝑥) are symmetric matrices and 𝑆(𝑥) is a matrix with suitable dimensions is equivalent to one of the following conditions:(i)𝑄(𝑥)>0, 𝑅(𝑥)−𝑆(𝑥)^𝑇𝑄(𝑥)⁻¹𝑆(𝑥)>0;(ii)𝑅(𝑥)>0, 𝑄(𝑥)−𝑆(𝑥)𝑅(𝑥)⁻¹𝑆(𝑥)>0.

Lemma 2.6 ((the Lyapunov-Krasovskii stability theorem). (Kolmanovskii and Myshkis, Hale and Verduyn Lunel [16])). Consider the delayed differential equation 𝑥(𝑡)=𝑓̇(𝑡,𝑥(𝑡)), (2.6) where 𝑓∶𝑅×𝐶→𝑅^𝑛 is continuous and takes 𝑅× (bounded subsets of 𝐶) into bounded subsets of 𝑅^𝑛, and let 𝑢,𝑣,𝑤∶𝑅⁺→𝑅⁺ be continuous and strictly monotonically nondecreasing functions with 𝑢(𝑠), 𝑣(𝑠), 𝑤(𝑠) being positive for 𝑠>0 and 𝑢(0)=𝑣(0)=0. If there exists a continuous functional 𝑉∶𝑅×𝐶→𝑅 such that 𝑢𝑥()𝑥,𝑉̇(((𝑡𝑥(𝑡,(‖‖)≤𝑉𝑡,𝑥≤𝑣(‖‖)𝑡,𝑥𝑡,𝑥)))≤−𝑤(‖)‖) (2.7) where 𝑉̇ is the derivative of 𝑉 along the solutions of the above delayed differential equation, then the solution 𝑥=0 of this equation is uniformly asymptotically stable.

Remark 2.7. The functional 𝑉 is called a Lyapunov-Krasovskii functional.

Lemma 2.8 (Moon et al. [22]). Let 𝑎(⋅)∈𝑅^𝑛𝑎, 𝑏(⋅)∈𝑅^𝑛𝑏 and 𝑀(⋅)∈𝑅^{𝑛𝑎×𝑛𝑏} be defined on an interval Ω. Then, for any matrices 𝑋∈𝑅^{𝑛𝑎×𝑛𝑎}, 𝑌∈𝑅^{𝑛𝑎×𝑛𝑏}, and 𝑍∈𝑅^{𝑛𝑏×𝑛𝑏}, one has −2_Ω𝑎(𝑥)^𝑇𝑀𝑏(𝑥)𝑑𝑥≤_Ω𝑎(𝑥)𝑏(𝑥)^𝑇𝑌𝑋𝑌−𝑀_𝑇−𝑀^𝑇𝑍𝑎(𝑥)𝑏(𝑥)𝑑𝑥, (2.8) where 𝑌𝑋𝑌_𝑇𝑍≥0. (2.9)

Lemma 2.9. For all positive-definite matrices 𝑃 and vectors 𝑥 and 𝑦, one has −2𝑥^𝑇𝑦≤inf_𝑃>0𝑥^𝑇𝑃𝑥+𝑦^𝑇𝑃⁻¹𝑦. (2.10)

Lemma 2.10 (see [16]). Consider the delayed dynamical network (2.2). Let 0=𝜆₁>𝜆₂≥𝜆₃≥⋯≥𝜆_𝑁,0=𝜇₁>𝜇₂≥𝜇₃≥⋯≥𝜇_𝑁 (2.11) be the eigenvalues of the outer-coupling matrices 𝐵 and 𝐵′^, respectively. If the 𝑛-dimensional linear time-delayed and nontime delayed system 𝑤̇_𝑖(𝑡)=𝐽(𝑡)𝑤_𝑖(𝑡)+𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡)+𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖(𝑡−𝜏),𝑘=2,3,…,𝑁, (2.12) of 𝑁−1 differential equations is asymptotically stable about their zero solutions for some Jacobian matrix 𝐽(𝑡)∈𝑅^𝑛×𝑛 of 𝑓(𝑥(𝑡)) at 𝑠(𝑡), then the synchronized states (2.3) are asymptotically stable.

3. The Criteria for Asymptotic Synchronization

In this section, we derive the conditions for the asymptotic synchronization of time-delayed coupled dynamical networks when they are either time-dependent or time-independent.

3.1. Case 1: The Time Delay-Independent Stability Criterion

Theorem 3.1. Consider the general time delayed and non-time delayed complex dynamical network (2.2). If there exist two positive definite matrices 𝑃 and 𝑄>0 such that 𝐽(𝑡)^𝑇𝑃+𝑃𝐽(𝑡)+2𝑐₁𝜆_𝑖𝐴+𝑄𝑐₂𝜇_𝑁𝑐𝑃𝐴′₂𝜇_𝑁𝐴′𝑃−𝑄>0, (3.1) then the synchronization manifold (2.3) of network (2.2) can be asymptotically synchronized for all fixed time delay 𝜏>0.

Proof. For each fixed 𝑖=1,2,…,𝑁, choose the Lyapunov-Krasovskii functional 𝑉_𝑖(𝑡)=𝑤_𝑖(𝑡)^𝑇𝑃𝑤_𝑖(𝑡)+^𝑡_𝑡−𝜏𝑤_𝑖(𝑠)^𝑇𝑄𝑤_𝑖(𝑠)𝑑𝑠 (3.2) for some matrices 𝑃>0 and 𝑄>0 to be determined. Then the derivative of 𝑉_𝑖(𝑡) along the trajectories of (3.2) is 𝑑𝑉_𝑖(𝑡)=𝑤̇𝑑𝑡^𝑖(𝑡)^𝑇𝑃𝑤_𝑖(𝑡)+𝑤_𝑖(𝑡)^𝑇𝑃𝑤̇_𝑖(𝑡)+𝑤_𝑖(𝑡)^𝑇𝑄𝑤_𝑖(𝑡)−𝑤_𝑖(𝑡−𝜏)^𝑇𝑄𝑤_𝑖(𝑡−𝜏) (3.3) which, upon substitution of (2.12), gives 𝑉̇_𝑖(𝑡)=𝐽(𝑡)𝑤_𝑖(𝑡)+𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡)+𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖(𝑡−𝜏)^𝑇𝑃𝑤_𝑖(𝑡)+𝑤_𝑖(𝑡)^𝑇𝑃×𝐽(𝑡)𝑤_𝑖(𝑡)+𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡)+𝑐₂𝜇_𝑖𝐴′𝑤_𝑖(𝑡−𝜏)+𝑤_𝑖(𝑡)^𝑇𝑄𝑤_𝑖(𝑡)−𝑤_𝑖(𝑡−𝜏)^𝑇𝑄𝑤_𝑖(𝑡−𝜏)=𝑤_𝑖(𝑡)^𝑇𝐽(𝑡)^𝑇𝑤_𝑖(𝑡)+𝑤_𝑖(𝑡)^𝑇𝜆_𝑖𝐴^𝑇𝑃𝑤_𝑖(𝑡)+𝑤_𝑖(𝑡−𝜏)^𝑇𝜇_𝑖𝐴^′𝑃𝑤_𝑖(𝑡)+𝑤_𝑖(𝑡)^𝑇𝑃𝐽(𝑡)𝑤_𝑖(𝑡)+𝑤_𝑖(𝑡)^𝑇𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡)+𝑤_𝑖(𝑡)^𝑇𝑃𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖(𝑡−𝜏)+𝑤_𝑖(𝑡)^𝑇𝑄𝑤_𝑖(𝑡)−𝑤_𝑖(𝑡−𝜏)^𝑇𝑄𝑤_𝑖(𝑡−𝜏)=𝑤_𝑖(𝑡)^𝑇𝐽(𝑡)^𝑇𝑃+𝑃𝐽(𝑡)+2𝑐₁𝜆_𝑖𝐴+𝑄𝑤_𝑖(𝑡)+2𝑤_𝑖(𝑡)^𝑇𝑐₂𝜇_𝑖𝑃𝐴^′𝑤_𝑖(𝑡−𝜏)−𝑤_𝑖(𝑡−𝜏)^𝑇𝑄𝑤_𝑖(𝑡−𝜏). (3.4) Now, by using the inequality in Lemma 2.9, we have 2𝑤_𝑖(𝑡)^𝑇𝑐₂𝜇_𝑖𝑃𝐴^′𝑤𝑖(𝑡−𝜏)≤𝑤_𝑖(𝑡−𝜏)^𝑇𝑄𝑤_𝑖(𝑡−𝜏)+𝑤_𝑖(𝑡)^𝑇𝑐₂²𝜇_𝑖²𝑃𝐴^′𝑄𝐴^′𝑃𝑤_𝑖(𝑡), (3.5) which, upon substituting (3.5) into (3.2), gives 𝑉̇_𝑖(𝑡)≤𝑤_𝑖(𝑡)^𝑇𝐽(𝑡)^𝑇𝑃+𝑃𝐽(𝑡)+2𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖²𝑤_𝑖(𝑡)^𝑇𝑃𝐴^′𝑄𝐴^′𝐴𝑤(𝑡)+𝑄𝑤_𝑖(𝑡). (3.6) It therefore follows from the Schur complement (Lemma 2.5) and the linear matrix inequality (3.1) that 𝑉̇_𝑖(𝑡)<0 for all the 𝑁−1 equations in the general time delayed and non-time delayed system (2.12) and hence the system (2.12) is asymptotically synchronized by the Lyapunov-Krasovskii stability theorem. So, by Theorem 3.1, the synchronization manifold (2.3) of the network(2.2) is asymptotically synchronized. This completes the proof of the theorem.

The following corollaries follow immediately from the above theorem.

Corollary 3.2. Consider the general non-time delayed complex dynamical network 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+𝑐₁^𝑁_𝑗=1𝑏_𝑖𝑗𝐴𝑥_𝑗(𝑡). (3.7) If there exists a positive definite matrix 𝑃>0 such that 𝐽(𝑡)^𝑇𝑃+𝑐₁𝜆_𝑖𝐴𝑃<0, (3.8) then the synchronization manifold (2.3) of network (3.7) can be asymptotically synchronized.

Proof. From Lemma 2.10, we have 𝑤̇_𝑖(𝑡)=𝐽(𝑡)𝑤_𝑖(𝑡)+𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡) (3.9) and the result follows by choosing the Lyapunov functional 𝑉_𝑖(𝑡)=(1/2)𝑤_𝑖(𝑡)^𝑇𝑃𝑤_𝑖(𝑡).

Corollary 3.3. Consider the general time delayed complex dynamical network 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+𝑐₂^𝑁_𝑗=1𝑏^′_𝑖𝑗𝐴^′𝑥_𝑗(𝑡−𝜏). (3.10) If there exist two positive definite matrices 𝑃>0 and 𝑄>0 such that 𝐽(𝑡)^𝑇𝑃+𝑃𝐽(𝑡)+𝑄𝑐₂𝜇_𝑁𝑃𝐴^′𝑐₂𝜇_𝑁𝐴′𝑃−𝑄<0, (3.11) then the synchronization manifold (2.3) of network (3.10) can be asymptotically synchronized.

Remark 3.4. The results of [16] are obtainable as particular cases of Theorem 3.1.

Remark 3.5. The above analysis is applicable to a general system with arbitrary time delays. A simpler synchronization scheme, however, could be applied to systems with time delays that are already known and are small in value.

3.2. Case 2: The Criterion for Time Delay-Dependent Stability

Theorem 3.6. Consider the general time delayed and non-time delayed complex dynamical network (2.2) with a fixed time delay 𝜏∈(0,ℎ] for some small ℎ. If there exist three positive definite matrices 𝑃,𝑄,𝑍>0, such that (1,1)𝑃𝑐₂𝜇_𝑖𝐴′−𝑌+ℎ𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴^𝑇𝑍𝑐₂𝜇_𝑖𝐴^′𝑐₂𝜇_𝑖𝐴^′𝑇𝑃−𝑌^𝑇+ℎ𝑐₂𝜇_𝑖𝐴^′𝑇𝑍𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴ℎ𝑐₂𝜇_𝑖²𝐴^′𝑇𝑍𝐴′−𝑄<0 (3.12) with (1,1)=𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′𝑇𝑃+𝑃𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′+ℎ𝑋+𝑌^𝑇+𝑌+𝑄+ℎ𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴^𝑇𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴𝑌𝑋𝑌_𝑇𝑍≥0, (3.13) then the synchronization manifold (2.3) of network (2.2) can be asymptotically synchronized.

Proof. For each fixed 𝑖=1,2,…,𝑁, choose the Lyapunov-Krasovskii functional 𝑉_𝑖(𝑡)=𝑤_𝑖(𝑡)^𝑇𝑃𝑤_𝑖(𝑡)+^𝑡_𝑡−𝜏𝑤_𝑖(𝑠)^𝑇𝑄𝑤_𝑖(𝑠)+⁰_−𝜏^𝑡_𝑡+𝛽𝑤̇_𝑖(𝑠)^𝑇𝑍𝑤̇_𝑖(𝑠)𝑑𝑠𝑑𝛽 (3.14) for some matrices 𝑃,𝑄,𝑍>0 to be determined and let 𝑉₁=𝑤̇_𝑖(𝑡)^𝑇𝑃𝑤_𝑖(𝑡),𝑉₂=^𝑡_𝑡−𝜏𝑤_𝑖(𝑠)^𝑇𝑄𝑤_𝑖(𝑠),𝑉₃=⁰_−𝜏^𝑡_𝑡+𝛽𝑤̇_𝑖(𝑠)^𝑇𝑍𝑤̇_𝑖(𝑠)𝑑𝑠𝑑𝛽. (3.15) Then, 𝑉_𝑖(𝑡)=𝑉₁+𝑉₂+𝑉₃ and it follows from the Newton-Leibniz equation that 𝑤̇^𝑡_{𝑡−𝜏𝑖}(𝜉)𝑑𝜉=𝑤_𝑖(𝑡)−𝑤_𝑖(𝑡−𝜏) (3.16) so that (2.12) can be transformed into 𝑤̇_𝑖(𝑡)=𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖(𝑡)−𝑐₂𝜇_𝑖𝐴^{′∫𝑡𝑡−𝜏̇𝑤𝑖}(𝑠)𝑑𝑠. (3.17) Hence 𝑉₁=𝑤̇_𝑖(𝑡)^𝑇𝑃𝑤_𝑖(𝑡)+𝑤_𝑖(𝑡)^𝑇𝑃𝑤̇_𝑖(𝑡)=𝑤_𝑖(𝑡)^𝑇𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′𝑇𝑃+𝑃𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖(𝑡)−2𝑤_𝑖(𝑡)^𝑇𝑃𝑐₂𝜇_𝑖𝐴′^𝑡_𝑡−𝜏𝑤̇_𝑖(𝑠)𝑑𝑠 (3.18) and so, by Lemma 2.9, we have −2𝑤_𝑖(𝑡)^𝑇𝑃𝑐₂𝜇_𝑖𝐴′^𝑡_𝑡−𝜏𝑤̇_𝑖(𝑠)𝑑𝑠=−2^𝑡_𝑡−𝜏𝑤_𝑖(𝑡)^𝑇𝑃𝑐₂𝜇_𝑖𝐴′𝑤̇_𝑖(𝑠)𝑑𝑠≤^𝑡_𝑡−𝜏𝑤_𝑖𝑤̇(𝑡)_𝑖(𝑠)^𝑇𝑋𝑌−𝑃𝑐₂𝜇_𝑖𝑌𝐴′_𝑇−𝐴^′𝑇𝑐₂𝜇_𝑖𝑤𝑃𝑍^𝑖(𝑡)𝑤̇_𝑖(𝑠)=𝑑𝑥_{𝑡𝑡−𝜏}𝑤_𝑖(𝑡)^𝑇𝑋𝑤_𝑖(𝑡)𝑑𝑠+^𝑡_𝑡−𝜏𝑤̇_𝑖(𝑠)^𝑇𝑍𝑤̇_𝑖(𝑠)𝑑𝑠+2^𝑡_𝑡−𝜏𝑤_𝑖(𝑡)^𝑇𝑌−𝑃𝑐₂𝜇_𝑖𝐴^′𝑤̇_𝑖(𝑠)𝑑𝑠=𝜏𝑤_𝑖(𝑡)^𝑇𝑋𝑤_𝑖(𝑡)+2𝑤_𝑖(𝑡)^𝑇𝑌−𝑃𝑐₂𝜇_𝑖𝐴^′𝑡_𝑡−𝜏𝑤̇_𝑖(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏𝑤̇_𝑖(𝑠)^𝑇𝑍𝑤̇_𝑖(𝑠)=𝜏𝑤_𝑖(𝑡)^𝑇𝑋𝑤_𝑖(𝑡)+2𝑤_𝑖(𝑡)^𝑇𝑌−𝑃𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖(𝑡)−2𝑤_𝑖(𝑡)^𝑇𝑌−𝑃𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖()+𝑡−𝜏_{𝑡𝑡−𝜏}𝑤̇_𝑖(𝑠)𝑍𝑤̇_𝑖(𝑠) (3.19) and so 𝑉₁≤𝑤_𝑖(𝑡)^𝑇𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′𝑇𝑃+𝑃𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖(𝑡)+2𝑤_𝑖(𝑡)^𝑇𝑃𝑐₂𝜇_𝑖𝐴^′−𝑌𝑤_𝑖(𝑡−𝜏)+^𝑡_𝑡−𝜏𝑤̇_𝑖(𝑠)^𝑇𝑍𝑤_𝑖(𝑠)𝑑𝑠. (3.20) Similarly, we have 𝑉₂=𝑤_𝑖(𝑡)^𝑇𝑄𝑤_𝑖(𝑡)−𝑤_𝑖(𝑡−𝜏)^𝑇𝑄𝑤_𝑖(),𝑉𝑡−𝜏₃=𝜏𝑤̇_𝑖(𝑡)^𝑇𝑍𝑤̇_𝑖(𝑡)−^𝑡_𝑡−𝜏𝑤̇_𝑖(𝑠)^𝑇𝑍𝑤̇_𝑖(𝑠)𝑑𝑠≤ℎ𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡)+𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖(𝑡−𝜏)^𝑇×𝑍𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡)+𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖(𝑡−𝜏)−^𝑡_𝑡−𝜏𝑤̇_𝑖(𝑠)^𝑇𝑍𝑤̇_𝑖(𝑠)𝑑𝑠=ℎ𝑤_𝑖(𝑡)^𝑇𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴^𝑇𝑍𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡)+ℎ𝑤_𝑖(𝑡)^𝑇×𝐽(𝑡)𝑡+𝑐₁𝜆_𝑖𝐴^𝑇𝑍𝑐₂𝜇_𝑖𝐴′𝑤_𝑖(𝑡−𝜏)+ℎ𝑤_𝑖(𝑡−𝜏)^𝑇𝑐₂𝜇_𝑖𝐴^′𝑇𝑍^𝑇×𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡)+ℎ𝑤_𝑖(𝑡−𝜏)^𝑇𝜇_𝑖²𝐴^′𝑇𝑍𝐴^′𝑤_𝑖(𝑡−𝜏)−^𝑡_𝑡−𝜏𝑤̇_𝑖(𝑠)^𝑇𝑍𝑤̇_𝑖(𝑠)𝑑𝑠=ℎ𝑤_𝑖(𝑡)^𝑇𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴^𝑇𝑍𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡)+2ℎ𝑤_𝑖(𝑡)^𝑇×𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴^𝑇𝑍𝑐₂𝜇_𝑖𝐴′𝑤_𝑖(𝑡−𝜏)+ℎ𝑤_𝑖(𝑡−𝜏)^𝑇𝜇_𝑖²𝐴^′𝑇𝑍𝐴^′𝑤_𝑖()−𝑡−𝜏_{𝑡𝑡−𝜏}𝑤̇_𝑖(𝑠)^𝑇𝑍𝑤̇_𝑖(𝑠)𝑑𝑠 (3.21) and so 𝑉̇_𝑖(𝑡)=𝑉₁+𝑉₂+𝑉₃≤𝑤_𝑖(𝑡)^𝑇𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′𝑇𝑃+𝑃𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′+ℎ𝑋+𝑌^𝑇+𝑌+𝑄+ℎ𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴^𝑇𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴𝑤_𝑖(𝑡)+𝑤_𝑖(𝑡−𝜏)^𝑇ℎ𝜇_𝑖²𝐴^′𝑇𝑍𝐴^′−𝑄𝑤_𝑖(𝑡−𝜏)+2𝑤_𝑖(𝑡)^𝑇𝑃𝑐₂𝜇_𝑖𝐴^′−𝑌+ℎ𝐽(𝑡)+𝑐₁𝜆_𝑖^𝑇𝑍𝑐₂𝜇_𝑖𝐴^′𝑤_𝑖(𝑡−𝜏). (3.22)
Finally, we have 𝑉̇_𝑖(𝑡)≤𝑤_𝑖(𝑡)𝑤_𝑖(𝑡−𝜏)^𝑇()()1,1)(1,22,1ℎ𝑐₂𝜇²𝐴^′𝑇𝑍𝐴^′_𝑄𝑤_𝑖(𝑡)𝑤_𝑖(),𝑡−𝜏 (3.23) where (1,1)=𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′𝑇𝑃+𝑃𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴+𝑐₂𝜇_𝑖𝐴^′+ℎ𝑋+𝑌^𝑇+𝑌+𝑄+ℎ𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴^𝑇𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴(1,2)=𝑃𝑐₂𝜇_𝑖𝐴^′−𝑌+ℎ𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴^𝑇𝑍𝑐₂𝜇_𝑖𝐴^′(2,1)=𝑐₂𝜇_𝑖𝐴^′𝑇𝑃−𝑌^𝑇+ℎ𝑐₂𝜇_𝑖𝐴^′𝑇𝑍𝐽(𝑡)+𝑐₁𝜆_𝑖𝐴𝑌𝑋𝑌_𝑇𝑍≥0. (3.24) It now follows from Lemma 2.5 that the conditions of the theorem are equivalent to 𝑉̇_𝑖(𝑡)<0 and that by the Lyapunov-Krasovskii Stability Theorem all the nodes of the system (2.12) are asymptotically stable when (3.12) and (3.13) hold for 𝑖=1,2,…,𝑁. This completes the proof of Theorem 3.6.

Corollary 3.7. Consider the general time delayed complex dynamical network (3.10) with a fixed time delay 𝜏∈(0,ℎ]𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+𝑐₂^𝑁_𝑗=1𝑏^′_𝑖𝑗𝐴^′𝑥_𝑗(𝑡−𝜏) (3.25) for some ℎ<+∞. If there exist two positive definite matrices, 𝑃,𝑄>0, 𝑋, 𝑌, and 𝑍 such that (1,1)𝑐₂𝜇_𝑖𝑃𝐴^′−𝑌+ℎ𝐽(𝑡)^𝑇𝑍𝑐₂𝜇_𝑖𝐴^′𝑐₂𝜇_𝐴^′𝑇𝑃−𝑌^𝑇+ℎ𝑐₂𝜇_𝑖𝐴^′𝑇𝑍𝐽(𝑡)ℎ𝑐₂²𝜇_𝑖²𝐴^′𝑇𝑍𝐴^′−𝑄<0, (3.26) where (1,1)=𝑃𝐽(𝑡)+𝐽(𝑡)^𝑇+ℎ𝑋+𝑌^𝑇+𝑌+𝑄+ℎ𝐽(𝑡^𝑇𝑍𝐽(𝑡)), then the synchronization manifold (2.3) of network (3.10) is asymptotic synchronization.

Remark 3.8. The proof can be found in [16]. Those are the two results of general complex dynamical network with fixed time-invariant delay 𝜏∈(0,ℎ] for some ℎ<+∞; the conclusions are less conservative than the time-independent delay. The delay-dependent stability is another method applying to the delayed system. And it could provide a useful and meaningful upper bound of the delay ℎ, which could ensure the delayed system achieves asymptotic synchronization only if the time delay is less than ℎ.

4. Numerical Simulations

The above criteracould be applied to networks with different topologies and different size. We put two examples to illustrate the validity of the theories.

Example 4.1. We use a three-dimensional stable nonlinear system as an example to illustrate the main results, Theorem 3.1, of our paper; this is the time delay-independent situation. The model could be described as follows: 𝑥̇_𝑖1𝑥̇_𝑖2𝑥̇_𝑖3=−𝑥_𝑖1+𝑥²_𝑖2−2𝑥_𝑖2−3𝑥_𝑖3+𝑥_𝑖2𝑥_𝑖3,𝑖=1,2,3. (4.1) The solution of the 3-dimensional stable nonlinear system equations can be written as 𝑥_𝑖1=𝑐₁𝑒^−𝑡−𝑐𝑒^−4𝑡,𝑥_𝑖2=𝑐₂𝑒^−2𝑡,𝑥_𝑖3=𝑐₃𝑒^−3𝑡−𝑐₂𝑒^2𝑡2, (4.2) which is asymptotically stable at the equilibrium point of the system 𝑠(𝑡)=0, where 𝑐=−𝑐²₂/3 and 𝑐₁, 𝑐₂, 𝑐₃ are all constants. It is easy to see that the Jacobian matrix is 𝐽=diag{−1,−2,−3}. We assume the inner-coupling matrices 𝐴, 𝐴^′ are all identity matrices, namely, 𝐴=𝐴′=diag{1,1,1}, and the outer coupling configuration matrices 𝐵=𝐵^′=−21011−21001−211012. (4.3) The eigenvalues of the coupling matrices are 𝜆(𝐵)=𝜆(𝐵^′)={0,−1.5,−1.5}. We choose the coupling strength 𝑐₁=0.5, 𝑐₂=1. By using Theorem 3.1 and the LMI Toolbox in MATLAB, we obtained the following common two positive-definite matrices: 𝑃=diag{1.1204,12.3091,10.165},𝑄=diag{2.3710,22.5713,26.0849}. (4.4) According to the conditions in Theorem 3.1, we know the synchronized state 𝑠(𝑡) is global asymptotically stable for any fixed delay. The quantity 𝐸(𝑡)=∑^𝑁_𝑖=1𝑥_𝑖(𝑡)−𝑠(𝑡)²𝑁 (4.5) is used to measure the quality of the synchronization process. We plot the evolution of 𝐸(𝑡) in the upper part in Figure 1. For the time delay here we choose 𝜏=0.1. The lower subplot indicates the synchronization results of the network.

(a)

(b)

Figure 1 

Synchronization evolution 𝐸(𝑡) for the delay-independent network with 𝑐1=0.5, 𝑐2=1, and 𝜏=0.1.

Example 4.2. We use a 4-nodes networks model as another example to illustrate the Theorem 3.6; this is the time delay-dependent situation. The model could be described as follows: 𝑥̇_𝑖1𝑥̇_𝑖2𝑥̇_𝑖3=−𝑥_𝑖1−2𝑥_𝑖2+𝑥²_𝑖3−3𝑥_𝑖3+𝑥_𝑖2𝑥_𝑖3,𝑖=1,2,3,4. (4.6) We choose the same coupling strength 𝑐₁=0.5, 𝑐₂=1; the eigenvalues of the coupling matrices are 𝜆(𝐵)=𝜆(𝐵^′)={0,−2,−2,−4}. By using Theorem 3.6 and the LMI Toolbox in MATLAB, we obtained the following matrices: {}{},{}{},{}.𝑃=diag1.4078,1.4057,1.4054,𝑄=1.4157,1.4157,1.4157𝑍=−1.4218,−1.4303,−1.4367,𝑋=18.3024,19.6808,21.0815𝑌=0.0221,0.0208,0.0196 (4.7)

By using Theorem 3.6 in this paper, it is found that the maximum delay bound for the complex dynamical network to form asymptotic synchronization is ℎ=1.𝐸(𝑡) are defined the same as in the example. We plot the evolution of 𝐸(𝑡) in upper part in Figure 2. The lower subplot indicates the synchronization results of the network. It can be seen from the figures that the network in this example can achieve asymptotic synchronization.

(a)

(b)

Figure 2 

Synchronization evolution 𝐸(𝑡) for the delay-dependent network with 𝑐1=0.5, 𝑐2=1, and 𝜏=0.1.

5. Conclusion

This paper considered a class of complex networks with both time delayed and non-time delayed coupling. We derived, respectively, a sufficient criterion for time delay-dependent and time delay-independent asymptotic synchronization which are more general than those obtained in previous works. These asymptotic synchronization results were obtained by using the Lyapunov-Krasovskii stability theorem and the linear matrix inequality. Two simple examples were also used to validate the theoretical analysis.

Acknowledgments

The authors thank the referees and the editor for their valuable comments on this paper. This work is supported by the National Natural Science Foundation of China (Grant no. 61273220), Guandong Education University Industry Cooperation Projects (Grant no. 2009B090300355), and the Shenzhen Basic Research Project (JC201006010743A, JC200903120040A).