Adaptive-Impulsive Control of the Projective Synchronization in Drive-Response Complex Dynamical Networks with Time-Varying Coupling

Abstract

This paper investigates the projective synchronization (PS) of drive-response time-varying coupling complex dynamical networks with time delay via an adaptive-impulsive controlling method, in which the weights of links are time varying. Based on the stability analysis of impulsive control system, sufficient conditions for the PS are derived, and a hybrid controller, that is, an adaptive feedback controller with impulsive control effects, is designed. Numerical simulations are performed to verify the correctness and effectiveness of theoretical result.

1. Introduction

Complex network models widely exist in the real world including the spread of infectious diseases, the World Wide Web, food webs, various wireless communication networks, metabolic networks, biological neural works, and scientific citation webs. Since the discovery of small-world effect [1] and scale-free feature [2] of complex networks, many scientists and engineers from various disciplines, such as mathematics, physics, biology, engineering, have paid increasing attention to the studies of complex networks.

Synchronization, one of the typical collective behaviours of complex dynamical networks, has received rapidly increasing attention from different fields in recent years [3–24]. And different control schemes including adaptive control [10–12], pinning control [13–16], and impulsive control [17–22] have been used to study the above problem. Recently, some authors presented hybrid control strategy to investigate the synchronization of complex networks [23–26]. Compared to the conventional control method, hybrid control method is more effective to the networks with evolutionary features [23]. In many practical situations, some complex networks may change suddenly and sharply and thus the modes switch simultaneously. This kind of networks can be found in many evolutionary processes, such as optimal control model in economics, bursting rhythm model in pathology, mobile communication networks, and social networks.

The typical configuration of chaotic synchronization consists of drive and response systems and has been widely investigated. The PS, a new chaos synchronization phenomenon, was first studied in two coupled partially linear systems by Mainieri and Rehacek in [27]. Later, the PS between two complex networks has obtained much more attention [28–32]. In [28], Guo et al. studied the PS in drive-response networks via impulsive control. In [29, 30], authors discussed the PS of a drive-response dynamical network model without the time delay. Sun et al. studied the PS in drive-response dynamical networks of partially linear systems with time-varying coupling delay in [31]. Xu et al. [32] investigated the PS of a class of the drive-response dynamical networks with coupling delays. However, the time-varying coupling was not considered in the above studies. To simulate more realistic networks, time-varying coupling [33–35] should be taken into account. Motivated by the above discussions, this paper will focus on the adaptive-impulsive PS problem of drive-response time-varying coupling dynamical networks with time delay. Based on the stability theory of the impulsive differential equation, some criteria for the PS are derived. Furthermore, analytical results show that the drive-response networks can realize the PS.

The rest of this paper is organized as follows. In Section 2, we present the model of drive-response time-varying coupling dynamical networks with time delay and a hybrid controller is designed. In Section 3, synchronization criteria for PS are derived. Numerical simulations are shown in Section 4. The conclusion is finally drawn in Section 5.

2. Model Description and Preliminaries

Consider the following drive-response time-varying coupling dynamical network model with time delay:𝑢̇_𝑑(𝑡)=𝑀(𝑧)⋅𝑢_𝑑(𝑡),𝑧̇(𝑡)=𝑓𝑢_𝑑(𝑡),𝑧(𝑡),𝑢̇_𝑟𝑖(𝑡)=𝑀(𝑧)⋅𝑢_𝑟𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑐_𝑖𝑗(𝑡)Γ(𝑡)𝑢_𝑟𝑗(𝑡−𝜏),𝑖=1,2,…,𝑁, (2.1) where the drive system and the response network systems are linked through the variable 𝑧(𝑡)∈𝑅¹, 𝑢_𝑑(𝑡)=(𝑢¹_𝑑(𝑡),𝑢²_𝑑(𝑡),…,𝑢^𝑛_𝑑(𝑡))^𝑇∈𝑅^𝑛,  𝑢_𝑟𝑖(𝑡)=(𝑢¹_𝑟𝑖(𝑡),𝑢²_𝑟𝑖(𝑡),…,𝑢^𝑛_𝑟𝑖(𝑡))^𝑇∈𝑅^𝑛 and the 𝑑 and 𝑟 stand for the drive system and response system, respectively. The constant 𝑐>0 is the coupling strength to be adjusted, 𝜏≥0 is the time-delay. 𝑀(𝑧)∈𝑅^𝑛×𝑛 is a matrix which depends on the variable 𝑧(𝑡). Γ(𝑡)∈𝑅^𝑛×𝑛 is the time-varying inner-coupling link matrix at time 𝑡. 𝐶(𝑡)=(𝑐_𝑖𝑗(𝑡))_𝑁×𝑁 is the outer-coupling configuration matrix, in which 𝑐_𝑖𝑗(𝑡)≠0 if there is a link from node 𝑖 to node 𝑗(𝑖≠𝑗), and 𝑐_𝑖𝑗(𝑡)=0(𝑖≠𝑗) otherwise, the diagonal elements of matrix 𝐶(𝑡) are given by 𝑐_𝑖𝑖(𝑡)=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑐_𝑖𝑗(𝑡),𝑖=1,2,…,𝑁. (2.2)

If there exists a constant 𝛼(𝛼≠0) such that lim_𝑡→∞=‖𝑒_𝑖(𝑡)‖=‖𝑢_𝑟𝑖(𝑡)−𝛼𝑢_𝑑(𝑡)‖=0 for 𝑖=1,2,…,𝑁, then the PS of network (2.1) is achieved. 𝛼 is a desired scaling factor. The aim of this paper is to discuss the adaptive-impulsive PS in the drive-response time-varying coupling complex dynamical network with time delay. We choose the linear impulsive controller 𝐵_𝑖𝑘 which is a 𝑛×𝑛 constant matrix, and design an adaptive feedback controller 𝑈_𝑖. Therefore, the network (2.1) can be rewritten as the following impulsive differential equations:𝑢̇_𝑑(𝑡)=𝑀(𝑧)⋅𝑢_𝑑(𝑡),𝑧̇(𝑡)=𝑓𝑢_𝑑(𝑡),𝑧(𝑡),𝑢̇_𝑟𝑖(𝑡)=𝑀(𝑧)⋅𝑢_𝑟𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑐_𝑖𝑗(𝑡)Γ(𝑡)𝑢_𝑟𝑗(𝑡−𝜏)+𝑈_𝑖,𝑡≠𝑡_𝑘,Δ𝑢_𝑟𝑖=𝑢_𝑟𝑖𝑡⁺_𝑘−𝑢_𝑟𝑖𝑡⁻_𝑘=𝐵_𝑖𝑘𝑢_𝑟𝑖−𝛼𝑢_𝑑,𝑡=𝑡_𝑘, (2.3) where 𝑢_𝑟𝑖(𝑡⁺_𝑘)=lim_𝑡→𝑡⁺_𝑘𝑢_𝑟𝑖(𝑡),𝑢_𝑟𝑖(𝑡⁻_𝑘)=lim_𝑡→𝑡⁻_𝑘𝑢_𝑟𝑖(𝑡). Moreover, any solution of (2.3) is left continuous at each 𝑡_𝑘, that is, 𝑢_𝑟𝑖(𝑡⁻_𝑘)=𝑢_𝑟𝑖(𝑡_𝑘).

Letting the PS error 𝑒_𝑖(𝑡)=𝑢_𝑟𝑖(𝑡)−𝛼𝑢_𝑑(𝑡), the adaptive controllers 𝑈_𝑖 and updating laws are designed as follows:𝑈_𝑖=−𝑑_𝑖𝑒_𝑖(𝑡),𝑑_𝑖̇=𝑘_𝑖𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)=𝑘_𝑖𝑒_𝑖(𝑡)²,𝑘_𝑖>0. (2.4)

The system (2.3) is said to be synchronized if lim_𝑡→∞‖𝑒_𝑖(𝑡)‖=0.

Under the adaptive-impulsive control, the error dynamical network is characterized by𝑒_𝑖̇(𝑡)=𝑀(𝑧)⋅𝑒_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑐_𝑖𝑗(𝑡)Γ(𝑡)𝑒_𝑗(𝑡−𝜏)+𝑈_𝑖,𝑡≠𝑡_𝑘,𝑧̇(𝑡)=𝑓𝑢_𝑑(𝑡),𝑧(𝑡),Δ𝑒_𝑖=𝐵_𝑖𝑘𝑒_𝑖,𝑡=𝑡_𝑘,𝑘=1,2,…. (2.5) Before proceeding, we give some necessary assumptions and lemmas to derive the main results of the paper.

Lemma 2.1. The matrix inequality 2𝑥^𝑇𝑦≤𝑥^𝑇𝑄𝑥+𝑦^𝑇𝑄⁻¹𝑦 holds, for any vectors 𝑥,𝑦∈𝑅^𝑛 and a positive-definite matrix 𝑄∈𝑅^𝑛×𝑛.

Assumption 2.2. Suppose there exists a positive constant 𝜉, such that ‖𝑒_𝑖(𝑡−𝜏)‖²≤𝜉‖𝑒_𝑖(𝑡)‖² holds.

3. PS Analysis

In this section, we will make drive-response time-varying coupling complex dynamical networks achieve the PS by using the adaptive-impulsive controlling method.

Theorem 3.1. For given synchronization scaling factor 𝛼, the PS in drive-response dynamical networks will occur if the following conditions hold:(i)If there exist two constants 𝑎,𝜆 satisfying 0≤𝑎<𝜆 and 𝛿(𝑡)≤−𝜆<0 such thatln𝛽_𝑘−𝑎𝑡_𝑘−𝑡_𝑘−1≤0,𝑘=1,2,…, (3.1) then the trivial solution of error system (2.5) is global asymptotically stable, which implies drive-response networks achieve the projective synchronization under the adaptive-impulsive control.(ii)If 𝛿(𝑡)≥0 and there exists a constant 𝑎≥1 such thatln𝑎𝛽_𝑘+^𝑡𝑘+1_𝑡𝑘𝛿(𝑠)𝑑𝑠≤0,𝑘=1,2,…, (3.2) then 𝑎=1 implies that the trivial solution of error system (2.5) is stable and 𝑎>1 implies that the trivial solution of error system (2.5) is global asymptotically stable where 𝛿(𝑡)=max_{1≤𝑖≤𝑁}1𝜆_min(𝑃)sup𝜆_max𝑃𝑀(𝑧)+𝑀^𝑇(𝑧)𝑃−2𝑑^∗𝑃++𝑁𝜉,_𝑁𝑗=1𝑐𝑐_𝑖𝑗(𝑡)²‖𝑃Γ(𝑡)‖²,max_𝑘𝐼+𝐵_𝑖𝑘²=𝜌_𝑘<1,𝛽_𝑘≥𝜌_𝑘‖𝑃‖𝜆_min(𝑃), (3.3)𝑃 is a positive-definite matrix. 𝑑^∗ is the minimum value of the initial feedback strength 𝑑_𝑖0(𝑑_𝑖0≤𝑑_𝑖),𝑖=1,2,…,𝑁.

Proof. Consider the following Lyapunov functional: 𝑉(𝑡)=12^𝑁𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡). (3.4) For 𝑡≠𝑡_𝑘, the derivative of 𝑉(𝑡) along the trajectories of (2.5) is 𝑉̇(𝑡)=12^𝑁𝑖=1𝑒^𝑇_𝑖̇(𝑡)𝑃𝑒_𝑖(𝑡)+12^𝑁𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖̇(𝑡)=12^𝑁𝑖=1𝑀(𝑧)𝑒_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑐_𝑖𝑗(𝑡)Γ(𝑡)𝑒_𝑗(𝑡−𝜏)−𝑑_𝑖𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)+12^𝑁𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃𝑀(𝑧)𝑒_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑐_𝑖𝑗(𝑡)Γ(𝑡)𝑒_𝑗(𝑡−𝜏)−𝑑_𝑖𝑒_𝑖(𝑡)=12^𝑁𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃𝑀(𝑧)+𝑀^𝑇(𝑧)𝑃−2𝑑_𝑖𝑃𝑒_𝑖(𝑡)+12^{𝑁𝑖=1𝑁𝑗=1}𝑒^𝑇_𝑗(𝑡−𝜏)𝑐𝑐_𝑖𝑗(𝑡)Γ^𝑇(𝑡)𝑃𝑒_𝑖(𝑡)+12^{𝑁𝑖=1𝑁𝑗=1}𝑒^𝑇_𝑖(𝑡)𝑐𝑐_𝑖𝑗(𝑡)𝑃Γ(𝑡)𝑒_𝑗()≤1𝑡−𝜏2^𝑁𝑖=1sup𝜆_max𝑃𝑀(𝑧)+𝑀^𝑇(𝑧)𝑃−2𝑑_𝑖𝑃𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+12^{𝑁𝑖=1𝑁𝑗=1}𝑒^𝑇_𝑖(𝑡)𝑐𝑐_𝑖𝑗(𝑡)²𝑃Γ(𝑡)Γ^𝑇(𝑡)𝑃^𝑇𝑒_𝑖(𝑡)+12^{𝑁𝑖=1𝑁𝑗=1}𝑒^𝑇_𝑗(𝑡−𝜏)𝑒_𝑗()≤1𝑡−𝜏2^𝑁𝑖=1sup𝜆_max𝑃𝑀(𝑧)+𝑀^𝑇(𝑧)𝑃−2𝑑_𝑖0𝑃𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)+12^{𝑁𝑖=1𝑁𝑗=1}𝑒^𝑇_𝑖(𝑡)𝑐𝑐_𝑖𝑗(𝑡)²𝑃Γ(𝑡)Γ^𝑇(𝑡)𝑃^𝑇𝑒_𝑖(𝑡)+𝑁2^𝑁𝑖=1𝑒^𝑇_𝑖(𝑡−𝜏)𝑒_𝑖(𝑡−𝜏). (3.5) From Assumption 2.2, we get 𝑁2^𝑁𝑖=1𝑒^𝑇_𝑖(𝑡−𝜏)𝑒_𝑖(𝑡−𝜏)≤𝑁𝜉2^𝑁𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡). (3.6) Thus we have 𝑉̇(𝑡)≤12^𝑁𝑖=1sup𝜆_max𝑃𝑀(𝑧)+𝑀^𝑇(𝑧)𝑃−2𝑑^∗𝑃+𝑁𝜉+^𝑁_𝑗=1𝑐𝑐_𝑖𝑗(𝑡)²‖𝑃Γ(𝑡)‖²𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)≤max_{1≤𝑖≤𝑁}1𝜆_min(𝑃)sup𝜆_max𝑃𝑀(𝑧)+𝑀^𝑇(𝑧)𝑃−2𝑑^∗𝑃+𝑁𝜉+^𝑁_𝑗=1𝑐𝑐_𝑖𝑗(𝑡)²‖𝑃Γ(𝑡)‖^2𝑁_𝑖=112𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡)=𝛿(𝑡)^𝑁_𝑖=112𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡)=𝛿(𝑡)𝑉(𝑡). (3.7) This implies that 𝑉(𝑡)≤𝑉𝑡⁺_𝑘−1exp^𝑡_𝑡𝑘−1𝛿(𝑠)𝑑𝑠,𝑘=1,2,…,𝑡∈𝑡_𝑘−1,𝑡_𝑘. (3.8) When 𝑡=𝑡_𝑘, we have 𝑉𝑡⁺_𝑘=12^𝑁𝑖=1𝑒^𝑇_𝑖(𝑡)𝐼₂+𝐵_𝑖𝑘^𝑇𝑃𝐼₂+𝐵_𝑖𝑘𝑒_𝑖(𝑡)≤𝜌_𝑘||𝑃||𝜆_min(𝑃)𝑉𝑡_𝑘≤𝛽_𝑘𝑉𝑡_𝑘,𝑘=1,2,…. (3.9) When 𝑘=1 in inequality (3.8), then for any 𝑡∈(𝑡₀,𝑡₁], 𝑉(𝑡)≤𝑉𝑡⁺₀exp^𝑡_𝑡0𝛿(𝑠)𝑑𝑠. (3.10) This leads to 𝑉𝑡₁≤𝑉𝑡⁺₀exp^𝑡1_𝑡0𝛿(𝑠)𝑑𝑠. (3.11) Also from (3.9) we have 𝑉𝑡⁺₁≤𝛽₁𝑉𝑡₁≤𝛽₁𝑉𝑡⁺₀exp^𝑡1_𝑡0𝛿(𝑠)𝑑𝑠. (3.12) In the same way for 𝑡∈(𝑡₁,𝑡₂], we have 𝑉(𝑡)≤𝑉𝑡⁺₁exp^𝑡_𝑡1𝛿(𝑠)𝑑𝑠≤𝛽₁𝑉𝑡⁺₀exp^𝑡1_𝑡0𝛿(𝑠)𝑑𝑠exp^𝑡_𝑡1𝛿(𝑠)𝑑𝑠,=𝛽₁𝑉𝑡⁺₀exp^𝑡_𝑡0𝛿(𝑠)𝑑𝑠. (3.13) In general for any 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], one finds that 𝑉(𝑡)≤𝛽₁𝛽₂⋯𝛽_𝑘𝑉𝑡⁺₀exp^𝑡_𝑡0𝛿(𝑠)𝑑𝑠. (3.14)(i)If there exist two constants 𝑎,𝜆 satisfying 0≤𝑎<𝜆,𝛿(𝑡)≤−𝜆<0 and (3.1), we have 𝛽_𝑘≤𝑒^{𝑎(𝑡𝑘−𝑡𝑘−1)},𝑘=1,2,…. (3.15) From (3.14), one finds that 𝑉(𝑡)≤𝛽₁𝛽₂⋯𝛽_𝑘𝑉𝑡⁺₀𝑒^{−𝜆(𝑡−𝑡0)}=𝛽₁𝛽₂⋯𝛽_𝑘𝑉𝑡⁺₀𝑒^{−𝑎(𝑡−𝑡0)−(𝜆−𝑎)(𝑡−𝑡0)}≤𝛽₁𝛽₂⋯𝛽_𝑘𝑉𝑡⁺₀𝑒^{−𝑎(𝑡𝑘−𝑡0)−(𝜆−𝑎)(𝑡−𝑡0)}≤𝑉𝑡⁺₀𝑒^{𝑎(𝑡1−𝑡0)}𝑒^{𝑎(𝑡2−𝑡1)}⋯𝑒^{𝑎(𝑡𝑘−𝑡𝑘−1)}𝑒^{−𝑎(𝑡𝑘−𝑡0)−(𝜆−𝑎)(𝑡−𝑡0)}=𝑉𝑡⁺₀𝑒^{−(𝜆−𝑎)(𝑡−𝑡0)},𝑡∈𝑡_𝑘,𝑡_𝑘+1. (3.16) Then the trival solution of error system (2.5) is global asymptotically stable, which implies drive-response time-varying coupling dynamical networks (2.3) achieve the PS under the adaptive-impulsive control.(ii)If 𝛿(𝑡)≥0 and there exists a constant 𝑎≥1 satisfying (3.2), we have𝛽_𝑘≤1𝑎exp^𝑡𝑘_𝑡𝑘+1𝛿(𝑠)𝑑𝑠. (3.17) From (3.14), we have 𝑉(𝑡)≤𝛽₁𝛽₂⋯𝛽_𝑘𝑉𝑡⁺₀exp^𝑡_𝑡0𝛿(𝑠)𝑑𝑠≤𝛽₁𝛽₂⋯𝛽_𝑘𝑉𝑡⁺₀exp^𝑡𝑘+1_𝑡0𝛿(𝑠)𝑡𝑑𝑠≤𝑉⁺₀1𝑎exp^𝑡1_𝑡2𝛿(𝑠)1𝑑𝑠𝑎exp^𝑡2_𝑡3𝛿(𝑠)1𝑑𝑠⋯𝑎exp^𝑡𝑘_𝑡𝑘+1𝛿(𝑠)𝑑𝑠exp^𝑡𝑘+1_𝑡0𝛿(𝑠)𝑡𝑑𝑠=𝑉⁺₀1𝑎^𝑘exp^𝑡1_𝑡0𝛿(𝑠)𝑑𝑠,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (3.18)
This implies that error system (2.5) is global asymptotically stable about zero. Therefore, the PS of the drive-response dynamical networks (2.3) is achieved. The proof is completed.

Remark 3.2. The conditions given by Theorem 3.1 do not require the network configuration to be symmetric and irreducible, which can be applied to more real-world dynamical networks. Moreover, Theorem 3.1 does not impose any bound on the time-delay constant 𝜏. Thus, our synchronization results are a time-delay-independent stability criteria.

4. Numerical Simulation

In this section, to verify and demonstrate the effectiveness of the proposed methods, we consider the unified chaotic system as the drive system. It is well known that the unified chaotic system is described by 𝑥̇₁=(25𝜃+10)𝑥₂−𝑥₁,𝑥̇₂=(28−35𝜃)𝑥₁−𝑥₁𝑧+(29𝜃−1)𝑥₂,𝑧̇=𝑥₁𝑥₂−8+𝜃3𝑧, (4.1) where 𝑀(𝑧)=^{−(25𝜃+10)25𝜃+10}_{28−35𝜃−𝑧29𝜃−1}, 𝑓(𝑥,𝑧)=𝑥₁𝑥₂−(8+𝜃/3)𝑧, 𝑥=(𝑥₁,𝑥₂)^𝑇, 𝜃∈[0,1]. System (4.1) especially is always chaotic in the whole interval 𝜃∈[0,1].

The drive-response time-varying coupling dynamical networks with time delay are described as follows:(),()()̇𝑥̇=25𝜃+10𝑦−𝑥𝑦̇=28−35𝜃−𝑧𝑥+29𝜃−1𝑦,𝑧=𝑥𝑦−8+𝜃3𝑧,𝑥̇_𝑖=(25𝜃+10)𝑦_𝑖−𝑥_𝑖+𝑐⁵_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥_𝑗(𝑡−𝜏)+𝑢_𝑖1,𝑦_𝑖̇=(28−35𝜃−𝑧)𝑥_𝑖+(29𝜃−1)𝑦_𝑖+𝑐⁵_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥_𝑗(𝑡−𝜏)+𝑢_𝑖2. (4.2) Choose the time-varying coupling configuration matrix:𝐶(𝑡)=−2sin𝑡−102sin𝑡1sin𝑡−sin𝑡−cos𝑡00cos𝑡0cos𝑡0−cos𝑡00sin𝑡cos𝑡1−sin𝑡cos𝑡−1−12sin𝑡0−12−2sin𝑡. (4.3)

For simplicity, in the numerical simulations, we assume 𝑃=Γ(𝑡)=𝐼₂,𝑐=0.1,𝑘_𝑖=1,  𝜉=1,𝜃=0.03, the time-delay 𝜏=0.07, and the impulsive interval 𝑡_𝑘+1−𝑡_𝑘=0.08. The initial values of the drive systems and the response systems are chosen as −2, 1, 2, 3.8, −2, 4.2, −6, 4.6, −10, 5, −14, 5.4, −18, 𝐵_𝑖𝑘=diag{−0.79,−0.79},𝛽_𝑘=0.0441>0,𝑑_𝑖0=5. After calculations, we get 𝛿(𝑡)=max_{1≤𝑖≤𝑁}{(1/𝜆_min(𝑃))sup[𝜆_max(𝑃𝑀(𝑧)+𝑀^𝑇(𝑧)𝑃−2𝑑^∗𝑃)]+𝑁𝜉+∑^𝑁_𝑗=1(𝑐𝑐_𝑖𝑗(𝑡))²‖𝑃Γ(𝑡)‖²]=31.667>0. Let 𝑎=1.02. Then ln(𝑎𝛽_𝑘)+∫^𝑡𝑘−1_𝑡𝑘𝛿(𝑠)𝑑𝑠=−0.5681<0, according to Theorem 3.1, and the trivial solution of error system (2.5) is global asymptotically stable. Therefore, the adaptive-impulsive PS of the drive-response dynamical networks (2.3) is achieved. Figure 1 displays the trajectories of PS in the 𝑥−𝑦 plane when  𝛼=0.5. The synchronization errors 𝑒_𝑖1(𝑡)=𝑥_𝑖1−0.5𝑥₁ and 𝑒_𝑖2(𝑡)=𝑥_𝑖2−0.5𝑥₂  (𝑖=1,2,…,5) are shown, respectively, in Figure 2. Figure 3 show the evolution of the feedback strength 𝑑_𝑖. The numerical results show that the adaptive-impulsive controlling scheme for the drive-response complex network is effective in Theorem 3.1.

Figure 1 

The trajectories of PS in the 𝑥-𝑦 plane.

(a)

(b)

Figure 2 

PS errors under the adaptive-impulsive control: (a) 𝑒𝑖1(𝑡)=𝑥𝑖1−0.5𝑥1, (b) 𝑒𝑖2(𝑡)=𝑥𝑖2−0.5𝑥2.

Figure 3 

The evolution of the feedback strength 𝑑𝑖,𝑖=1,2,…,5.

Remark 4.1. We only consider the impulsive control; the other conditions are chosen as above, and simulation results are shown in Figure 4. From Figures 2 and 4, it is easy to find that the control effect is not as well as the adaptive-impulsive control method.

(a)

(b)

Figure 4 

PS errors under the impulsive control.

Remark 4.2. In [31, 32], the authors investigated the PS of the drive-response dynamical networks model, but the time-varying coupling was not taken into account. Here, the PS of the drive-response time-varying coupling dynamical networks with time delay is studied by employing the adaptive-impulsive control. Furthermore, the proposed adaptive-impulsive control scheme is more effective than the adaptive control scheme in [31] and the impulsive control scheme in [32].

5. Conclusion

In this paper, the adaptive-impulsive PS of the drive-response time-varying coupling dynamical networks with time delay has been investigated. Based on the stability analysis of impulsive functional differential equations, some sufficient conditions for realizing the PS are established under the adaptive-impulsive control. And the results are a time delay independent stability criteria. Finally, numerical simulations have also been given to show effectiveness of the proposed method by an example of the unified chaotic systems. In the near future, we will extend the proposed method to discrete complex dynamical networks and discuss the delay-dependent case.

Acknowledgments

The author would like to thank the associate editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper and Professor Qinsheng Bi for his help. This work was supported by the National Natural Science Foundation of China (nos. 10872080, 20976075, 10972091).