Synchronization of Intermittently Coupled Dynamical Networks

Abstract

This paper investigates the synchronization phenomenon of an intermittently coupled dynamical network in which the coupling among nodes can occur only at discrete instants and the coupling configuration of the network is time varying. A model of intermittently coupled dynamical network consisting of identical nodes is introduced. Based on the stability theory for impulsive differential equations, some synchronization criteria for intermittently coupled dynamical networks are derived. The network synchronizability is shown to be related to the second largest and the smallest eigenvalues of the coupling matrix, the coupling strength, and the impulsive intervals. Using the chaotic Chua system and Lorenz system as nodes of a dynamical network for simulation, respectively, the theoretical results are verified and illustrated.

1. Introduction

A complex dynamical network is a large set of interconnected nodes, in which each node is typically a nonlinear dynamical system. Many real systems in nature and engineering, such as physical, biological, technological, and social systems, can be described by various models of complex dynamical networks. Complex dynamical networks, therefore, have become a significant research topic for studying nonlinear dynamics in various fields of sciences and humanities today [1–7].

One of the most remarkable phenomena in complex dynamical networks is the synchronization of dynamical nodes, which has been extensively investigated in recent years [8–32]. Wang and Chen [8, 9] presented a unified dynamical network model and investigated its synchronization in small-world and scale-free networks. Belykh et al. [11, 12] proposed an effective method for determining the global stability of synchronization in dynamical networks with different topologies, which combines the Lyapunov function approach with graph-theoretic reasoning. Restrepo et al. [19] studied the emergence of coherence in large-scale complex networks of interacting heterogeneous dynamical systems and showed that the largest eigenvalue of the network adjacency matrix plays a key role in determining the transition to coherence. Zhou et al. [20–22] derived some synchronization criteria for general complex delayed dynamical networks. Recently, synchronization of complex dynamical networks with impulsive control has been extensively studied [23–32]. For example, Guan et al. [23, 24] proposed a hybrid impulsive and switching control strategy and investigated the stabilization of complex networks. Liu et al. [25] proposed an impulsive synchronization scheme for an uncertain dynamical network. Zhang et al. [26] designed an effective impulsive controller to achieve impulsive synchronization for a complex dynamical network with unknown coupling. Lu et al. [27] investigated the problem of globally exponential synchronization of impulsive dynamical network, with a unified impulsive synchronization criterion derived by proposing a concept of average impulsive interval. Tang et al. [29] investigated the pinning synchronization problem of stochastic impulsive networks. Zhou et al. [30] proposed an impulsive control approach for analyzing pinning stability in a complex delayed dynamical network comprised of linearly coupled dynamical systems with coupling delays. Han et al. [31] and Sun et al. [32] derived some distributed impulsive control schemes for various impulsively coupled complex dynamical systems with or without delays.

However, the aforementioned work and most other existing research focus on the networks whose couplings are invariable or continuously varying in time. In the real world, there are many networks in which the coupling among nodes is intermittent. For example, in neural networks, the connections among neurons are usually cut-off type or extremely faint. There are prominent impulsive interactions among neurons when they are stimulated by certain signals, and the interactions are not identical due to the differences of stimulating signals. This type of networks is referred to as intermittently coupled dynamical networks. Note that the structure of such networks is time varying, and the interactions among nodes can only take place when some conditions are satisfied which are usually described by a group of discrete time sequences. To the best of our knowledge, there are few theoretical results about intermittently coupled dynamical networks in the current literature.

In this paper, an intermittently coupled dynamical network consisting of identical nodes is investigated. In the network, the coupling among nodes can only occur at discrete instants, and the coupling configuration of the network is varying at different instants. Based on the stability theory for impulsive differential equations, some synchronization criteria are obtained, showing that the synchronizability of the intermittently coupled dynamical network is related to the second largest and the smallest eigenvalues of the coupling matrix, the coupling strength, and the impulsive intervals. It turns out that the analytical results about the second largest and the smallest eigenvalues of the coupling matrix are consistent with other known results about complex dynamical networks [8, 9, 19].

The rest of the paper is organized as follows. In Section 2, an intermittently coupled dynamical network model is introduced, and some necessary definitions and preliminary lemmas are presented. The main results of the paper are given in Section 3, where some synchronization criteria for the network model are derived. In Section 4, using the chaotic Chua system and Lorenz system as nodes of a dynamical network, respectively, numerical simulations are performed to illustrate and verify the theoretical results. Finally, conclusions are drawn in Section 5.

2. Model Description and Preliminaries

Consider an intermittently coupled dynamical network consisting of 𝑁 identical nodes, with each node being an n-dimension dynamical system, described by 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+^∞_𝑘=1𝑈_𝑖𝑡;𝑥₁,…,𝑥_𝑁𝛿𝑡−𝑡_𝑘,𝑖=1,2,…,𝑁,(1) where 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛)^𝑇∈𝑅^𝑛 is the state vector of node 𝑖, 𝑓∶𝑅^𝑛→𝑅^𝑛 is continuously differentiable, 𝑈_𝑖(𝑡;𝑥₁,𝑥₂,…,𝑥_𝑁)∶𝑅×𝑅^𝑛𝑁→𝑅^𝑛 are coupling functions, time sequence {𝑡_𝑘}^∞_𝑘=1 satisfies 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<𝑡_𝑘+1<⋯, lim_𝑘→∞𝑡_𝑘=∞, and 𝛿(⋅) is a Dirac delta function defined by 𝛿𝑡−𝑡_𝑘=∞,𝑡=𝑡_𝑘,0,𝑡≠𝑡_𝑘,(2) satisfying the identities ∫^+∞_−∞𝛿(𝑡)𝑑𝑡=1 and ∫^+∞_−∞𝑔(𝑡)𝛿(𝑡−𝑡₀)𝑑𝑡=𝑔(𝑡₀) for any continuous function 𝑔(𝑡). Suppose that the function 𝑓 satisfies the following condition.

(A1) There exists a positive scalar 𝐿 such that, for any 𝑥,𝑦∈𝑅^𝑛, 𝑓(𝑥)−𝑓𝑦≤𝐿𝑥−𝑦.(3) In this paper, we consider the dynamical network with time-varying diffusive coupling at discrete instants. Let 𝑈_𝑖𝑡_𝑘;𝑥₁,…,𝑥_𝑁=𝑐^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑡_𝑘Γ𝑥_𝑗𝑡_𝑘−𝑥_𝑖𝑡_𝑘,(4) where constant 𝑐>0 is the coupling strength, 𝐴(𝑡_𝑘)=(𝑎_𝑖𝑗(𝑡_𝑘))∈𝑅^𝑁×𝑁 is the outer-coupling matrix which represents the coupling configuration of network (1) at instant 𝑡_𝑘, and 𝑎_𝑖𝑗(𝑡_𝑘) is defined as follows: if there is a connection between nodes 𝑖 and j (𝑖≠𝑗), then 𝑎_𝑖𝑗(𝑡_𝑘)=𝑎_𝑗𝑖(𝑡_𝑘)=1, otherwise, 𝑎_𝑖𝑗(𝑡_𝑘)=𝑎_𝑗𝑖(𝑡_𝑘)=0 (𝑖≠𝑗); the diagonal elements are defined by 𝑎_𝑖𝑖𝑡_𝑘=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑡_𝑘=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑗𝑖𝑡_𝑘,𝑖=1,2,…,𝑁,(5) and Γ=diag{𝛾₁,𝛾₂,…,𝛾_𝑛}∈𝑅^𝑛×𝑛 is the inner-linking matrix. Let Γ=diag{1,1,…,1}. Thus, (4) becomes 𝑈_𝑖𝑡_𝑘;𝑥₁,𝑥₂,…,𝑥_𝑁=𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑡_𝑘Γ𝑥_𝑗𝑡_𝑘.(6) Assume that network (1) is connected at instants 𝑡₁,𝑡₂,…, in the sense that there is no isolated cluster; that is, 𝐴(𝑡_𝑘) is an irreducible matrix.

Lemma 1 (see [14]). Suppose that the outer-coupling matrix 𝐴=(𝑎_𝑖𝑗)∈𝑅^𝑁×𝑁 satisfies the above-mentioned conditions. Then, (i)0 is an eigenvalue of matrix 𝐴 of multiplicity 1, associated with eigenvector (1/√𝑁,1/√𝑁,…,1/√𝑁)^𝑇;(ii) all the other eigenvalues of 𝐴 are real-valued and are strictly negative.

Lemma 1 implies that the outer-coupling matrix 𝐴(𝑡_𝑘) of network (1) at instant 𝑡_𝑘 has a 0 eigenvalue of multiplicity 1, with 𝜆_𝑁(𝑡_𝑘)≤⋯≤𝜆₂(𝑡_𝑘)<𝜆₁(𝑡_𝑘)=0.

Remark 2. According to matrix theory, there exists an orthogonal matrix 𝐵(𝑡_𝑘) such that 𝐴(𝑡_𝑘)=𝐵^𝑇(𝑡_𝑘)Λ(𝑡_𝑘)𝐵(𝑡_𝑘), where Λ(𝑡_𝑘)=diag{𝜆₁(𝑡_𝑘),…,𝜆_𝑁(𝑡_𝑘)}. It is easy to see that the first column of 𝐵^𝑇(𝑡_𝑘) can be chosen as the eigenvector (1/√𝑁,1/√𝑁,…,1/√𝑁)^𝑇 corresponding to the zero eigenvalue of 𝐴(𝑡_𝑘).
Next, network (1) is rewritten as the following impulsive differential equations: 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑖=1,2,…,𝑁,Δ𝑥_𝑖𝑡_𝑘=𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑡_𝑘𝑥_𝑗𝑡_𝑘,𝑡=𝑡_𝑘,𝑘=1,2,…,(7) where Δ𝑥_𝑖(𝑡_𝑘)=𝑥_𝑖(𝑡⁺_𝑘)−𝑥_𝑖(𝑡⁻_𝑘) is the “jump” in the state variable at instant 𝑡_𝑘, with 𝑥_𝑖(𝑡⁺_𝑘)=lim_𝑡→𝑡⁺_𝑘𝑥(𝑡), 𝑥_𝑖(𝑡⁻_𝑘)=lim_𝑡→𝑡⁻_𝑘𝑥(𝑡). For simplicity, assume that 𝑥_𝑖(𝑡⁻_𝑘)=𝑥_𝑖(𝑡_𝑘), which means 𝑥_𝑖(𝑡) is continuous from the left.

Definition 3. The synchronization manifold is presented as 𝐒={(𝑥^𝑇₁,𝑥^𝑇₂,…,𝑥^𝑇_𝑁)^𝑇∈𝑅^𝑛𝑁∶𝑥_𝑖=𝑥_𝑗,𝑖,𝑗=1,2,…𝑁}, where 𝑥_𝑖∈𝑅^𝑛,𝑖=1,2,…,𝑁.

Definition 4. The coupled system (6) is said to achieve synchronization if, forall𝑖,𝑗=1,2,…,𝑁, lim_𝑡→∞||𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)||=0.
From Definitions 3 and 4, it can be easily seen that the intermittently coupled system (7) achieves synchronization if and only if the synchronization manifold S for the coupled system (7) is globally asymptotically stable.

3. Synchronization of the Intermittently Coupled Network

Let 𝑥(𝑡)=(1/𝑁)∑^𝑁_𝑗=1𝑥_𝑗(𝑡) and 𝑋(𝑡)=[𝑥^𝑇(𝑡),…,𝑥^𝑇(𝑡)]^𝑇∈𝐒, which can be regarded as a projection of 𝑋(𝑡)=[𝑥^𝑇₁(𝑡),…,𝑥^𝑇_𝑁(𝑡)]^𝑇 on the synchronization manifold. The dynamical equation of   𝑥(𝑡) can be written as 𝑥̇‌(𝑡)=1𝑁^𝑁_𝑖=1𝑥̇_𝑖(𝑡)=1𝑁^𝑁_𝑖=1𝑓𝑥_𝑖(𝑡),𝑡≠𝑡_𝑘.(8) On the other hand, 𝑥𝑡⁺_𝑘=1𝑁^𝑁_𝑖=1𝑥_𝑖𝑡⁺_𝑘=1𝑁^𝑁_𝑖=1𝑥_𝑖𝑡_𝑘+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑡_𝑘𝑥_𝑗𝑡_𝑘=1𝑁^𝑁_𝑖=1𝑥_𝑖𝑡_𝑘+𝑐𝑁^𝑁_𝑗=1^𝑁_𝑖=1𝑎_𝑖𝑗𝑡_𝑘𝑥_𝑗𝑡_𝑘=𝑥𝑡_𝑘,𝑘=1,2,….(9) Define the synchronization error of node 𝑖 as 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥(𝑡),𝑖=1,2,…,𝑁. Obviously, ∑^𝑁_𝑖=1𝑒_𝑖(𝑡)=0.

For 𝑡≠𝑡_𝑘,𝑘=1,2,…, one has 𝑒_𝑖̇(𝑡)=𝑓𝑥_𝑖(𝑡)−1𝑁^𝑁_𝑗=1𝑓𝑥_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)−𝑓(𝑥(𝑡+1))𝑁^𝑁_𝑗=1𝑓(𝑥(𝑡))−𝑓𝑥_𝑗(𝑡)=𝑓𝑒_𝑖(𝑡)−1𝑁^𝑁_𝑗=1𝑓𝑒_𝑗(𝑡),(10) where 𝑓(𝑒_𝑖(𝑡))=𝑓(𝑥(𝑡)+𝑒_𝑖(𝑡))−𝑓(𝑥(𝑡)).

According to assumption (A1), it can be easily shown that 𝑓𝑒_𝑖(𝑡)≤𝐿𝑒_𝑖(𝑡),𝑖=1,2,…,𝑁.(11) For 𝑡=𝑡_𝑘,𝑘=1,2,…, one has 𝑒_𝑖𝑡⁺_𝑘=𝑥_𝑖𝑡⁺_𝑘−𝑥𝑡⁺_𝑘=𝑥_𝑖𝑡_𝑘+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑡_𝑘𝑥_𝑗𝑡_𝑘−𝑥𝑡_𝑘=𝑒_𝑖𝑡_𝑘+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑡_𝑘𝑒_𝑗𝑡_𝑘.(12) Therefore, the error dynamical system can be described as 𝑒_𝑖̇(𝑡)=𝑓𝑒_𝑖(𝑡)−1𝑁^𝑁_𝑗=1𝑓𝑒_𝑗(𝑡),𝑡≠𝑡_𝑘,𝑒_𝑖𝑡⁺_𝑘=𝑒_𝑖𝑡_𝑘+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑡_𝑘𝑒_𝑗𝑡_𝑘,𝑘=1,2,…,(13) where 𝑖=1,2,…,𝑁.

Let 𝑒(𝑡)=[𝑒^𝑇₁(𝑡),…,𝑒^𝑇_𝑁(𝑡)]^𝑇∈𝑅^𝑛𝑁. It is easily seen that the stability of the synchronization manifold is equivalent to 𝑒(𝑡)→0 as 𝑡→∞. In the following, we directly investigate the dynamical behaviors of the error dynamical system (13).

Theorem 5. Let 𝑃∈𝑅^𝑛×𝑛 be a symmetric and positive definite matrix, with 𝜆_max(𝑃) and 𝜆_min(𝑃) being the largest and the smallest eigenvalues, respectively. Suppose that there exists a constant 𝜉>1 and, for all 𝑘=1,2,…, 𝜆_min(𝑃)exp^{−2𝐿(𝜆max(𝑃)/𝜆min(𝑃))(𝑡𝑘+1−𝑡𝑘)}𝜆_max(𝑃)𝜆𝑡_𝑘≥𝜉>1,(14) where 𝜆(𝑡_𝑘)=max{(1+𝑐𝜆_𝑖(𝑡_𝑘))²∣𝑖=2,𝑁}, 𝜆₂(𝑡_𝑘) and 𝜆_𝑁(𝑡_𝑘) are the second largest and the smallest eigenvalues of matrix 𝐴(𝑡_𝑘), respectively, and the constant 𝑐>0 is the coupling strength. Then, the trivial solution of the error dynamical system (21) is asymptotically stable, implying that network (1) achieves synchronization.

Proof. Construct a Lyapunov function as 𝑉(𝑡,𝑒)=𝑒^𝑇(𝑡)𝐼_𝑁⊗𝑃𝑒(𝑡)=^𝑁_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡),(15) where ⊗ denotes the Kronecker product operator.
For 𝑡≠𝑡_𝑘, taking Dini’s derivative along the trajectories of (13) gives 𝐷⁺𝑉(𝑡,𝑒)=^𝑁_𝑖=12𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖̇(𝑡)=^𝑁_𝑖=12𝑒^𝑇_𝑖(𝑡)𝑃𝑓𝑒_𝑖(𝑡)−1𝑁^𝑁_𝑗=1𝑓𝑒_𝑗(𝑡)=^𝑁_𝑖=12𝑒^𝑇_𝑖(𝑡)𝑃𝑓𝑒_𝑖(𝑡)≤^𝑁_𝑖=11𝐿𝜆_min(𝑃)𝜆_max(𝑃)𝑓^𝑇𝑒_𝑖(𝑡)𝑓𝑒_𝑖(𝑡)+𝐿𝜆_min(𝑃)𝜆_max(𝑃)𝑒^𝑇_𝑖(𝑡)𝑃^𝑇𝑃𝑒_𝑖(𝑡)≤^𝑁_𝑖=1𝐿𝜆_min(𝑃)𝜆_max(𝑃)𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡)𝜆_min(𝑃)+𝐿𝜆_max(𝑃)𝜆_min(𝑃)𝑒^𝑇_𝑖(𝑡)𝑃𝑒_𝑖(𝑡)=2𝐿𝜆_max(𝑃)𝜆_min(𝑃)𝑉(𝑡,𝑒).(16)
For 𝑡=𝑡_𝑘, one has 𝑒𝑡⁺_𝑘=𝑒^𝑇₁𝑡⁺_𝑘,𝑒^𝑇₂𝑡⁺_𝑘,…,𝑒^𝑇_𝑁𝑡⁺_𝑘^𝑇=𝑒𝑡_𝑘+𝑐𝐴^𝑇𝑡_𝑘⊗𝐼_𝑛𝑒𝑡_𝑘=𝐼_𝑁⊗𝐼_𝑛+𝑐𝐴^𝑇𝑡_𝑘⊗𝐼_𝑛𝑒𝑡_𝑘.(17) Therefore, 𝑉𝑡⁺_𝑘,𝑒=𝑒^𝑇𝑡⁺_𝑘𝐼_𝑁⊗𝑃𝑒𝑡⁺_𝑘=𝑒^𝑇𝑡_𝑘𝐼_𝑁+𝑐𝐴𝑡_𝑘⊗𝐼_𝑛^𝑇𝐼_𝑁×𝐼⊗𝑃_𝑁+𝑐𝐴𝑡_𝑘⊗𝐼_𝑛𝑒𝑡_𝑘≤𝜆_max(𝑃)𝑒^𝑇𝑡_𝑘𝐼_𝑁+𝑐𝐴𝑡_𝑘⊗𝐼_𝑛^𝑇×𝐼_𝑁+𝑐𝐴𝑡_𝑘⊗𝐼_𝑛𝑒𝑡_𝑘=𝜆_max(𝑃)𝑒^𝑇𝑡_𝑘𝐼_𝑁+𝑐𝐴𝑡_𝑘²⊗𝐼_𝑛𝑒𝑡_𝑘.(18) Based on the property of the matrix 𝐴(𝑡_𝑘) mentioned in Remark 2, there exits an orthogonal matrix 𝐵(𝑡_𝑘) such that 𝐴(𝑡_𝑘)=𝐵^𝑇(𝑡_𝑘)Λ(𝑡_𝑘)𝐵(𝑡_𝑘), where Λ𝑡_𝑘=diag𝜆₁𝑡_𝑘,…,𝜆_𝑁𝑡_𝑘(19) and the first column of 𝐵^𝑇(𝑡_𝑘) is the eigenvector (1/√𝑁,…,1/√𝑁)^𝑇 corresponding to the zero eigenvalue of 𝐴(𝑡_𝑘). Let 𝜂(𝑡_𝑘)=(𝜂^𝑇₁(𝑡_𝑘),…,𝜂^𝑇_𝑁(𝑡_𝑘))^𝑇=(𝐵(𝑡_𝑘)⊗𝐼_𝑛)𝑒(𝑡_𝑘). Then, 𝜂₁𝑡_𝑘=1√𝑁^𝑁_𝑗=1𝑒_𝑗𝑡_𝑘=𝟎.(20) Therefore, 𝜂𝑡_𝑘=𝟎,𝜂^𝑇₂𝑡_𝑘,…,𝜂^𝑇_𝑁𝑡_𝑘^𝑇.(21) Let 𝜆(𝑡_𝑘)=max{(1+𝑐𝜆_𝑖(𝑡_𝑘))²∣𝑖=2,𝑁}, where 𝜆₂(𝑡_𝑘) and 𝜆_𝑁(𝑡_𝑘) are the second largest and the smallest eigenvalues of matrix 𝐴(𝑡_𝑘), respectively. It follows from (18) and (21) that 𝑉𝑡⁺_𝑘,𝑒≤𝜆_max(𝑃)𝑒^𝑇𝑡_𝑘×𝐵^𝑇𝑡_𝑘𝐼_𝑁+𝑐Λ𝑡_𝑘²𝐵𝑡_𝑘⊗𝐼_𝑛𝑒𝑡_𝑘=𝜆_max(𝑃)𝜂^𝑇𝑡_𝑘𝐼_𝑁+𝑐Λ𝑡_𝑘²⊗𝐼_𝑛𝜂𝑡_𝑘=𝜆_max(𝑃)^𝑁_𝑖=1𝜂^𝑇_𝑖𝑡_𝑘1+𝑐𝜆_𝑖𝑡_𝑘²𝜂_𝑖𝑡_𝑘≤𝜆_max(𝑃)𝜆𝑡_𝑘^𝑁_𝑖=1𝜂^𝑇_𝑖𝑡_𝑘𝜂_𝑖𝑡_𝑘=𝜆_max(𝑃)𝜆𝑡_𝑘𝑒^𝑇𝑡_𝑘𝐵^𝑇𝑡_𝑘⊗𝐼_𝑛×𝐵𝑡_𝑘⊗𝐼_𝑛𝑒𝑡_𝑘≤𝜆_max(𝑃)𝜆_min(𝑃)𝜆𝑡_𝑘𝑒^𝑇𝑡_𝑘𝐼_𝑁⊗𝑃𝑒𝑡_𝑘=𝜆_max(𝑃)𝜆_min(𝑃)𝜆𝑡_𝑘𝑉𝑡_𝑘,𝑒.(22) Let 𝑆_𝜌={𝑥∈𝑅^𝑛∶||𝑥||<𝜌}. Since ||𝑒(𝑡⁺_𝑘)||=||[(𝐼_𝑁+𝑐𝐴(𝑡_𝑘))⊗𝐼_𝑛]𝑒(𝑡_𝑘)||≤||(𝐼_𝑁+𝑐𝐴(𝑡_𝑘))⊗𝐼_𝑛||||𝑒(𝑡_𝑘)|| and ||(𝐼_𝑁+𝑐𝐴(𝑡_𝑘))⊗𝐼_𝑛||<∞, there exists a 𝜌₀>0 such that 𝑒(𝑡_𝑘)∈𝑆_𝜌0 implies 𝑒(𝑡⁺_𝑘)∈𝑆_𝜌 for all 𝑘.
Since 𝑃 is a symmetric and positive definite matrix, one has 𝜆_min(𝑃)^𝑛_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡)≤𝑉(𝑡,𝑒)≤𝜆_max(𝑃)^𝑛_𝑖=1𝑒^𝑇_𝑖(𝑡)𝑒_𝑖(𝑡).(23)
Thus, by the well-known comparison Theorem [33], the asymptotic stability of the trivial solution of the impulsive dynamical system (13) follows from the comparison system below: 𝜔̇(𝑡)=2𝐿𝜆_max(𝑃)𝜆_min(𝑃)𝜔(𝑡),𝑡≠𝑡_𝑘,𝜔𝑡⁺_𝑘=𝜆_max(𝑃)𝜆_min(𝑃)𝜆𝑡_𝑘𝜔𝑡_𝑘,𝑡=𝑡_𝑘𝜔𝑡,𝑘=1,2,…,₊₀=𝜔₀.(24) Therefore, by the stability criterion for impulsive differential equations [34], there exists a constant 𝜉>1 such that 𝜆ln^max(𝑃)𝜆_min(𝑃)𝜆𝑡_𝑘𝜉+2𝐿𝜆_max(𝑃)𝜆_min(𝑃)𝑡_𝑘+1−𝑡_𝑘≤0,(25) namely, 𝜆_min(𝑃)exp^{−2𝐿(𝜆max(𝑃)/𝜆min(𝑃))(𝑡𝑘+1−𝑡𝑘)}𝜆_max(𝑃)𝜆𝑡_𝑘≥𝜉>1.(26) Hence, the trivial solution of error dynamical system (13) is asymptotically stable, implying that network (1) achieves synchronization.

Remark 6. For a dynamical network consisting of 𝑁 nodes, the number of possible coupling matrices which represent the coupling configurations of the network is finite. Therefore, one may define 𝜆^max=max_𝑘=1,2,…𝜆₂𝑡_𝑘,𝜆^min=min_𝑘=1,2,…𝜆_𝑁𝑡_𝑘,(27) and 𝜆=max{(1+𝑐𝜆^max)²,(1+𝑐𝜆^min)²}. It is obvious that 𝜆(𝑡_𝑘)≤𝜆 for all 𝑘=1,2,….

Remark 7. The synchronizability of network (1) is determined by the second largest eigenvalue 𝜆₂(𝑡_𝑘) and the smallest eigenvalue 𝜆_𝑁(𝑡_𝑘) of the coupling matrix, the coupling strength 𝑐, and impulsive intervals 𝜎_𝑘=𝑡_𝑘+1−𝑡_𝑘, 𝑘=1,2,….

4. Numerical Simulations

In this section, two illustrative examples about the chaotic Chua system and Lorenz system, respectively, are given to demonstrate the theoretical results obtained above. Without loss of generality, let the impulses be equidistant and separated by a constant interval 𝜎>0.

Example 1. The chaotic Chua system is used as nodes of a dynamical network, which is described by 𝑥̇𝑦̇𝑧̇=𝛼𝑦−𝑥−𝜑(𝑥)𝑥−𝑦+𝑧−𝛽𝑦,(28) where 𝛼,𝛽 are two parameters and 𝜑(𝑥)=𝑏𝑥+0.5(𝑎−𝑏)(|𝑥+1|−|𝑥−1|) represents a piecewise-linear diode, where 𝑎<𝑏<0 are two constants. It is well known that Chua system is chaotic when 𝛼=9.22, 𝛽=15.99, 𝑎=−1.25, and 𝑏=−0.76.
The intermittently coupled dynamical network is described by 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡≠𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘=𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑡_𝑘𝑥_𝑗𝑡_𝑘,𝑡=𝑡_𝑘,𝑘=1,2,…,(29) where 𝑓𝑥_𝑖(𝑡)=𝛼𝑥_𝑖2−𝑥_𝑖1−𝜑𝑥_𝑖1𝑥_𝑖1−𝑥_𝑖2+𝑥_𝑖3−𝛽𝑥_𝑖2.(30) The outer-coupling matrix 𝐴(𝑡_𝑘)=(𝑎_𝑖𝑗(𝑡_𝑘))∈𝑅^𝑁×𝑁 represents the coupling configuration of the dynamical network at instant 𝑡_𝑘 and is defined as follows.
When 𝑘 is odd, the dynamical network has nearest-neighbor coupling, so 𝐴𝑡_𝑘=𝐴₁=−210⋯011−21⋯00⋮⋮⋮⋮⋮⋮100⋯1−2.(31)
When 𝑘 is even, the dynamical network has star coupling, so 𝐴𝑡_𝑘=𝐴₂=−𝑁+111⋯111−10⋯00⋮⋮⋮⋮⋮⋮100⋯0−1.(32) In the simulation, we choose 𝑁=100. The initial values of these systems are chosen randomly from interval [−20,20]. For a small impulsive interval 𝜎>0, the synchronization of network (29) can be achieved by choosing an appropriate coupling strength 𝑐. The numerical simulation results are shown in Figures 1–5, where 𝑒_𝑖𝑗=∑^𝑁_𝑖=2|𝑥_𝑖𝑗−𝑥_1𝑗|/(𝑁−1),𝑗=1,2,3.
In Figure 1, 𝜎=0.01 and 𝑐=0.01. One can see that the state errors between node 1 and node 𝑖, 𝑖=2,3,…,100, tend to zero asymptotically as time evolves, implying that network (29) achieves synchronization. When 𝜎=0.05 and 𝑐=0.01, network (29) cannot achieve synchronization, as shown in Figure 2. Figure 3 shows that network (29) achieves synchronization with 𝜎=0.05 and 𝑐=0.05. Figure 4 shows that network (29) cannot achieve synchronization where 𝜎=0.05 and 𝑐=0.06. Figure 5 shows that network (29) achieves synchronization with 𝜎=0.01 and 𝑐=0.06.

Figure 1 

𝑁=100, 𝜎=0.01, and 𝑐=0.01.

Figure 2 

𝑁=100, 𝜎=0.05, and 𝑐=0.01.

Figure 3 

𝑁=100, 𝜎=0.05, and 𝑐=0.05.

Figure 4 

𝑁=100, 𝜎=0.05, and 𝑐=0.06.

Figure 5 

𝑁=100, 𝜎=0.01, and 𝑐=0.06.

Example 2. The chaotic Lorenz system is used as nodes of a dynamical network, which is described by 𝑥̇𝑦̇𝑧̇=𝑎𝑦−𝑥𝑐𝑥−𝑥𝑧−𝑦𝑥𝑦−𝑏𝑧,(33) with a chaotic attractor when 𝑎=10, 𝑏=8/3, and 𝑐=28.
The intermittently coupled dynamical network is described by 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑘=1,2,…,Δ𝑥_𝑖𝑡_𝑘=𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝑡_𝑘𝑥_𝑗𝑡_𝑘,𝑡=𝑡_𝑘,(34) where 𝑓𝑥_𝑖(𝑡)=𝑎𝑥_𝑖2−𝑥_𝑖1𝑐𝑥_𝑖1−𝑥_𝑖1𝑥_𝑖3−𝑥_𝑖2𝑥_𝑖1𝑥_𝑖2−𝑏𝑥_𝑖3.(35)
Let 𝑙≡𝑘mod3. The outer-coupling matrix 𝐴(𝑡_𝑘)=(𝑎_𝑖𝑗(𝑡_𝑘))∈𝑅^𝑁×𝑁 is defined as follows.
When 𝑙=1, the dynamical network has global coupling, and its outer-coupling matrix is 𝐴𝑡_𝑘=𝐴_𝑔𝑐=−𝑁+111⋯11−𝑁+11⋯1⋮⋮⋮⋮⋮111⋯−𝑁+1.(36)
When 𝑙=2, the dynamical network has nearest-neighbor coupling, with 𝐴𝑡_𝑘=𝐴₁=−210⋯011−21⋯00⋮⋮⋮⋮⋮⋮100⋯1−2.(37)
When 𝑙=0, the dynamical network has star coupling, with 𝐴𝑡_𝑘=𝐴₂=−𝑁+111⋯111−10⋯00⋮⋮⋮⋮⋮⋮100⋯0−1.(38)
In the simulation, we choose 𝜎=0.1. The initial values of these systems are chosen randomly from interval [−20,20]. The numerical simulation results indicate that the synchronizability of network (34) is related to the coupling strength 𝑐 and the number of nodes 𝑁, as shown in Figures 6–10, where 𝑒_𝑖𝑗=∑^𝑁_𝑖=2|𝑥_𝑖𝑗−𝑥_1𝑗|/(𝑁−1),𝑗=1,2,3. For the coupling matrices of this network (34), the spectral width of eigenvalues will be broadened with increasing 𝑁. Therefore, the synchronizability of the network will be weakened if the node number increases sharply.
In Figure 6, 𝑁=200 and 𝑐=0.01. One can see that the state errors tend to zero asymptotically as time evolves, implying that network (34) achieves synchronization. When 𝑁=200 and 𝑐=0.015, network (34) cannot achieve synchronization, as illustrated in Figure 7. When 𝑁=50 and 𝑐=0.015, network (34) achieves synchronization, as shown in Figure 8. When 𝑁=50 and 𝑐=0.055, network (34) cannot achieve synchronization, as shown in Figure 9. Figure 10 demonstrates that network (34) achieves synchronization when 𝑁=50 and 𝑐=0.048.

Figure 6 

𝑁=200, 𝜎=0.1, and 𝑐=0.01.

Figure 7 

𝑁=200, 𝜎=0.1, and 𝑐=0.015.

Figure 8 

𝑁=50, 𝜎=0.1, and 𝑐=0.015.

Figure 9 

𝑁=50, 𝜎=0.1, and 𝑐=0.055.

Figure 10 

𝑁=50, 𝜎=0.1, and 𝑐=0.048.

5. Conclusions

This paper investigates the synchronization phenomenon of an intermittently coupled dynamical network, in which the coupling among nodes can occur only at discrete instants and the coupling configuration of the network is varying at different instants. For such an intermittently coupled dynamical network consisting of identical nodes, based on the stability theory for impulsive differential equations, some synchronization conditions are derived. It is shown that the synchronizability of the network is related to the second largest eigenvalue and the smallest eigenvalue of the coupling matrix, the coupling strength, and the impulsive intervals. It is worth noting that the analytical results about the second largest and the smallest eigenvalues of the coupling matrix are consistent with most existing criteria for complex dynamical networks. Finally, by employing the chaotic Chua system and Lorenz system as nodes of a dynamical network, respectively, numerical simulations are carried out to illustrate and verify the theoretical results.

Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSF) through Grant no. 10972011, the Natural Science Foundation of Hebei Province of China through Grant no. A2010001943, and the Hong Kong Research Grants Council through GRF Grant City U1114/11E.