Graph-theoretic approach to exponential synchronization of coupled systems on networks with mixed time-varying delays

Communicated by Dr. James Lam

Accepted Manuscript

Graph-theoretic approach to exponential synchronization of coupled systems on networks with mixed time-varying delays Beibei Guo, Yu Xiao, Chiping Zhang PII: DOI: Reference:

S0016-0032(17)30264-8 10.1016/j.jfranklin.2017.05.029 FI 3003

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

17 August 2016 16 March 2017 20 May 2017

Please cite this article as: Beibei Guo, Yu Xiao, Chiping Zhang, Graph-theoretic approach to exponential synchronization of coupled systems on networks with mixed time-varying delays, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.05.029

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Graph-theoretic approach to exponential synchronization of coupled systems on networks with mixed time-varying delays Beibei Guoa , Yu Xiaoa,∗, Chiping Zhanga Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, PR China

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Abstract

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In this paper, the issue of exponential synchronization for coupled systems on networks with mixed time-varying delays is concerned. An approach combining Kirchhoff’s matrix tree theorem in graph theory with Lyapunov method and periodically intermittent control is taken to investigate the problem. This method is different from the corresponding previous works. Two different kinds of synchronization conditions in the form of Lyapunovtype theorem and coefficients-type criterion are derived. They both reveal synchronization has a close relation with the topology structure of the network. Finally, the feasibility and effectiveness of the proposed method are illustrated by several numerical simulation figures. Keywords: Exponential synchronization, Coupled systems, Mixed time-varying delays, Graph-theoretic method, Periodically intermittent control

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1. Introduction

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In the past few decades, the coupled systems on networks (CSNs) have received intensive interest due to their extensive applications in diverse areas such as communication networks, neural networks, World Wide Web, power grids and others fields [1, 2]. As is well known, synchronization, as a typical kind of dynamics, is an important group behavior of networks and has drawn considerable researchers’ attentions since the synchronization mechanism can explain well many natural phenomena, including the synchronous information exchange in the Internet and worldwide web, and the synchronous transfer of digital or analog signals in communication networks. Thus, the synchronization analysis for CSNs is of great significance both in theory and in practice. Time delay is frequently encountered in information transmission between subsystems [3, 4]. According to the way it occurs, time-delay can be classified as two types: discrete and distributed. Discrete time-delay is relatively easier to be identified in practice and, therefore, synchronization analysis for coupled system with discrete delays has become a hot research topic in the past few years [5, 6, 7]. On the other hand, since there may exist a distribution of propagation delays over a period of time in some cases, thus, it is necessary to introduce continuously distributed delays over a certain duration of time such that the distant past has less influence compared with the recent behavior of the state. Recently, the synchronization analysis problem for general complex systems with both discrete and distributed delays (or called mixed time-delays) has received increasing research attention and many relevant results have been reported in the literature (see e.g. [8, 9, 10, 11, 12] and the references therein). Song [8] investigated the global exponential synchronization of coupled connected ∗

Corresponding author. Tel.: +86 150 0464 6332. Email address: [email protected] (Yu Xiao)

Preprint submitted to Elsevier

May 30, 2017

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neural networks with both discrete and distributed delays. Zhu et al. [9] studied the adaptive synchronization of chaotic Cohen-Crossberg neural networks with mixed time delays. Generally, time delay may degrade the performance of the system or even cause instability of the closed-loop system. For this reason, many efforts have been done on the problem of controller design for time delay systems. Consequently, many approaches and control techniques have been proposed covering linear matrix inequality approach [13], adaptive control [14], fuzzy control [15], impulsive control [16], intermittent control [17] and so on. Recently, intermittent control has attracted considerable research attention since it has wide applications in many fields, such as manufacturing, transportation, communication and signal processing [18, 19, 20]. A significant merit of intermittent control is that it has a nonzero control width, and can be easily implemented in practice. For example, in communications, when the strength of the system signal is below the required level at the terminal, the external control signal can be added to achieve the desired result or requirement [21]. And then, in order to reduce the control cost, the external control can be removed. In view of those merits, some stability and synchronization criteria of networks have been obtained by utilizing intermittent control. For instance, in [22], Hu et al. analyzed the exponential lag synchronization for neural networks with mixed delays via periodically intermittent control and analysis technique; in [23], combining Lyapunov stability theory with stochastic analysis approaches and periodically intermittent control, Gan et al. investigated exponential synchronization of stochastic Cohen-Grossberg neural networks with mixed time-varying delays and reactiondiffusion; in [24], by introducing multi-parameters and using Lyapunov functional theory, Hu et al. considered the globally exponential synchronization for a class of reaction-diffusion neural networks. For coupled systems on networks with mixed time-varying delays (CSNMTVDs) under intermittent control, it is easy to see in the above literature that Lyapunov method provides a powerful tool to explore the synchronization laws. However, it is quite tough to construct a proper global Lyapunov function due to the intricate relations between the topological structure of networks and divergent dynamical properties of the coupled nodes. Fortunately, new techniques have been proposed to cope with this issue. For example, Herzog et al. [25] developed a method to construct a sequence of ”optimal” Lyapunov functions by algorithmic procedure. Athreya et al. [26] proposed a method to construct a Lyapunov function by patching together functions which are locally Lyapunov in a collection of regions whose union covers all of the possible routes to infinity. With the help of some results in graph theory, Li et al. developed a systematic method, namely graph-theoretic approach, to construct global Lyapunov functions for large-scale system. They obtained some stability criteria by using the graph theory and applied the theoretical results to coupled oscillators, epidemic models and predator-prey models. Since then, many researchers have joined the study fields and plenty of results were gained (see [29, 30, 31] and the references therein). However, as far as we know, for synchronization problem of CSNMTVDs based on graph-theoretic approach and periodically intermittent control scheme, few results are found in the literature. This situation motivates our present investigation. As is shown in [29], the CSNMTVDs can be described in a directed graph. In this paper, we address the exponential synchronization for CSNMTVDs via periodically intermittent control and first attempt to use some graph theoretical results to explore the relationship between synchronization and topology property of the directed graph. The main contributions are listed as follows. Firstly, combining graph-theoretic approach with Lyapunov method, two different kinds of sufficient laws for exponential synchronization of CSNMTVDs are derived, of which one is given in the form of Lyapunov functions and network topology, while the other is given by the means of coefficients of system. Secondly, some results in graph theory are effectively utilized to avoid directly finding the global Lyapunov function of CSNMTVDs. Thirdly, two numerical examples are given to illustrate and visualize the effectiveness and feasibility of these derived criteria in this work. The rest of this paper is organized as follows. Some preliminaries and model formulations are presented in 2

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Section 2. The exponential synchronization between two different CSNMTVDs is investigated in Section 3. Section 4 provides two numerical examples to certify the effectiveness of the proposed theory. Finally, we use a conclusion to close the paper. 2. Preliminaries and models

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Some useful notations and some concepts on graph theory are stated in Subsection 2.1. And then the model formulations are introduced in Subsection 2.2.

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2.1. Preliminaries The following notations will be used throughout this paper. Let R1+ and Rn be the set of all nonnegative real numbers and n−dimensional Euclidean space, respectively, and | · | be the Euclidean norm in Rn . Also, let P N+ = {0, 1, 2, · · · }, ` = {1, 2, · · · , l}, Z+ = {1, 2, · · · }, m = li=1 mi for mi ∈ Z+ . The superscript 0 T0 stands for the transpose. In addition, C 1,1 (Rn × R1+ ; R1+ ) represents for the family of all nonnegative functions V(x, t) on Rn × R1+ which are continuously differentiable in x and t, and C([−η, 0]; Rn ) is the space of continuous functions x : [−η, 0] → Rn with norm kxk = sup−η≤t≤0 |x(t)|. For the sake of better comprehension, some basic concepts on graph theory and a useful lemma are stated as follows. And more details can be found in [27]. A digraph G = (`, E) contains a set ` of vertices and a set E of arcs (i, j) leading from initial vertex j to terminal vertex i. A subgraph H of G is said to be spanning if H and G have the same vertex set. A digraph G is weighted if a positive weight ai j is assigned to each arc (i, j). Here ai j > 0 if and only if there exists an arc from vertex j to vertex i in G, and we call A = (ai j )l×l the weight matrix. The weight W(G) of G is the product of the weights on all its arcs. A directed path P in G is a subgraph with distinct vertices {i1 , i2 , · · · , i s } such that its set of arcs is {(ik , ik+1 ) : k = 1, 2, · · · , s − 1}. If i s = i1 , we call P a directed cycle. A tree T is rooted at vertex i, called the root, if i is not a terminal vertex of any arcs, and each of the remaining vertices is a terminal vertex of exactly one arc. A subgraph Q is unicyclic if it is a disjoint union of rooted trees whose roots form a directed cycle. A digraph G is strongly connected if for any pair of distinct vertices, there exists a directed path from one to the other. Denote the digraph with weight matrix A as (G, A). A weighted digraph (G, A) is said to be balanced if W(C) = W(−C) for all directed cycles C. Here −C denotes the reverse of C and is constructed by reversing the direction of all arcs in C. For a unicyclic graph Q with cycle CQ , let Q˜ be the unicyclic graph ˜ The Laplacian matrix obtained by replacing CQ with −CQ . Suppose that (G, A) is balanced, then W(Q) = W(Q). P of (G, A) is defined as L = (pkh )l×l , where pkh = −akh for k , h and pk j = j,k ak j for k = h.

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Lemma 1. Suppose that l ≥ 2. Let ck denote the cofactor of the k-th diagonal element of L. Then the following identity holds: l X X X ck akh Fkh (xk , xh ) = W(Q) Frs (xr , x s ), k,h=1

Q∈Q

(s,r)∈E(CQ )

where k, h ∈ `, Fkh (xk , xh ) is an arbitrary function, Q is the set of all spanning unicyclic graphs of (G, A), W(Q) is the weight of Q and CQ denotes the directed cycle of Q. Particularly, if (G, A) is strongly connected, then ck > 0 for k ∈ `.

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2.2. Model formulation In this paper, we will make drive-response CSMTVDs achieve exponential synchronization via periodically intermittent control. Moreover, we try to utilize graph-theoretic approach and Lyapunov method to establish two kinds of synchronization laws. Thus, we need to design control input for a driven system in order to achieve synchronization with the response system, provided that the two systems start from different initial conditions. Based on graph theory, a network is built on a digraph G with l (l ≥ 2) vertices, then the coupled system is constructed as follows: Z t l l l X X X dxk (t) vkh mh (xh (s))ds + Ik , k ∈ `, (1) dkh gh (xh (t − τkh (t))) + ckh fh (xh (t)) + = bk (xk (t)) + dt t−σkh (t) h=1 h=1 h=1

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where t ≥ 0, xk (t) ∈ Rmk corresponds to the state of the k-th vertex at time t; bk (·) : Rmk → Rmk is an appropriate behaved function; fh (·), gh (·) and mh (·): Rmh → Rmk are continuously activation functions; the coefficients ckh , dkh and vkh denote, respectively, the strength of the coupling, the discrete time-delay connection strength of the coupling and the distributed delay strength of the coupling of the h-th vertex to the k-th vertex; Here ckh , dkh , vkh is defined as follows: if there is a connection from vertex k to vertex h, then the coupling ckh , 0, dkh , 0, vkh , 0, otherwise, ckh = 0, dkh = 0, vkh = 0; τkh (t) and σkh (t) are the time-varying delays and distributed time-varying delays and satisfies 0 ≤ τkh (t) ≤ τ and 0 ≤ σkh (t) ≤ σ for all t ≥ 0, respectively; Ik represents an external input or bias. We treat network (1) as the drive system, the response system can be designed as Z t l l l X X X dzk (t) = bk (zk (t)) + ckh fh (zh (t)) + dkh gh (zh (t − τkh (t))) + vkh hh (zh (s))ds + Ik + uk (t), k ∈ `, (2) dt t−σkh (t) h=1 h=1 h=1

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where t ≥ 0, uk (t) is also an intermittent control defined by  l X      Rkh (zh (t) − xh (t)) , nT ≤ t < (n + θ)T,  uk (t) =   h=1     0, (n + θ)T ≤ t < (n + 1)T.

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here n ∈ N+ , Rkh (k, h ∈ `) are the constant control strengths, T is the control period, and 0 < θ < 1 is the rate of control duration. It is evident that the coupling structure of drive system (1) and response system (2) are the same. For the convenient of the reader, we here draw the architecture for systems (1) and (2) shown in Fig.1. Let ek (t) = zk (t) − xk (t) be the k-th synchronization error state vector between drive-response systems (1) and (2), by subtracting (1) from (2), we get the error system below:  l l  X X    ˜ ˜  e˙ k (t) = bk (ek (t)) + ckh fh (eh (t)) + dkh g˜ h (eh (t − τkh (t)))      h=1 h=1     Z t l l  X X      + v m ˜ (e (s))ds + Rkh eh (t), nT ≤ t < (n + θ)T, kh h h     t−σ (t) kh  h=1 h=1 (4)   l l  X X     e˙ k (t) = b˜ k (ek (t)) + ckh f˜h (eh (t)) + dkh g˜ h (eh (t − τkh (t)))      h=1 h=1     Z t  l X      + vkh m ˜ h (eh (s))ds, (n + θ)T ≤ t < (n + 1)T,    t−σkh (t) h=1

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6 Response system

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1 Drive system

Fig. 1: Example of coupling structure of drive-response networks. The solid lines represent the deterministic coupling among the nodes within a network, while dashed ones represent the coupling between the drive network and response network.

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where t ≥ 0, b˜ k (ek (t)) = bk (zk (t)) − bk (xk (t)), f˜h (eh (t)) = fh (zh (t)) − fh (xh (t)), g˜ h (zh (t − τkh (t))) = gh (zh (t − τkh (t))) − gh (xh (t − τkh (t))), m ˜ h (eh (s)) = mh (zh (s)) − mh (xh (s)). Furthermore, we assume that systems (1) and (2) have different initial values given, respectively, by x(t) = ϕ(t), z(t) = φ(t),

t ∈ [−η, 0],

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where ϕ(t) = (ϕT1 (t), ϕT2 (t), · · · , ϕTl (t))T ∈ C([−η, 0], Rm ), φ(t) = (φT1 (t), φT2 (t), · · · , φTl (t))T ∈ C([−η, 0], Rm ) and η = max{τ, σ}. Before ending this section, we introduce a hypothesis and definition [31] for derive our main results.

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Assumption 1. For any k, h ∈ `, τkh (t) and σkh (t) satisfy τ˙ kh (t) < 1 and σ ˙ kh (t) ≤ a < 1 for all t,respectively, where a is a constant.

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Definition 1. Drive-respond systems (1) and (2) are said to be exponentially synchronized if there exist W(||ϕ − φ||) > 0, γ > 0 such that |x(t) − z(t)| ≤ W(||ϕ − φ||)e−γt , t > 0.

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3. Synchronization Analysis

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In this section, our aim is to explore some novel synchronization criteria for drive-response systems (1) and (2) by the means of combining graph theory with Lyapunov method. The main results are stated as follows. 3.1. Lyapunov-type theorem We first prepare a definition about k-th vertex-Lyapunov function following [32]. Definition 2. For k, h ∈ ` and p ≥ 2, functions Vk (ek , t) ∈ C 1,1 (Rmk × R1+ ; R1+ ), as Vk (ek , t) = Vk(1) (ek , t) + Vk(2) (t),

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in which Vk(2) (t) ≥ 0, are called k-th vertex-Lyapunov function for system (4) if the following conditions hold: 5

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A1. There exist positive constants αk and βk , such that αk |ek | p ≤ Vk(1) (ek ) ≤ βk |ek | p .

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(2) A2. There exist constants σ(1) k > 0, σk > 0, function Mkh (E k , E h ) and matrix D = (δkh )l×l , δkh ≥ 0 satisfying,

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dVk (ek , t) ∂Vk (ek , t) ∂Vk (ek , t) ∂ , + ek dt ∂t ∂ek ∂t  l l X X    (1) p p   δkh Mkh (Ek , Eh ), nT ≤ t < (n + θ)T, δkh |ek (t − τhk (t))| + −σk |ek (t)| +      h=1 h=1 ≤   l l  X X   (2)  p p  δkh Mkh (Ek , Eh ), (n + θ)T ≤ t < (n + 1)T, δkh |ek (t − τhk (t))| +    −σk |ek (t)| + h=1

h=1

(h,k)∈E(C)

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Pl (2) where σ(1) k = σk − ςk,p and ςk,p = pRkk + (p − 1) h=1,h,k |Rkh |. A3. Along each directed cycle C of weighted digraph (G, D), for all Ek ∈ R pmk , Yh ∈ R pmh , there is X Mkh (Ek , Eh ) ≤ 0.

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The following two hypotheses are essential for our later study.

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Assumption 2. For any k ∈ `, system (4) admits vertex-Lyapunov function Vk (ek , t), the following condition holds, Pl σ(1) δkh k > h=1 . βk αk

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For each k ∈ `, consider the following function

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σ(1) G(εk ) = −εk + k − βk

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eεk η ,

where εk ≥ 0. It is obvious that, for k ∈ `

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σ(1) G(0) = k − βk

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δkh

˙ k ) = −1 − η > 0 and G(ε

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eεk η < 0.

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In terms of G(εk ) is continuous on [0, ∞) and G(εk ) → −∞ as εk → +∞. Thus for each k ∈ `, there exists a positive constant ε∗k such that G(ε∗k ) ≥ 0 and G(εk ) > 0 for εk ∈ (0, ε∗k ). Letting ε = mink∈` {ε∗k }, then for all k ∈ ` we have Pl σ(1) δkh k −ε + − h=1 eεη ≥ 0. (8) βk αk Remark 1. Assumption 2 presents a restriction on the coupling strength δkh . As is evident from condition A2 we know that δkh ≥ 0, so the sum of δkh (h = 1, 2, · · · , l) can not be sufficient large to ensure the vertex-Lyapunov function V˙ k (ek , t) ≤ 0. In fact, the restriction of coupling strength provides a valuable guarantee for the stability of error system. 6

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Assumption 3. ςk,p < 0 and ε − ζ(1 − θ) > 0, where ζ = maxk∈` {|ςk,p |/βk }. Remark 2. Assumption 3 shows that the rate of control duration θ can be obtained as θ > 1 − ε/ζ if the parameter ε and control strengths Rkh are chosen properly. In other words, Assumption 3 provides a way to design the time interval of periodically intermittent control. Its validity is verified in Section 4 numerical test.

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The following theorem is provided to guarantee exponential synchronization of drive-response systems (1) and (2) under periodically intermittent control (3).

V(e(t), t) =

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ck αk |ek (t)| p +

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Theorem 1. Let digraph (G, D) be strongly connected. Suppose that system (4) admits vertex-Lyapunov function Vk (ek , t) for any k ∈ `, Assumptions 2 and 3 hold. Then the drive-respond systems (1) and (2) can achieve exponential synchronization under intermittent control (3). P Proof. First, we set V(e(t), t) = lk=1 ck Vk (ek (t), t), where ck is the cofactor of the k-th diagonal element of the Laplacian matrix of (G, D). Noting that (G, D) is strongly connected, we have ck > 0 for any k ∈ ` by Lemma 1. Next, according to (5), (6) and condition Vk(2) (t) ≥ 0, we get that

l X k=1

ck Vk(2) (t)

ck Vk(2) (t) ≥

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ck αk |ek (t)| p

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k=1 ck αk

k∈`

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 2p 1− 2p  min{ck αk } > 0. In addition, when t ∈ [−η, 0], from (6) we have k∈`

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V(e(t), t) ≤

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where α =

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  2p  l  l l l X X X  2p  X    ck αk  ck αk  2 2   ≥   P = ci αi |e (t)| |e (t)| c α  P k k i i  l l c α c α j j j j j=1 j=1 i=1 i=1 k=1 k=1  l 1− 2p  p 2 X  ≥  ck αk  min{ck αk } |e(t)| p = α|e(t)| p ,

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 l  X  ≤  ck βk  |e(t)| p + M = β|e(t)| p + M, k=1

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P P in which β = lk=1 ck βk > 0, M = lk=1 ck sup−η≤t≤0 Vk(2) (t). In the sequel, we divide the proof into the following three steps for ease of exposition. Step 1. Prove H1 (t) = eεt V(e(t), t) − hM0 < 0, for all t ∈ [−η, θT ), where h > 1 is a constant, M0 = sup−η≤s≤0 {β|x(s) − z(s)| p + M}. It is not difficult to verify that, for all t ∈ [−η, 0], H1 (t) < 0. Next, we prove that, for t ∈ [0, θT ), H1 (t) < 0.

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Otherwise, there exists a t0 ∈ [0, θT ) such that H1 (t0 ) = 0, D+ H1 (t0 ) ≥ 0,

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and H1 (t) < 0

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where D+ denotes Dini derivative. Applying the conditions A1-A3, (11) and (12), we can derive that l X

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εt0 εt0 ˙ D+ H1 (t0 ) = εeεt0 V(e(t0 ), t0 ) + eεt0 V(e(t 0 ), t0 ) = εe V(e(t0 ), t0 ) + e

ck V˙ k (ek (t0 ), t0 )

k=1

  l l l l   X X X X     (1) p p δ M (E , E ) c c δ |e (t − τ (t ))| + |e (t )| + eεt0  c σ εV(e(t ), t ) −  kh kh k h k k kh k 0 hk 0 k 0 k 0 0 k     h=1 k=1 k,h=1 k=1   l l   X X   σ(1) δkh  k εt0  (e ) (e e  εV(e(t ), t ) − c c V (t ), t + V (t − τ (t )), t − τ (t ))  0 0 k k k k 0 0 k k 0 hk 0 0 hk 0     βk αk k,h=1 k=1 Pl σ(1) δkh k εt0 εt0 e V (e(t0 ), t0 ) + h=1 eεt0 V(e(t0 − τhk (t0 )), t0 − τhk (t0 )) εe V(e(t0 ), t0 ) − βk αk  Pl (1)   σ  δkh ε − k  hM0 + h=1 eεη hM0 βk αk   P l  σ(1) δkh εη  h=1 k − e  hM0 . − −ε + βk αk

≤

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≤ = <

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=

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From (8) we can obtain that D+ H1 (t0 ) < 0, which leads to a contradiction with D+ H1 (t0 ) ≥ 0. Hence, inequality (10) holds.

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Step 2. Prove H2 (t) = eεt V(e, t) − hM0 eζ(t−θT ) < 0, for all t ∈ [θT, T ).

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If this is not true, there exists a t1 ∈ [θT, T ) such that and

H2 (t1 ) = 0, D+ H2 (t1 ) ≥ 0, H2 (t) < 0

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For k, h ∈ `, if θT ≤ t1 − τhk (t1 ) ≤ t1 , we get

for t ∈ [θT, t1 ).

eεt1

l X k=1

ck Vk (ek (t1 − τhk (t1 )), t1 − τhk (t1 )) = eεt1 V (e(t1 − τhk (t1 )), t1 − τhk (t1 )) < eεη hM0 eζ(t1 −θT ) ,

and if −η ≤ t1 − τhk (t1 ) ≤ θT , from step 1, we have εt1

e

l X k=1

ck Vk (ek (t1 − τhk (t1 )), t1 − τhk (t1 )) = eεt1 V (e(t1 − τhk (t1 )), t1 − τhk (t1 )) < eεη hM0 < eεη hM0 eζ(t1 −θT ) . 8

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Hence, for any k ∈ `, we always have eεt1

l X k=1

ck Vk (ek (t1 − τhk (t1 )), t1 − τhk (t1 )) < eεη hM0 eζ(t1 −θT ) .

Then, we can arrive at

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ζ(t1 −θT ) ˙ D+ H2 (t1 ) = εeεt1 V(e(t1 ), t1 ) + eεt1 V(e(t 1 ), t1 ) − ζhM0 e l X = εeεt1 V(e(t1 ), t1 ) + eεt1 ck V˙ k (ek (t1 ), t1 ) − ζhM0 eζ(t1 −θT ) k=1

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  l l   X X    (2) p p |e (t )| + c δ |e (t − τ (t ))| − ζhM0 eζ(t1 −θT ) ≤ eεt1  c σ εV(e(t ), t ) −  k 1 k kh k 1 hk 1 k 1 1 k     k,h=1 k=1  l  l     X X  c δ M (E , E ) +eεt1   k kh kh k h     k=1 h=1   l l   X X   σ(2) δkh  k εt1  (e ) (e εV(e(t ), t ) − c ≤ e  V (t ), t + c V (t − τ (t )), t − τ (t )) 1 1 k k k 1 1 k k k 1 hk 1 1 hk 1      β α k k k=1 k,h=1

ED

M

−ζhM0 eζ(t1 −θT )   Pl   σ(2) δkh k ζ(t −θT ) 1  hM0 e < ε − + h=1 eεη hM0 eζ(t1 −θT ) − ζhM0 eζ(t1 −θT ) βk αk   P l  σ(2) ςk,p ςk,p δkh εη  h=1 k − − e  hM0 eζ(t1 −θT ) − hM0 eζ(t1 −θT ) − ζhM0 eζ(t1 −θT ) = − −ε + βk βk αk βk   Pl  σ(1) ςk,p δkh εη  k h=1 = − −ε + − e  hM0 eζ(t1 −θT ) − hM0 eζ(t1 −θT ) − ζhM0 eζ(t1 −θT ) . βk αk βk

CE

PT

It follows from Assumption 2 and (8) that D+ H2 (t1 ) < 0, which contradicts D+ H2 (t1 ) ≥ 0. Hence, H2 (t) < 0 for all t ∈ [θT, T ). According to Step 1 that, for t ∈ [−η, θT ), eεt V(e, t) < hM0 < hM0 eζ(1−θ)T .

AC

From Step 2 that, for t ∈ [θT, T ), eεt V(e, t) < hM0 eζ(t−θT ) < hM0 eζ(1−θ)T .

Thus, for all t ∈ [−η, T ), the following inequality holds, eεt V(e, t) < hM0 eζ(1−θ)T . Similar to the proof of (10), we can obtain that eεt V(e, t) < hM0 eζ(1−θ)T , 9

ACCEPTED MANUSCRIPT

for T ≤ t < (1 + θ)T . Analogous to the proof of Step 2, one sees that eεt V(e, t) < hM0 eζ(1−θ)T eζ[t−(1+θ)T ] = hM0 eζ(t−2θT ) , for (1 + θ)T ≤ t < 2T .

and for (n + θ)T ≤ t < (n + 1)T ,

CR IP T

Step 3. By mathematical induction method, we can certificate the following estimates are true for any integer n. For nT ≤ t < (n + θ)T , eεt V(e, t) < hM0 enζ(1−θ)T , (13)

eεt V(e, t) < hM0 enζ(1−θ)T eζ[t−(n+θ)T ] = hM0 eζ[t−(n+1)θT ] . Consequently, if nT ≤ t < (n + θ)T , we have n ≤ t/T , then

(14)

AN US

eεt V(e, t) < hM0 enζ(1−θ)T ≤ hM0 eζ(1−θ)t . Moreover, when (n + θ)T ≤ t < (n + 1)T , and n + 1 > t/T , we get

eεt V(e, t) < hM0 eζ[t−(n+1)θT ] < hM0 eζ(1−θ)t .

That is, for any t ≥ 0, eεt V(e, t) < hM0 eζ(1−θ)t always holds. Finally, let h → 1, from (9), it is readily seen that

M

eεt α|e(t)| p ≤ eεt V(e, t) ≤ M0 eζ(1−θ)t , which implies that

ED

|x(t) − z(t)| = |e(t)| ≤

 M  1p 0

α

− 1p [ε−ζ(1−θ)]t

e

βkϕ − φk p + M = α

! 1p

e−λt = W(kϕ − φk)e−λt ,

1

CE

PT

for any t ≥ 0, where λ = ε−ζ(1−θ) and W (kϕ − φk) = ((βkϕ − φk p + M)/α) p . Hence, according to Definition 1, we p can conclude that exponential synchronization between systems (1) and (2) can be realized under periodically intermittent control (3). The proof is therefore complete.

AC

Remark 3. The feature of this paper is that the CSNMTVDs is described by a digraph (G, D) shown in Fig.2, in which each vertex of digraph stands for an individual system called vertex system, the directed arcs of digraph indicate inter-connections and interactions among vertex systems. In Theorem 1, we establish a kind of synchronization criteria for drive-respond systems (1) and (2) in the form of Lyapunov-type theorem by the topology structure of networks and Lyapunov function, which implies that synchronization has a close relationship with topology property of the corresponding digraph (G, D). However, this relationship is not reflected in most published works [22, 23, 24]. Consequently, this paper compensates for the inadequacy of this aspect through utilizing some graph theoretical results.

10

ACCEPTED MANUSCRIPT

1

δ13 δ31 3

δ23

δ21 δ12 2

CR IP T

δ32 Fig. 2: Example of a digraph (G, D) with three vertices.

AN US

Remark 4. By using some results in graph theory, a global Lyapunov function V of system (4) is successfully P constructed via the vertex-Lyapunov functions Vk . That is, V(e, t) = lk=1 ck Vk (ek , t), where ck is the cofactor of the k-th diagonal element of the Laplacian matrix of (G, D). Thus, this avoids to directly seek for the Lyapunov function of CSNMTVDs. In fact, the practical coupled systems are very complex. To make progress, different fields have suppressed certain complications. For example, in nonlinear dynamics the simple and nearly identical dynamical systems are coupled together in simple, regular ways. These simplifications make that any issues of structural complexity are avoided and the systems potentially formidable dynamics could be studied intensively. Thus, with the help of graph theory, CSs can be studied much more easily. The validity of the technique is presented in Theorem 3.

ED

M

Remark 5. Condition A3 restricts mutual influence properties of the vertex systems in the network, and it limits the form of function Mkh strongly. It seems to be too complicated to check A3 for infinite times for the large number of directed cycles. However, this problem can be successfully solved if we find some appropriate functions Mkh , k, h = 1, 2, · · · , l. In what follows, we take one further step to study others easily verifiable conditions based on some graph theoretical results. The outcomes are listed as follows. Corollary 1. (1) Suppose that (G, D) is balanced, we can readily get

k,h=1

ck δkh Mkh (Ek , Eh ) =

PT

l X

X 1X [Mkh (Ek , Eh ) + Mhk (Eh , Ek )] . W(Q) 2 Q∈Q (h,k)∈E(C ) Q

CE

Thus, (7) can be replaced by

X

(h,k)∈E(CQ )

[Mkh (Ek , Eh ) + Mhk (Eh , Ek )] ≤ 0.

(15)

AC

Therefore, the conclusion of Theorem 1 holds if (7) is replaced by (15). (2) In view that if for every Mkh (Ek , Eh ) there exist functions Pk (Ek ) and Ph (Eh ), such that Mkh (Ek , Eh ) ≤ Pk (Ek ) − Ph (Eh ),

(16)

then (7) can be readily verified. That is, the result of Theorem 1 holds if (7) is replaced by (16). Based on graph-theoretic approach and Lyapunov method as well as periodically intermittent control, Theorem 1 provides some sufficient laws to synchronize the systems (1) and (2). As an application of Theorem 1, we would like to explore other kind of sufficient conditions by using the coefficients of systems (1) and (2), which is more convenient to be tested in practice since it mainly depends on the coefficients in systems. 11

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3.2. Coefficients-type criterion The main purpose of this subsection is to form another exponential synchronization criterion through the coefficients of systems (1) and (2). We now state the results. Theorem 2. Suppose that the following conditions hold: B1. There exist positive constants ηk , Ah , Bh and Ch such that (xk − zk )T (bk (xk ) − bk (zk )) ≤ −ηk |xk − zk |2 , |gh (xh ) − gh (zh )| ≤ Bh |xh − zh |,

CR IP T

And let

| fh (xh ) − fh (zh )| ≤ Ah |xh − zh |, |mh (xh ) − mh (zh )| ≤ Ch |xh − zh |.

(17)

 n o 2 2 2 2 2   max |c |A + σ C + B + |R |, |d |B , h , k,  kh kh kh h h h h  k∈` δkh ,     0, h = k.

AN US

B2. Digraph (G, D) is strongly connected and the following inequality holds:

l l X 1 X 2 v +2 δkh . 2ηk > |ckh | + |dkh | + ςk,2 + 1 − a h=1 kh h=1 h=1 h=1 l X

l X

Proof. For any k ∈ `, denote Vk(2) (t)

2

= |ek | ,

=σ

l Z X

0

−σkh (t)

ED

Vk(1) (ek , t)

M

Then exponential synchronization between drive-respond systems (1) and (2) can be realized under intermittent control (3).

h=1

PT

Then we have

Vk (ek , t) = Vk(1) (ek , t) + Vk(2) (t) = |ek |2 + σ

l Z X h=1

Z

t

2

t+s

0 −σkh (t)

|m ˜ h (eh (θ))| dθds +

Z

t

l Z X h=1

|m ˜ h (eh (θ))|2 dθds +

t+s

t t−τkh (t)

l Z X h=1

|˜gh (eh (θ))|2 dθ.

t t−τkh (t)

|˜gh (eh (θ))|2 dθ.

CE

Obviously, condition A1 is satisfied. Differentiating with respect to time along the trajectories of system (4), when nT ≤ t < (n + θ)T , we can get that

AC

dVk (ek (t), t) dt

=

=

2eTk (t)e0k (t) 2eTk (t) +

l X h=1

+ σ

l Z X h=1

b˜ k (ek (t)) +

vkh

Z

l X

0

−σkh (t)

Z

t

t+s

2

|m ˜ h (eh (θ))| dθds +

ckh f˜h (eh (t)) +

h=1

t

t−σkh (t)

l X h=1

l Z X h=1

t t−τkh (t)

dkh g˜ h (eh (t − τkh (t))) +

! X Z l m ˜ h (eh (s))ds + σσ ˙ kh (t)

t

t−σkh (t)

h=1

12

2

|˜gh (eh (θ))| dθ l X

!0

Rkh eh (t)

h=1

|m ˜ h (eh (θ))|2 dθ + σkh (t)|m ˜ h (eh (t))|2

ACCEPTED MANUSCRIPT

0

!

2

−σkh (t)

|m ˜ h (eh (t + s))| ds +

≤ 2eTk (t)b˜ k (ek (t)) + 2eTk (t) +2eTk (t)

l X

l X

h=1

2

Rkh eh (t) +

ckh f˜h (eh (t)) + 2eTk (t)

2eTk (t)

l X

vkh

h=1

l Z t X

t−σkh (t)

h=1

2

|˜gh (eh (t))| − (1 − τ˙ kh (t))|˜gh (eh (t − τkh (t)))|

h=1

h=1,h,k

−(1 − a)σ

l X

|m ˜ h (eh (θ))|2 dθ +

Z

l X h=1

t

t−σkh (t)

l X h=1

!

dkh g˜ h (eh (t − τkh (t))) + 2eTk (t)Rkk ek (t)

m ˜ h (eh (s))ds +

l X

σσkh (t)|m ˜ h (eh (t))|2

h=1

CR IP T

−

Z

|˜gh (eh (t))|2 .

Applying the following inequality [33], "Z τ #T "Z τ # Z τ w(s)ds G w(s)ds ≤ τ wT (s)Gw(s)ds, 0

0

AN US

0

and the fact that

|b − c|2 = (b − c)T (b − c) = −2bT c + |b|2 + |c|2 , we get l X

vkh

h=1

h=1

t

t−σkh (t)

|m ˜ h (eh (θ))|2 dθ

Z t 2  t  m ˜ h (eh (s))ds − (1 − a) m ˜ h (eh (θ))dθ  t−σkh (t) t−σkh (t) h=1  2  Z t l Z t X   T −2ek (t)vkh − m ˜ h (eh (s))ds + (1 − a) m ˜ h (eh (θ))dθ  t−σkh (t) t−σkh (t) h=1 2 Z t l l X √ 1 1 X 2 − 1 − a m ˜ (e (s))ds − v e (t) vkh |ek (t)|2 + √ h h kh k 1 − a t−σkh (t) 1−a h=1 h=1 l X 1 v2 |ek (t)|2 . 1 − a h=1 kh l X

CE

≤

t−σkh (t)

 Z  T 2ek (t)vkh

m ˜ h (eh (s))ds − (1 − a)σ

l Z X

≤

(20)

ED

=

t

(19)

PT

≤

Z

M

2eTk (t)

(18)

(21)

AC

From (20), the inequality 2aT b < |a|2 + |b|2 holds obviously, then we have 2eTk (t)

l X

h=1,h,k

Rkh eh (t) ≤

l X

h=1,h,k

h i |Rkh | |ek (t)|2 + |eh (t)|2 .

By condition B1, we see that 2eTk (t)

l X h=1

ckh f˜h (eh (t)) ≤

l X h=1

l i X h i 2 ˜ |ckh | |ek (t)| + | fh (eh (t))| = |ckh | |ek (t)|2 + | fh (xh (t)) − fh (zh (t))|2

h

2

h=1

13

(22)

ACCEPTED MANUSCRIPT

≤

h=1

and 2eTk (t)

l X h=1

h i |ckh | |ek (t)|2 + A2h |eh (t)|2 ,

dkh g˜ h (eh (t − τkh (t))) ≤

l X

h i |dkh | |ek (t)|2 + |˜gh (eh (t − τkh (t)))|2

h=1 l X

h i |dkh | |ek (t)|2 + B2h |eh (t − τkh (t))|2 .

h=1 l X

= ≤

(23)

h=1

h i |dkh | |ek (t)|2 + |gh (xh (t − τkh (t))) − gh (zh (t − τkh (t)))|2

CR IP T

l X

Substituting (21)-(24) into the estimate (25), we readily obtain that

(24)

h=1

|ckh |A2h |eh (t)|2

+

l X

σ

2

h=1

Ch2 |eh (t)|2

 l l X X  ≤ − 2ηk − |ckh | − |dkh | − ςk,2 − h=1

=

l X

|Rkh ||eh (t)| +

h=1

v2kh −

l X h=1

l X

2 −σ(1) k |ek (t)|

in which

+

l X

2

h=1

δkh |ek (t − τhk (t))| +

h=1

!

δkh M(Ek , Eh ),

h=1

l X

l

l

1 X 2 X vkh − δkh , 1 − a h=1 h=1 h=1 h=1     2 2 2 M(Ek , Eh ) = |eh (t)| + |eh (t − τkh )| − |ek (t)| + |ek (t − τhk (t))|2 .

CE

σ(1) k = 2ηk −

AC

l X

l X

h=1

B2h |eh (t)|2

 l X  δkh  |ek (t)|2 + δkh |ek (t − τhk (t))|2

δkh |eh (t)|2 − |ek (t)|2 + |eh (t − τkh (t))|2 − |ek (t − τhk (t))|2

ED

h=1

h=1

2

h=1,h,k

1 1−a

PT

+

l X

+

l X

M

+

l X

AN US

dVk (ek (t), t) dt   l l l l l X X X X X   1 2 2     ≤ −2ηk + vkh  |ek (t)| + |ckh | + |dkh | + 2Rkk + |Rkh | + |dkh |B2h |eh (t − τkh (t))|2 1 − a h=1 h=1 h=1 h=1,h,k h=1

|ckh | −

|dkh | − ςk,2 −

Similarly, when (n + θ)T ≤ t < (n + 1)T , we have dVk (ek (t), t) dt

=

2eTk (t)e0k (t)

+ σ

l Z X h=1

=

2eTk (t)

b˜ k (ek (t)) +

l X h=1

0 −σkh (t)

Z

t

t+s

2

|m ˜ h (eh (θ))| dθds +

ckh f˜h (eh (t)) +

l X h=1

l Z X h=1

t t−τkh (t)

dkh g˜ h (eh (t − τkh (t))) + 14

2

|˜gh (eh (θ))| dθ l X h=1

vkh

Z

!0

t

t−σkh (t)

! m ˜ h (eh (s))ds

ACCEPTED MANUSCRIPT

2

t−σkh (t)

|m ˜ h (eh (θ))| dθ + σkh (t)|m ˜ h (eh (t))| −

2

2

|˜gh (eh (t))| − (1 − τ˙ kh (t))|˜gh (eh (t − τkh (t)))|

 l l X X   |dkh | + ≤ −2ηk + |ckh | + h=1

h=1

+

l X h=1

|ckh |A2h |eh (t)|2

+

l X

σ

h=1

2

 l l X X  ≤ 2ηk − |ckh | − |dkh | − h=1

h=1

+

l X h=1

=

2

!

Z

0

2

−σkh (t)

|m ˜ h (eh (t + s))| ds

!

 l l X 1 X 2  2 v  |ek (t)| + |dkh |B2h |eh (t − τkh (t))|2 1 − a h=1 kh  h=1

Ch2 |eh (t)|2 1 1−a

+

l X

l X h=1

v2kh

h=1

−

CR IP T

h=1

σσ ˙ kh (t)

t

B2h |eh (t)|2

l X h=1

 l X  δkh  |ek (t)|2 + δkh |ek (t − τhk (t))|2 h=1

  δkh |eh (t)|2 − |ek (t)|2 + |eh (t − τkh (t))|2 − |ek (t − τhk (t))|2

2 −σ(2) k |ek (t)|

+

l X h=1

in which σ(2) k

2

δkh |ek (t − τhk (t))| + l X

AN US

+

h=1 l X

Z

l X

δkh M(Ek , Eh ),

h=1

l X

l

l

1 X 2 X = 2ηk − |ckh | − |dkh | − v − δkh . 1 − a h=1 kh h=1 h=1 h=1

M

+

l X

ED

Hence, conditions A2 and A3 in Definition 2 are both fulfilled. Moreover, from condition B2, it can be checked easily that Assumption 1 holds. Therefore, we can know that exponential synchronization between derive-response systems (1) and (2) can be achieved. This completes the proof.

CE

4. Numerical examples

PT

Remark 6. Recently, the synchronization of systems with time delay has been widely studied, see [6, 7, 34]. In [6, 7], the authors obtain some sufficient conditions by using Wirtinger-type inequalities. It can relax the assumptions successfully. We will attempt to use the Wirtinger method to deal with the time-varying delay in the future to get better condition.

In this section, we will give two examples to demonstrate the effectiveness of our results.

AC

Example 1. Considering the following drive system: Z t 3 3 3 X X X dxk (t) = bk (xk (t))+ ckh fh (xh (t))+ dkh gh (xh (t−τkh (t)))+ vkh mh (xh (s))ds+ Ik , k ∈ {1, 2, 3}. (25) dt t−σkh (t) h=1 h=1 h=1 The parameters of system (25) are chosen as • c11 = −0.6, c12 = 0.8, c13 = −0.45, c21 = −0.05, c22 = 1.0, c23 = 2.0, c31 = 0, c32 = −1.8, c33 = 0. • d11 = −1.0, d12 = 0, d13 = −0.01, d21 = −0.002, d22 = −0.8, d23 = 0, d31 = −0.035, d32 = −0.8, d33 = 0. 15

ACCEPTED MANUSCRIPT

• v11 = 0.065, v12 = 0.24, v13 = 0.31, v21 = −0.01, v22 = −0.12, v23 = −0.07, v31 = 0.021, v32 = 0.045, v33 = 0.01. Furthermore, assume that bk (xk ) = −0.1xk , fh (xh ) = gh (xh ) = mh (xh ) = tanh(xh ) and Ik = 1 for all k, h = 1, 2, 3. For convenience, we choose τkh (t) = | sin(t)| and σkh (t) = | cos(t)| (k, h = 1, 2, 3) for all t ≥ 0. The response system is expressed by

where

Z

t t−σkh (t)

mh (zh (s))ds + Ik + uk (t),

(26)

CR IP T

3

3

3

X X X dzk (t) vkh ckh fh (zh (t)) + dkh gh (zh (t − τkh (t))) + = bk (zk (t)) + dt h=1 h=1 h=1

 3 X      Rkh (zh (t) − xh (t)) , nT ≤ t < (n + θ)T,  uk (t) =   h=1     0, (n + θ)T ≤ t < (n + 1)T.

(27)

AN US

The initial conditions of systems (25) and (26) are given, respectively, by

x1 (t) = 0.3 cos(π(t − cos(t))) − 0.2, x2 (t) = −0.3 cos(0.5π(t − cos(t))) + 0.2, x3 (t) = −0.2 cos(π(t − cos(t))), z1 (t) = 0.11 cos(0.5π(t − cos(t))), z2 (t) = −0.55 cos(π(t − cos(t))) + 0.2,

z3 (t) = −0.9 cos(0.5π(t − cos(t))),

PT

ED

M

where t ∈ [−1, 0]. By simple calculations, we can get ηk = 0.1, Ah = Bh = Ch = 1 for k, h = 1, 2, 3. Choosing R11 = −10, R22 = −11.5, R33 = −10.5, Rkh = 0 (k, h = 1, 2, 3, k , h). Then ζ = 11.5, ε∗1 = 0.982, ε∗2 = 1.227, and ε∗3 = 0.962 by computation. Hence, we choose ε = 0.962, T = 8. And from Assumption 2 that θ > 0.917. Selecting θ = 0.92. The conditions in Theorem 2 are all fulfilled. Then drive-response systems (25) and (26) are synchronized under intermittent control (27). The synchronization simulation results of systems (25) and (26) are shown in Fig. 3-5. And we also plot the synchronization errors between systems (25) and (26) in Fig. 6. These simulation results confirm the effectiveness of the derived results. 3 2.5

0.5

1.5

AC

x1(z1)

1

0

−1

−1.5

2

0.5

−0.5

−1

0

1

0

−0.5

−2

z2

2

x2(z2)

1.5

x2

2.5

z1

CE

2

3 x1

4

6

8

−1.5

10

t

0

5

10

15

t

Fig. 3: Synchronization curves of x1 and z1 .

Fig. 4: Synchronization curves of x2 and z2 .

16

20

ACCEPTED MANUSCRIPT

1 x3

2

e1 e2

z3

1.5

e3

0.5

1

0 −0.5

0

−0.5

−1 −1.5

−1

−2 −2.5

0

5

10

15

−1.5 −1

20

0

t

CR IP T

ei(i=1,2,3)

x3(z3)

0.5

1

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6

t

Fig. 6: Synchronization errors between systems (25) and (26).

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Fig. 5: Synchronization curves of x3 and z3 .

Example 2. Consider the following delayed coupled oscillators on networks with four nodes as a drive system: Z t 4 4 l X X X tanh(xh (s))ds = 1, k ∈ {1, 2, 3, 4}. x¨k (t)+0.1 x˙k (t)+xk (t)+ ckh 0.1xh (t)+ dkh xh (t−0.5(sin(t)−1))+ vkh h=1

h=1

t−1

h=1

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The parameters of (28) are assumed that

(28)

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• c11 = 0, c12 = 0.01, c13 = 0, c14 = 0.01, c21 = 0.03, c22 = 0, c23 = 0.1, c24 = 0.03, c31 = 0, c32 = 0.22, c33 = 0, c34 = 0.43, c41 = 0.01, c42 = 0.63, c43 = 0.031, c44 = 0. • d11 = 0, d12 = 0.2, d13 = 0.015, d14 = 0.015, d21 = 0.02, d22 = 0, d23 = 0.035, d24 = 0.001, d31 = 0.055, d32 = 0.002, d33 = 0, d34 = 0.001, d41 = 0.01, d42 = 0.06, d43 = 0.01, d44 = 0.

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• v11 = −0.09, v12 = 0.2, v11 = 0.03, v14 = 0.014, v21 = 0.03, v22 = −0.07, v23 = 0.03, v24 = 0.04, v31 = 0.02, v32 = 0.01, v33 = −0.05, v34 = 0.009, v41 = 0.001, v42 = 0.02,v43 = 0.03, v44 = −0.009.

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The corresponding response system is described by z¨k (t) + 0.1˙zk (t) + yk (t) +

4 X h=1

ckh 0.1zh (t) +

4 X h=1

dkh zh (t − 0.5(sin(t) − 1)) +

l X h=1

vkh

Z

t t−1

tanh(zh (s))ds + uk (t) = 1, (29)

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where ckh , dkh and vkh are the same as defined in system (28), and the periodically intermittent controller is designed as  4 X      Rkh (xh (t) − zh (t)) , 8n ≤ t < 8(n + 0.8),  uk (t) =  (30)  h=1     0, 8(n + 0.8) ≤ t < 8(n + 1),

where R11 = −0.9, R12 = 0.002, R11 = 0.0003, R14 = 0.00014, R21 = 0.00011, R22 = −0.9, R23 = 0.0003, R24 = 0.0004, R31 = 0.0001, R32 = 0.012, R33 = −0.8, R34 = 0.0009, R41 = 0.0001, R42 = 0.001, R43 = 0.003, R44 = −0.9. 17

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Consider p = 2, we can know that conditions B1 and B2 in Theorem 2 are all fulfilled by some simple calculations. In addition, What is more, we can get ζ = 1.79919 and ε = 0.9 by computation. It follows from Assumption 2 that θ > 0.4998. Therefore, it can be concluded from Theorem 2 that the drive-response systems (28) and (29) can realize exponential synchronization under periodically intermittent control (30). Additionally, the simulation results are shown in Figs 7-9. These numerical simulations support our theoretical results. 15

10

zk(t) (k=1,2,3,4)

5

0

−5

5

0

−5

0

50

100

150

t

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xk(t) (k=1,2,3,4)

15

x1 x2 x3 x4

−10

0

50

100

z1 z2 z3 z4

150

t

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Fig. 7: The paths of the solution to drive system (28) with iniFig. 8: The paths of the solution to drive system (29) with initial tial values (x1 (t) = 3.1, x2 (t) = 3.5, x3 (t) = −3.2, x4 (t) = 2.4, t ∈ values (z1 (t) = 3.2, z2 (t) = 4.1, z3 (t) = −4.8, z4 (t) = 0.7, t ∈ [−1, 0]) [−1, 0])

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5. Conclusions

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Remark 7. In the previous study of synchronization, the authors mainly used LMI and Lyapunov method [34]. However, the delay-coupled bring great difficulties in setting up linear matrix in equality criteria for the synchronization. Constructing an appropriate Lyapunov function is not an easy thing which is an obstacle to using Lyapunov method. It can be clearly seen that the conditions in Theorem 2 mainly depend on the coefficients of system. That means, one can directly check the synchronization laws by using the coefficients of system. Thus, the derived results in this paper can be much more easily applied in the practice.

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In this paper, the exponential synchronization problem for a class of CSNMTVDs has been discussed via periodically intermittent control. By using graph-theoretic method and Lyapunov function method, some novel exponential synchronization conditions have been obtained. These criteria are given in two different kinds. One is presented in the form of Lyapunov functions and topology property of the system, and the other is given by the coefficients in network. They both show that synchronization has a close relationship with the topology property of networks. It should be especially pointed out that a Lyapunov function for the general CSNMTVDs has been developed in graph theory, which means the difficulty of constructing Lyapunov function is overcome. The method proposed in this paper builds a platform for the study of exponential synchronization of CSNMTVDs under periodically intermittent control. These synchronization criteria also provide some new insight for the CSNMTVDs from the view of topology property of networks. In addition, this is the first time for graph theory to study the exponential synchronization of CSNMTVDs under periodically intermittent control. What is more, 18

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4 e1 e2 e3 e4

3

1 0 −1 −2 −3 −4

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t

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ek(t) (k=1,2,3,4)

2

Fig. 9: The paths of synchronization error.

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numerical simulations are given to verity the effectiveness of the proposed theory. We expect that the coupled systems with periodically intermittent control can gain many valuable applications. In future work, we will consider to extend the proposed method for solving other systems, e.g. stochastic coupled systems, fractional order complex networks and so on. Acknowledgements

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The authors are very grateful to the reviewers for carefully reading the paper and for their valuable comments and suggestions which have improved the paper.

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