Global exponential stability for multi-group neutral delayed systems based on Razumikhin method and graph theory

Available online at www.sciencedirect.com

Journal of the Franklin Institute 355 (2018) 3122–3144 www.elsevier.com/locate/jfranklin

Global exponential stability for multi-group neutral delayed systems based on Razumikhin method and graph theory Ying Guo a,∗, Yida Wang b, Xiaohua Ding b a Department b Department

of Mathematics, Qingdao Technological University, Qingdao 266520, PR China of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, PR China

Received 22 July 2017; received in revised form 9 December 2017; accepted 16 February 2018 Available online 2 March 2018

Abstract This paper is concerned with the global exponential stability for an original class called coupled systems of multi-group neutral delayed differential equations (MNDDEs). By employing Razumikhin method along with graph theory, sufficient conditions are established to guarantee the global exponential stability of MNDDEs, which are in the form of Razumikhin theorem. For the convenience of use, sufficient conditions in the form of coefficients are also obtained. Furthermore, coefficient-type criterion is employed to study the stability of coupled neutral delay oscillators which shows the applicability of our findings. Finally, two numerical examples are given to demonstrate the validity and feasibility of the theoretical results. © 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction It is well known that, in the real world, many phenomena can be modeled by neutral differential equations (NDEs) which play an important role not only in control models [1], but also in population models [2]. Thus, NDEs are widely employed to many areas such as economics, biology, physics, mechanics. Furthermore, many attractive results have been ∗

Corresponding author. E-mail address: [email protected] (Y. Guo).

https://doi.org/10.1016/j.jfranklin.2018.02.010 0016-0032/© 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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obtained (See [3–8] and the references therein). The properties of NDEs have drawn considerable attention of researchers, among which emphasis is placed on the stability that served as a main character of a system. It is taken for granted that the stability analysis for NDEs has earned the interest of an increasing number of scientists [9–12]. As is well known to all that time delays often exist in many practical situations. For example, in control systems, signals transmitting from the sensor to the controller and then from the controller to the actuator, can induce time delays. Again for instance, in epidemiology, there usually exists an interval between a man being infected and getting ill. It is worth noting that time delays can affect the dynamic behaviors of a system, such as instability, oscillation, divergence, and chaos [13–15]. Therefore, considering the effect of time delays to the dynamics of NDEs, neutral delayed differential equations (NDDEs) have been widely studied [16,17]. On the other hand, during the past few years, multi-group models have been an active research topic, because it holds wide applications in lots of fields comprising mathematical ecology and especially in epidemiology [18,19]. For example, multi-group models have been used to describe the dynamic transmission of many infectious diseases in heterogeneous host populations, such as measles, mumps, and HIV/AIDS [20–23]. Multi-group models have started to attract people’s interest since it is more suitable for realistic situations. A lot of investigations about multi-group models have been reported, however, in the scope of authors’ knowledge, there are few researches for multi-group neutral delayed differential equations (MNDDEs). Except Lyapunov method, as an efficient tool to study the stability, Razumikhin method is also commonly used and has an outstanding advantage from the perspective of authors, i.e., we are able to utilize simple functions rather than functionals, which implies the imposed conditions of Razumikhin method have fewer limitations. During the past few years, Razumikhin-type stability theorem for various kinds of single-group models was established, such as delayed models [24,25], discrete models [26], and stochastic models [27,28]. According to the references mentioned above, it is easy to know that Razumikhin method brings a powerful tool to study the stability of delayed differential equations. To the best of authors’ knowledge, Razumikhin method has never been applied to MNDDEs so far. This paper fills this gap. In recent years, coupled systems have been extensively studied and also have widespread applications developed in mechanical, electronic, and biological fields [29–32]. All of these applications mainly depend on the dynamical characteristics. Studying the dynamic behaviors of coupled systems, especially the stability, is one of the dominant themes. The Lyapunov method is a feasible method as well as a significant tool in the theory of stability. However, for special coupled systems, it is quite tough to consider the direct construction of an appropriate Lyapunov function, because their dynamics base on both the individual vertex-dynamics and the coupling topology. Fortunately, a novel approach combined with graph theory was proposed to construct a Lyapunov function, which was firstly applied to the investigation for stability of coupled systems by Li et al.[33,34]. By using this method, the global stability for some classes of coupled systems have been studied successfully, see [35–40]. Due to the growing ties among people in different countries and regions, absolutely independent places are hard to find, so it is the same with ecological communities. For example, dispersal is an important factor determining community composition and community turnover. In order to give a better description of some situations, dispersal should be incorporated into multi-group models. The majority of researches are based on a single digraph

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[41–44], however, when we research a model of multi-group with multi-dispersal, the graphtheoretic method based on a single digraph may be inapplicable any more. So Zhang et al.[45] extended this method to multi-graph and studied the stability of multi-group models with multi-dispersal. Motivated by the discussion above, the main results in this paper will be acquired depending on two principal theories, namely, Razumikhin method and Kirchhoff’s Matrix Tree Theorem in graph theory. By constructing an appropriate Lyapunov function, two different types of sufficient criteria are obtained to ensure the global exponential stability for coupled systems of MNDDEs, which are in the form of Razumikhin theorem and coefficient-type theorem, respectively. The main contributions of this paper are listed below. • •

MNDDEs as a new model is firstly studied according to our survey. Razumikhin method combined with graph theory is applied to discuss the global exponential stability for MNDDEs, and easily verifiable sufficient conditions are given.

The remainder of this paper is outlined as follows. In Section 2, the notations throughout this paper, some preliminary results and our mathematical model of MNDDEs are introduced. In Section 3, the exponential stability for MNDDEs is discussed by using the combination of Razumikhin method and graph theory. Besides, the Razumikhin-type stability theorem for MNDDEs is presented. Then in Section 4, a stability criterion in the form of coefficients is proposed. In Section 5, the stability of coupled neutral delay oscillators is discussed by applying the coefficient-type criterion. In the end, two numerical examples are provided to demonstrate the validity and feasibility of our theoretical results in Section 6. 2. Preliminaries and model description In this section, we first give some useful notations and some necessary knowledge of graph theory in Section 2.1. Then we present the model description, a needful assumption, and the definition of exponential stability for the proposed model in Section 2.2. 2.1. Preliminaries In this paper, we use the following notations for simplification. Write Rn and R1+ for n-dimensional Euclidean space and the set of non-negative real numbers, respectively. Z+ stands for positive integer and G = {1, 2, . . . , g}, where g ∈ Z+ . Let S = {1, 2, . . . , s}, where s ∈ Z+ . τ = max{τ1 , . . . , τg} for τi ∈ R1+ . Let IA be the indicator function of a set of A. The space of continuous functions is denoted by C([−τ, 0]; Rn ). In addition, x : [−τ, 0] → Rn with norm ||x|| = sup−τ ≤s≤0 |x(s)|. |x| denotes the Euclidean norm for vectors x. Let C 1 (Rn ; R1+ ) represent the family of all non-negative functions that are continuously differentiable in x. The transpose of vectors or matrices is denoted by superscript “T”. The following necessary concepts on graph theory can be found in Li et al.[33]. A digraph G = (V, E ) contains a set V = {1, 2, . . . , g} of vertices and a set E of arcs (i, j) leading from initial vertex i to terminal vertex j. A digraph G is weighted if each arc (j, i) is assigned a positive weight aij . Here aij > 0 if and only if there exists an arc from vertex j to vertex i in G, and we call A = (ai j )g×g as the weight matrix whose entry aij equals the weight of arc (j, i) if it exists, and 0 otherwise. A digraph G is strongly connected if, for any pair of distinct vertices, there exists a directed path from one to the other. Denote a digraph with weight

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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. The Laplacian matrix of (G, A ) is defined as ⎛ ⎞ k=1 a1k −a12 ··· −a1n ⎜ −a21 k=2 a2k · · · −a2n ⎟ ⎜ ⎟ L=⎜ ⎟. .. .. .. .. ⎝ ⎠ . . . . −an1

−an2

···

k=n ank

Lemma 1. [33] Suppose g ≥ 2. The cofactor of the kth diagonal element of Laplacian matrix of (G, A ) is denoted by ck . Then the following identity holds: g 

ck akh Fkh (xk , xh ) =

k,h=1

 Q∈Q

W (Q)



Fhk (xh , xk ), 1 ≤ k, h ≤ g.

(k,h)∈E (CQ )

Here the weight in Q is denoted by W(Q), Fkh (xk , xh ) are arbitrary functions, Q represents the set of all spanning unicyclic graphs of (G, A ), and CQ denotes the directed cycle of Q. Particularly, if (G, A ) is strongly connected, then ck > 0 for any k ∈ G. 2.2. Model formulation Consider the following coupled system of MNDDEs:

 (kh) (h) d[xi(k) (t ) − γk xi(k) (t − τk )] = fi(k) x (k) (t ), x (k) (t − τk ), t + Hi (xi (t ) dt h=1 g

−γh xi(h) (t − τh )), k ∈ G, i ∈ S, t ≥ 0,

(1)

 where xi(k) (t ) ∈ Rmi , x (k) (t )  ((x1(k) (t ))T , (x2(k) (t ))T , . . . , (xs(k) (t )T ))T ∈ Rm , i∈S mi = m, mi ∈ Z+ . Both τ k and γ k are non-negative constants, function fi(k) : Rm × Rm × R1+ → Rmi is continuous, which describes the state of the ith component of the kth group. Continuous function Hi(kh) : Rmi → Rmi represents the dispersal from the hth group to the kth group for the ith component. Hi(kh) = 0 if and only if there is no dispersal from the hth group to the kth group for the ith component. In order to obtain the main results, suppose that functions fi(k) and Hi(kh) meet Lipschitz condition. By Theorems 12.2.1 to 12.2.3 in [46] we know that system (1) has a unique solution for each initial state x0 =  ∈ C([−τ, 0]; Rmg ). Moreover the unique solution of system (1) can be denoted by x(t, )  ((x (1) (t, ))T , . . . , (x (g) (t, ))T )T = (x1(1) (t, ))T , (x2(1) (t, ))T , . . . , (xs(1) (t, ))T , . . . , (x1(g) (t, ))T , (x2(g) (t, ))T ,

T . . . , (xs(g) (t, ))T . To prove our result, the following assumption and definition are needed: Assumption 1. Functions fi(k) and Hi(kh) satisfy fi(k) (0, 0, t ) = 0, Hi(kh) (0) = 0. Remark 1. It is not difficult to see that Assumption 1 and Lipschitz condition guarantee that system (1) has a trivial solution x(t, 0) = 0.

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Definition 1. The trivial solution of system (1) is said to be exponentially stable if there exist positive constants γ , K, such that the following condition holds: g s  

|xi(k) (t, )| p ≤ Ke−γ t , t ≥ 0,

i=1 k=1

for some p > 0 and all  ∈ C([−τ, 0]; Rmg ). 3. Razumikhin theorem Based on Razumikhin method and graph theory the exponential stability for system (1) is investigated in this section, and some stability conditions for system (1) are given. In order to facilitate the rest of proof, the definition of vertex-Lyapunov functions set is given as follows: Definition 2. Set {Vi(k) (xi(k) ) ∈ C 1 (Rmi ; R1+ ), k ∈ G, i ∈ S} is said to be vertex-Lyapunov functions set for system (1) if the following conditions hold for any i ∈ S, k, h ∈ G. V1. There exist positive constants αi(k) , βi(k) , and p such that αi(k) |xi(k) | p ≤ Vi(k) (xi(k) ) ≤ βi(k) |xi(k) | p . V2. There exist constant ai(kh) ≥ 0, positive constants q > 1 and σi(k) , and functions Fi (kh) : R × Rmi → R1 , such that mi

s 

ci(k)V˙i(k) (xi(k) )

i



s 

ci(k) (Vi(k) )xk

fi(k) x (k) (t ), x (k) (t



− τk ), t +

i

≤−

s 

 ci(k) σi(k)Vi(k) xi(k) (t ) − γk xi(k) (t − τk )

g 

Hi(kh)

  (h) (h) xi (t ) − γh xi (t − τh )

h=1

i

+

s 

ci(k)

i=1

g 

 ai(kh) Fi (kh) xi(k) (t ) − γk xi(k) (t − τk ), xi(h) (t ) − γk xi(h) (t − τh ) ,

(2)

h=1

for t ≥ 0 and those xi(k) (t ) that satisfy   Vi(k) xi(k) (t − θ ) − γk xi(k) (t − θ − τk ) < qVi (k) xi(k) (t ) − γk xi(k) (t − τk ) , −τ ≤ θ ≤ 0. V3. Along each directed cycle CQi of weighted digraph (G, Ai ), in which A = (ai(kh) )g×g there is   Fi (kh) xi(k) , xi(h) ≤ 0, (3) (h,k)∈E (CQi )

for any xi(k) ∈ Rmi , xi(h) ∈ Rmi . Theorem 1. Suppose that system (1) admits a vertex-Lyapunov functions set {Vi(k) (xi(k) ), k ∈ G, i ∈ S} and digraph (G, Ai ) is strongly connected, in which Ai = (ai(kh) )g×g. If γkp < 21−p ,

(4)

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then the trivial solution of system (1) is exponentially stable. Proof. Define

i(k) (t ) = max



−τ ≤θ≤0

where γ <

min

1≤k≤g,1≤i≤s

 eγ (t+θ )Vi(k) xi(k) (t + θ ) − γk xi(k) (t + θ − τk ) , t ≥ τ,

  lg(q/τ ), σi(k) .

It is clearly that i(k) (t ) is continuous out of the continuity of xi(k) (t ) and Vi(k) (xi(k) (t )). Fix t ≥ τ and define    θ¯i(k) = max −τ ≤ θ ≤ 0 : eγ (t+θ )Vi(k) xi(k) (t + θ ) − γk xi(k) (t + θ − τk ) = i(k) (t ) . (5) It is easy to find that θ¯i(k) ∈ [−τ, 0] such that  ¯ (k) eγ (t+θi )Vi(k) xi(k) (t + θ¯i(k) ) − γk xi(k) (t + θ¯i(k) − τk ) = i(k) (t ). Then, we have a discussion in three cases: θ¯i(k) ∈ (−τ, 0), θ¯i(k) = −τ, and θ¯i(k) = 0. Fix i ∈ S, for simplicity, and write three sets and their corresponding indicator functions as 1i = {i ∈ S : −τ < θ¯i(k) < 0}, 2i = {i ∈ S : θ¯i(k) = −τ }, 3i = {i ∈ S : θ¯i(k) = 0}, and I1i (k) =

 1, 0,

 k ∈ 1i , 1, 2 I (k) = i k ∈¯ 1i , 0,

k ∈ 2i , I3i (k) = 1 − I1i − I2i . k ∈¯ 2i ,

Case 1. If k ∈ 1i , then  (k) ¯ (k) eγ (t+θi )Vi(k) xi(k) (t + θi(k) ) − γk xi(k) (t + θi(k) − τk ) < eγ (t+θi )Vi(k) xi(k) (t + θ¯i(k) )  −γk xi(k) (t + θ¯i(k) − τk ) , θ¯i(k) < θi(k) ≤ 0. By selecting θi(k) = 0, we can obtain that   ¯ (k) eγ t Vi(k) xi(k) (t ) − γk xi(k) (t − τk ) < eγ (t+θi )Vi(k) xi(k) (t + θ¯i(k) ) − γk xi(k) (t + θ¯i(k) − τk ) . In view of the continuity of Vi(k) (xi(k) (t ) − γk xi(k) (t − τk )), we can compute that  lim+ eγ (t+ )Vi(k) xi(k) (t + ) − γk xi(k) (t + − τk ) →0  ¯ (k) < eγ (t+θi )Vi(k) xi(k) (t + θ¯i(k) ) − γk xi(k) (t + θ¯i(k) − τk )



g    t+  (kh) (kh) (k) (k) (h) (h) γr xi (r) − γk xi (r − τk ), xi (r) − γh xi (r − τh ) dr . e ai Fi + lim+ →0

t

h=1

Thus, there exists a sufficiently small 1 > 0 and  eγ (t+ 1 )Vi(k) xi(k) (t + 1 ) − γk xi(k) (t + 1 − τk )  ¯ (k) < eγ (t+θi )Vi(k) xi(k) (t + θ¯i(k) ) − γk xi(k) (t + θ¯i(k) − τk )

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t+ 1

+

e

γr

t

g 

ai(kh) Fi (kh)

  (k) (k) (h) (h) xi (r) − γk xi (r − τk ), xi (r) − γh xi (r − τh ) dr.

(6)

h=1

Case 2. If k ∈ 2i , then   eγ t Vi(k) xi(k) (t ) − γk xi(k) (t − τk ) < eγ (t−τ )Vi(k) xi(k) (t − τ ) − γk xi(k) (t − τ − τk ) . As the discussion that is similar to Case 1, we have  lim+ eγ (t+ )Vi(k) xi(k) (t + ) − γk xi(k) (t + − τk ) →0  < eγ (t−τ )Vi(k) xi(k) (t − τ ) − γk xi(k) (t − τ − τk )



g    t+  (kh) (kh) (k) (k) (h) (h) γr xi (r) − γk xi (r − τk ), xi (r) − γh xi (r − τh ) dr . e ai Fi + lim+ →0

t

h=1

Hence, there exists a sufficiently small 2 > 0 and  eγ (t+ 2 )Vi(k) xi(k) (t + 2 ) − γk xi(k) (t + 2 − τk )  ¯ (k) < eγ (t+θi )Vi(k) xi(k) (t + θ¯i(k) ) − γk xi(k) (t + θ¯i(k) − τk )   t+ 2  g  (kh) (kh) (k) (k) (h) (h) γr xi (r) − γk xi (r − τk ), xi (r) − γh xi (r − τh ) dr. + e ai Fi t

(7)

h=1

Case 3. If k ∈ 3i , then   (k) eγ (t+θi )Vi(k) xi(k) (t + θi(k) ) − γk xi(k) (t + θi(k) − τk ) ≤ eγ t Vi(k) xi(k) (t ) − γk xi(k) (t − τk ) , −τ ≤ θi(k) ≤ 0, that is to say   Vi(k) xi(k) (t + θi(k) ) − γk xik (t + θi(k) − τk ) ≤ eγ t Vi(k) xi(k) (t ) − γk xi(k) (t − τk )  < qVi (k) xi(k) (t ) − γk xi(k) (t − τk ) . With the help of condition V2, we can make sure that there exists a sufficiently small 3 > 0 such that s   eγ (t+ 3 ) ci(k)Vi(k) xi(k) (t + 3 ) − γk xi(k) (t + 3 − τk ) i=1

= eγ t

s 



i=1 t+ 3

+ t

≤ eγ t

 ci(k)Vi(k) xi(k) (t ) − γk xi(k) (t − τk )

s  i=1

eγ r

s  i=1

  ci(k) Vi(k) xi(k) (r) − γk xi(k) (r − τk ) + γ Vi(k) xi(k) (r) − γk xi(k) (r − τk ) dr

 ci(k)Vi(k) xi(k) (t ) − γk xi(k) (t − τk )

Y. Guo et al. / Journal of the Franklin Institute 355 (2018) 3122–3144



t+ 3

+

e

γr

 −

t

+

g  (k)

ci

i=1 s 

+γ

i=1 s 



 ci(k) σi(k)Vi(k) xi(k) (r) − γk xi(k) (r − τk )

i=1

s 

≤ eγ t

s 

3129

 ai(kh) Fi (kh) xi(k) (r) − γk xi(k) (r − τk ), xi(h) (r) − γh xi(h) (r − τh )

h=1

ci(k)Vi(k)

 (k) (k) xi (r) − γk xi (r − τk ) dr

 ci(k)Vi(k) xi(k) (t ) − γk xi(k) (t − τk )

i=1

+

t+ 3

e

γr

t

s 

ci(k)

i=1

g 

 ai(kh) Fi (kh) xi(k) (r) −γk xi(k) (r −τk ), xi(h) (r) −γh xi(h) (r −τh ) dr.

(8)

h=1

By condition V3, Eqs. (6) and (7), it is not difficult to see that for a sufficiently small 0 < < min { 1 , 2 , 3 }, g s  

 ck(i) eγ (t+ )Vi(k) xi(k) (t + ) − γk xi(k) (t + − τk )

i=1 k=1

=

g s  

 ci(k) eγ (t+ )Vi(k) xi(k) (t + ) − γk xi(k) (t + − τk ) (I1i + I2i + I3i )

i=1 k=1

≤

g s  

 ¯ (k) ci(k) eγ (t+θi )Vi(k) xi(k) (t + θ¯i(k) ) − γk xi(k) (t + θ¯i(k) − τk ) I1i

i=1 k=1

   +eγ (t−τ )Vi(k) xi(k) (t − τ ) − γk xi(k) (t − τ − τk ) I2i + eγ t Vi(k) xi(k) (t ) − γk xi(k) (t − τk ) I3i  t+s g g s     + ci(k) eγ r ai(kh) Fi (kh) xi(k) (r) − γk xi(k) (r − τk ), xi(h) (r) − γh xi(h) (r − τh ) dr =

t i=1 k=1 g s   (k) (k) ci i (t ) i=1 k=1 s  t+   γr

+

e

i=1

≤

t

g s  

h=1

W (Qi )

Qi ∈Q i



 Fi (kh) xi(k) (r) −γk xi(k) (r −τk ), xi(h) (r) −γh xi(h) (r −τh ) dr

(h,k)∈E (CQi )

ci(k) i(k) (t ).

i=1 k=1

Therefore, by the definition of i(k) (t ), it can easily deduce that for 0 < < min { 1 , 2 , 3 }, g s   i=1 k=1

ci(k) i(k) (t + ) ≤

g s   i=1 k=1

ci(k) i(k) (t ).

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So, we have

s g 

 g s    (k) (k) 

i(k) (t + ) − i(k) (t ) (k) + D ci i (t )  ci lim sup ≤ 0. →0+ i=1 k=1 i=1 k=1

(9)

Then it follows easily that: g s  

ci(k) i(k) (t ) ≤

g s  

i=1 k=1

ci(k) i(k) (0), t ≥ τ.

i=1 k=1

Together with condition V1 we deduce that g s  

 p   ci(k) αi(k) eγ t xi(k) (t ) − γk xi(k) (t − τk )

i=1 k=1

≤

g s  

 ci(k) eγ t Vi(k) xi(k) (t ) − γk xi(k) (t − τk )

i=1 k=1

≤ =

g s   i=1 k=1 g s  

ci(k) max

−τ ≤θ≤0

  eγ (t+θ )Vi(k) xi(k) (t + θ ) − γk xi(k) (t + θ − τk )

ci(k) i(k) (t )

i=1 k=1

≤ = ≤

g s   i=1 k=1 g s   i=1 k=1 g s  

ci(k) i(k) (τ ) ci(k) max

−τ ≤θ≤0

  eγ (τ +θ )Vi(k) xi(k) (τ + θ ) − γk xi(k) (τ + θ − τk )

 p   ci(k) βi(k) eγ τ xi(k) (θ + τ ) − γk xi(k) (θ + τ − τk ) .

i=1 k=1

By using the following inequality: |a + b| p ≤ 2 p−1 (|a| p + |b| p ), we obtain ci(k) αi(k) eγ t |xi(k) (t )| p ≤ 2 p−1 ci(k) αi(k) eγ t |xi(k) (t ) −γk xi(k) (t −τk )| p + 2 p−1 ci(k) αi(k) eγ t |γk xi(k) (t − τk )| p . (10) Thus, it is not difficult to know that g 

ci(k) αi(k) eγ t |xi(k) (t )| p

k=1

≤ 2 p−1

g  k=1

ci(k) αi(k) eγ t |xi(k) (t ) − γk xi(k) (t − τk )| p + 2 p−1

g  k=1

ci(k) αi(k) eγ t |γk xi(k) (t − τk )| p

Y. Guo et al. / Journal of the Franklin Institute 355 (2018) 3122–3144

≤2

p−1

g   ci(k) βi(k) eγ τ sup |xi(k) (r + τ ) − γk xi(k) (r + τ − τk )| p −τ ≤r≤0

k=1

+ci(k) αi(k) γ p eγ t

sup

−τ ≤r≤t

|xi(k) (r)| p

 .

We can make a development to the inequality above and get another one g s  

≤

ci(k) αi(k) eγ t |xi(k) (t )| p

i=1 k=1 g  s   p−1 ci(k) βi(k) eγ τ sup |xi(k) (r 2 −τ ≤r≤0 i=1 k=1

+ci(k) αi(k) γ p eγ t

sup

−τ ≤r≤t

|xi(k) (r)| p

+ τ ) − γk xi(k) (r + τ − τk )| p

 ,

  where γ = max γ1 , γ2 , . . . , γg . From the fact that the function u(t ) = 2 p−1

g  s   ci(k) βi(k) eγ τ sup |xi(k) (r + τ ) − γk xi(k) (r + τ − τk )| p −τ ≤r≤0

i=1 k=1

+ci(k) αi(k) γ p eγ t

sup

−τ ≤r≤t

|xi(k) (r)| p



is increasing, one can compute that g s  

≤

ci(k) αi(k) eγ t sup |xi(k) (r)| p

−τ ≤r≤t i=1 k=1 g s   2 p−1 ci(k) βi(k) eγ τ sup |xi(k) (r + τ ) − −τ ≤r≤0 i=1 k=1 g s   (k) (k) +2 p−1 γ p ci αi eγ t sup |xi(k) (r)| p . −τ ≤r≤t i=1 k=1

γk xi(k) (r + τ − τk )| p

By using Eq. (4), one can derive that g s  

ci(k) αi(k) eγ t

i=1 k=1

sup

−τ ≤r≤t

|xi(k) (r)| p

≤

2 p−1

s

g

i=1

1−

(k) (k) γ τ k=1 ci βi e 2 p−1 γ p

sup |xi(k) (r + τ )

−τ ≤r≤0

−γk xi(k) (r + τ − τk )| p . Therefore, we have g s  

ci(k) αi(k) |xi(k) (r)| p ≤

2 p−1

i=1 k=1

−γk xi(k) (r + τ − τk )| p e−γ t .

s

g

i=1

1−

(k) (k) γ τ k=1 ci βi e 2 p−1 γ p

sup |xi(k) (r + τ )

−τ ≤r≤0

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Since the digraph (G, Ai ) is strongly connected, it yields that ci(k) > 0. Then, ci(k) αi(k) > 0. Consequently, g s  

ci(k) αi(k) |xi(k) (r)| p ≥ μ

i=1 k=1

g s  

|xi(k) (r)| p ,

i=1 k=1

where μ = min1≤k≤g,1≤i≤s {ci(k) αi(k) }. Set   2 p−1 si=1 gk=1 ci(k) βi(k) eγ τ K= sup |xi(k) (r + τ ) − γk xi(k) (r + τ − τk )| p . μ(1 − 2 p−1 γ p ) −τ ≤r≤0 It is easy to see that K > 0. Now the inequality is equivalent to the following one: g s  

|xi(k) (r)| p ≤ Ke−γ t .

i=1 k=1

By Definition 1, system (1) is exponentially stable. This completes the proof.  Remark 2. A sufficient criterion called Razumikhin theorem that guarantees the exponential stability of MNDDEs is presented in Theorem 1. The conditions of this theorem can be effectively checked in practice. The result shows that the exponential stability of system (1) has a close relationship to the connectedness of digraph (G, Ai ). The proposed model of MNDDEs in the paper takes into account the impacts of time delays and multiple dispersals on NDEs, so it contains some NDEs and non-neutral counterpart in the existing literature as its special cases and some criteria can be obtained when MNDDEs degrade into those systems. For example, MNDDEs will degrade into coupled systems of NDDEs in [47] when we only consider single dispersal, and the corresponding stability criterion will degenerate to Theorem 3.6 in [47]. The authors in [13,15] discussed the stability of delayed neural networks by means of Lyapunov–Krasovskii functional method, however, we investigate the stability of coupled neutral delayed systems by Razumikhin method and graph theory, which avoids the difficulty of finding Lyapunov-Krasovskii functional. In [45], the model of coupled systems with multi-dispersal was established and its stability was analyzed by multi-graph theory and Lyapunov-Krasovskii functional method. Compared with [45], considering both multiple dispersals and time delays, we establish the model of MNDDEs and study its stability by combining Razumikhin method and multi-digraph theory. Hence, the result of this paper has less conservative. Remark 3. It is noted that the stability of some coupled systems can be studied by Lyapunov– Krasovskii functional method [13–15]. In fact, it is possible to extend the proposed theorem with the help of Lyapunov–Krasovskii functional method together with graph theory. The difficulty is to construct a global Lyapunov–Krasovskii functional which is strictly decreasing on the whole range. Compared with the above method, the combined method of Razumikhin method and graph theory in the paper has less restriction, because it is not indispensable to be strictly decreasing on the whole range for the function V, i.e., the function V increasing on the local scale can not exert influence on our main result, and it should be decreasing when it meets certain limit. So it only needs a degression tendency generally. Hence, in Razumikhin-type theorem, it is needed that the derivative of function V is less than zero under the condition of certain restrictions.

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Remark 4. From Theorem 1, in the study of stability for MNDDEs, it is a key point to find a vertex-Lyapunov functions set. In practice, coupled systems may well be too obscure to construct their Lyapunov functions. To make a breakthrough, different fields have restrained certain complications. Like nonlinear dynamics, the simple and nearly identical dynamical systems are coupled together in simple, regular ways. These simplifications make that any problem of structural complexity is restricted and the system’s potentially formidable dynamics could be studied intensively. In many application domains, the Lyapunov functions Vi(k) (xi(k) ) for specific systems have been constructed by other researchers. Therefore, the vertex-Lyapunov functions set for specific MNDDEs can be selected as {Vi(k) (xi(k) )}. Note that if digraph (G, Ai ) is balanced, with the help of some properties in graph theory, then the result will be optimized as follows: n 

ci(k) ai(kh) Fi (kh) (xi(k) , xi(h) ) =

 1  W (Qi ) 2 (h,k)∈E (C Qi ∈Q i

k,h=1



 Fi (kh) (xi(k) , xi(h) ) + Fi (hk) (xi(h) , xi(k) ) .

Qi )

In this case, condition V3 can be replaced by   (kh) (k) (h) Fi (xi , xi ) + Fi (hk) (xi(h) , xi(k) ) ≤ 0.

(11)

(h,k)∈E (CQi )

Consequently, we get the following corollaries: Corollary 1. Suppose that digraph (G, Ai ) is balanced. Then the result of Theorem 1 still holds if Eq. (3) is replaced by Eq. (11). Corollary 2. If for every Fi (kh) (xi(k) , xi(h) ) there exist functions Pi(k) (xi(k) ) and Pi(h) (xi(h) ), such that Fi (kh) (xi(k) , xi(h) ) ≤ Pi(k) (xi(k) ) − Pi(h) (xi(h) ), k, h ∈ G, i ∈ S.

(12)

Then the conclusion of Theorem 1 holds if Eq. (3) is replaced by Eq. (12). 4. Coefficient-type theorem In this section, based on Razumikhin theorem, another sufficient stability criterion in the form of coefficients is proposed, with this handy criterion, the workload of checking can be lessened deeply. Theorem 2. The trivial solution of system (1) is exponentially stable if the following conditions hold for any k, h ∈ G, i, j ∈ S: B1. There is constant Ai(kh) , such that       (kh) (h)   xi (t ) − γh xi(h) (t − τh )  ≤ Ai(kh) xi(h) (t ) − γh xi(h) (t − τh ). Hi B2. The digraph (G, Ui ) is strongly connected, and there exists (ζi(k) ) j , such that

xi(k) (t ) − γk xi(k) (t − τk )

T

s

 (k) 2 fi(k) x (k) (t ), x (k) (t − τk ), t ≤ (ζi(k) ) j |x (k) j (t ) − γk x j (t − τk )| . j=1

B3. It holds that

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s 1 

ci(k) j=1

(k) c (k) j (ζ j )i +

g 

Ai(kh) < 0, γk2 <

h=1

1 . 2



Proof. Let Vi(k) (xi(k) ) = |xi(k) |2 . Write fi(k) = fi(k) x (k) (t ), x (k) (t − τk ), t Hi(kh) (xi(h) (t ) − γh xi(h) (t − τh )). According to conditions B1 and B2 we have

 g s s T    (k) ˙ (k) (k) (k) (k) (k) (kh) ci Vi = 2 ci xi (t ) − γk xi (t − τk ) fi + Hi i=1

i=1

≤2

s 

+2

s 

g   T   (kh)  ci(k) xi(k) (t ) − γk xi(k) (t − τk ) Hi 

i=1

≤2

s 

+2

h=1

s  (k)

ci

i=1

Hi(kh) =

h=1

T ci(k) xi(k) (t ) − γk xi(k) (t − τk ) fi(k)

i=1

and

 2   (k) (ζi(k) ) j x (k) j (t ) − γk x j (t − τk )

j=1 s 



 ci(k) xi(k) (t )

−

γk xi(k) (t

g      (kh)  − τk ) Hi .

i=1

h=1

It is easy to compute that s 

ci(k)

i=1

=

 2   (k) (ζi(k) ) j x (k) j (t ) − γk x j (t − τk )

j=1

s 

c (k) j

j=1

=

s 

s 

 2   (ζ j(k) )i xi(k) (t ) − γk xi(k) (t − τk )

i=1

s 

ci(k) ·

i=1

s 1 

ci(k)

 2  (k)  (k) (k) c (k) (ζ ) (t ) − γ x (t − τ ) x . i k k j j i i

(13)

j=1

Moreover, 2

s 

g        (kh)  ci(k) xi(k) (t ) − γk xi(k) (t − τk ) Hi 

i=1

≤2

s 

h=1

g         ci(k) xi(k) (t ) − γk xi(k) (t − τk ) Ai(kh) xi(h) (t ) − γh xi(h) (t − τh )

i=1

≤

s 

h=1

g  (k)

ci

i=1

g s  2   2      Ai(kh) xi(h) (t ) −γh xi(h) (t −τh ) + ci(k) Ai(kh) xi(k) (t ) −γk xi(k) (t −τk ) .

h=1

From Eqs. (13) and (14), it implies that s  i=1

ci(k)V˙i(k) (xi(k) )

i=1

h=1

(14)

Y. Guo et al. / Journal of the Franklin Institute 355 (2018) 3122–3144

3135

⎛

⎞ g s  2   2   (k) ≤ ci(k) ⎝ (k) c (k) Ai(kh) ⎠xi(k) (t ) − γk xi(k) (t − τk ) j (ζ j )i + ci j=1 i=1 h=1  g s 2     (k) (kh)  (h) (h) + ci Ai xi (t ) − γh xi (t − τh ) s 

i=1

h=1

⎛

⎞ g s  2   1  (k) (kh) ⎠ (k) (k) ≤2 ci(k) ⎝ (k) c (k) (ζ ) + A (t ) − γ x (t − τ ) x i k i k  j j i i c i i=1 j=1 h=1  g s 2  2   (k)  (kh)  (h)    (k) (h) (k) + ci Ai (t ) − γ x (t − τ ) − (t ) − γ x (t − τ ) xi xi h i h  k i k  s 

=−

i=1 s 

h=1

g s  2     ci(k) σi(k) xi(k) (t ) − γk xi(k) (t − τk ) + ci(k) ai(kh) Fi (kh) xi(k) (t ) − γk xi(k) (t − τk ),

i=1

xi(h) (t )

−

γh xi(h) (t

 − τh ) ,

i=1

h=1

where ⎛ σi(k)

⎞ g s    1 (k) (kh) ⎠ (kh) (kh) (kh) (k) (h) x = |xi(h) | − |xi(k) |. = −⎝ (k) c (k) (ζ ) + A , a = A , F , x i j j i i i i i i ci j=1 h=1

Till now, a conclusion can be made that conditions V1, V2, and V3 in Definition 2 have been met, which implies that {Vi(k) (xi(k) ), k ∈ G, i ∈ S} is a vertex-Lyapunov functions set for system (1). Together with conditions B2 and B3, all conditions of Corollary 2 are satisfied. This completes the proof.  Remark 5. On the basis of Razumikhin theorem, we establish a coefficient-type stability criterion. Compared with Theorem 1, the conditions in Theorem 2 are more convenient to verify, because they are expressed in the form of the coefficients of system (1), instead of the vertex-Lyapunov functions. In addition, it is easy to find from condition B2 that the exponential stability of system (1) is closely related to the topological structure of n digraphs (G, Ui ), i ∈ S. 5. An application to coupled neutral delay oscillators In this section, for the aim of applicability, the theoretical result will be applied to analyze the stability of coupled neutral delay oscillators. It is noted that a delay oscillator of the form x¨(t ) + ε x˙(t ) + x(t ) = 0, t ≥ 0, where ε ≥ 0 is the damping coefficient, is one of the commonly used models in many fields, such as physics [48], mechanics [49], and engineering [50]. To match our model, l neutral delay oscillators are taken into account and the kth neutral delay oscillator can be described as



x¨(k) (t ) − γk x¨(k) (t − τk ) + εk x˙(k) (t ) − γk x˙(k) (t − τk ) + x (k) (t ) − γk x (k) (t − τk ) = 0, t ≥ 0, (15)

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where γ k and τ k are non-negative constants. Let y (k) (t ) = x˙(k) (t ) + ηx (k) (t ), where η is a constant. Then, rewrite system (15) as ⎧

d[x (k) (t ) − γk x (k) (t − τk )] ⎪ ⎪ =y (k) (t ) − γk y (k) (t − τk ) − η x (k) (t ) − γk x (k) (t − τk ) , ⎪ ⎪ dt ⎨ 

 d[y (k) (t ) − γk y (k) (t − τk )] ⎪ = − εk y (k) (t ) − γk y (k) (t − τk ) − η x (k) (t ) − γk x (k) (t − τk ) ⎪ ⎪ dt ⎪

⎩ − x (k) (t ) − γk x (k) (t − τk ) , k ∈ G. Next, by coupling each differential equation of the system above on two digraphs (G, A ) and (G, B), respectively, where coupling matrices A = (akh )g×g and B = (bkh )g×g, we construct coupled neutral delay oscillators as below: ⎧ (k)   d[x (t )−γk x (k) (t−τk )] ⎪ = y (k) (t ) − ηx (k) (t ) − γk y (k) (t − τk ) − ηx (k) (t − τk ) ⎪ dt ⎪  ⎪ ⎪ + lh=1 akh (x (h) (t ) − γh x (h) (t − τh )), ⎨ 

 (k) (k) d[y (t )−γk y (t−τk )] (16) = −εk y (k) (t ) − γk y (k) (t − τk ) − η x (k) (t ) − γk x (k) (t − τk ) dt ⎪

 ⎪ l (k) (k) (h) ⎪ − x (t ) − γk x (t − τk ) + h=1 bkh (y (t ) ⎪ ⎪ ⎩ −γh y (h) (t − τh )), k ∈ G. Let X (k) (t ) = (x (k) (t ), y (k) (t ))T and T F (k) (X (k) (t ), X (k) (t − τk ), t ) = f1(k) (X (k) (t ), X (k) (t − τk ), t ), f2(k) (X (k) (t ), X (k) (t − τk ), t ) , in which

f1(k) (X (k) (t ), X (k) (t − τk ), t ) = y (k) (t ) − γk y (k) (t − τk ) − η x (k) (t ) − γk x (k) (t − τk ) and 

 f2(k) (X (k) (t ), X (k) (t − τ (t )), t ) = −εk y (k) (t ) − γk y (k) (t − τk ) − η x (k) (t ) − γk x (k) (t − τk )

− x (k) (t ) − γk x (k) (t − τk ) . Meanwhile, H1(kh) (x1(h) (t ) − γh x1(h) (t − τh )) = akh (x (h) (t ) − γh x (h) (t − τh )), H2(kh) (x2(h) (t ) − γh x2(h) (t − τh )) = bkh (y (h) (t ) − γh y (h) (t − τh )), where akh ≥ 0 and bkh ≥ 0. It is noted that akh = 0 if and only if there is no influence from x(h) to x(k) and bkh = 0 if and only if there is no influence from y(h) to y(k) . For the better understanding, a figure is drawn for system (16) when g = 4 (see Fig. 1). In this case, a14 = a24 = a41 = a42 = 0 and b13 = b14 = b31 = b41 = 0. In the remainder of this section, the global exponential stability of system (16) shall be proved. Theorem 3. Suppose that the following conditions hold for any k ∈ G. Then the trivial solution to system (16) is exponentially stable. C1. It holds that γk2 < 21 and g  h=1

akh <

g c1(k) (η − 1) − c2(k) |ηεk − 1|  −c1(k) + c2(k) (2εk − |ηεk − 1| ) , b < . kh 2c1(k) 2c2(k) h=1

C2. Digraphs (G, A ) and (G, B) are both strongly connected.

Y. Guo et al. / Journal of the Franklin Institute 355 (2018) 3122–3144

a21 x1

a31

a13

y1

a23 b32

b21 y2 b12

x4 a34

b42

b43

y3 b23

Group 2

a43 x3

x2 a12

Group 1

a32

3137

b24 Group 3

y4 b34

Group 4

Fig. 1. A sample description of the structure of system (16) with l = 4.

Proof. By fundamental inequality, we can derive that (k)

x (t ) − γk x (k) (t − τk ) f1(k) (X (k) (t ), X (k) (t − τk ), t )   (k)

(k)

(k) (k) (k) (k) = x (t ) − γk x (t − τk ) y (t ) − γk y (t − τk ) − η x (t ) − γk x (t − τk ) 1 1 − η (k) | x (t ) − γk x (k) (t − τk ) |2 + | y (k) (t ) − γk y (k) (t − τk ) |2 2 2 and (k)

y (t ) − γk y (k) (t − τk ) f2(k) (X (k) (t ), X (k) (t − τk ), t ) 



= y (k) (t ) − γk y (k) (t − τk ) − εk y (k) (t ) − γk y (k) (t − τk ) − η x (k) (t ) − γk x (k) (t − τk ) 

− x (k) (t ) − γk x (k) (t − τk )



= −εk | y (k) (t ) −γk y (k) (t −τk ) |2 +(ηεk −1 ) y (k) (t ) −γk y (k) (t −τk ) x (k) (t ) −γk x (k) (t −τk ) |ηεk − 1|  (k) ≤ −εk | y (k) (t ) − γk y (k) (t − τk ) |2 + | y (t ) − γk y (k) (t − τk ) |2 2  + | x (k) (t ) − γk x (k) (t − τk ) |2 |ηεk − 1| (k) |ηεk − 1| − 2εk = | x (t ) − γk x (k) (t − τk ) |2 + | y (k) (t ) − γk y (k) (t − τk ) |2 . 2 2 ≤

So, condition B2 in Theorem 2 is satisfied with (ζ1(k) )1 = 1−η , (ζ1(k) )2 = 21 , (ζ2(k) )1 = 2 (k) |ηεk −1| |ηεk −1|−2εk , (ζ2 )2 = . Obviously, the condition B1 in Theorem 2 is fulfilled. This, to2 2 gether with conditions C1 and C2, meets all the conditions of Theorem 2. Hence, system (16) is exponentially stable, which completes the proof.  Remark 6. Generally, by estimating the sign of the real part of eigenvalues, one can determine the stability of linear systems. However, it is rather difficult to calculate the eigenvalues of

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x1

a43

a21 x2

a32

y1

x4

a14

b41

b34

b12 y2

x3

y4

b23

y3

Fig. 2. Two digraphs of four coupled neutral delay oscillators in Example 1: (G, A ) (left) and (G, B) (right).

coupled systems, since such systems always have the very large dimension. In the paper, Theorem 3 provides a novel kind of sufficient criterion to determine the stability of delayed linear system (16) by graph theory. Remark 7. In recent years, many researchers have investigated such coupled oscillators and stochastic counterpart for their widespread applications, and some results about stability of these systems were reported [34,35,45]. In [34,35], the authors established coupled oscillators on a single digraph. In [45], coupled oscillators without considering time delays were studied by multi-digraph theory. Compared with the above literatures, taking into account the impacts of time delays and multiple dispersals, we construct coupled neutral delay oscillators (16) and obtain sufficient stability conditions which closely relate to the connectedness of two digraphs. 6. Numerical examples In this section, we provide two numerical examples to illustrate the effectiveness of the proposed results in this paper. Example 1. Let g = 4. Consider coupled neutral delay oscillators (16) on two digraphs (G, A ) and (G, B) in Fig. 2. Weighted matrixes A and B are defined as below: ⎛ ⎞ ⎛ ⎞ 0 0 0 0.1 0 0.2 0 0 ⎜0.1 ⎜ 0 0 0⎟ 0 0.1 0⎟ ⎟ , B=⎜ 0 ⎟. A=⎜ ⎝0 ⎝0 0.1 0 0⎠ 0 0 0.1⎠ 0 0 0.1 0 0.2 0 0 0 It is easy to calculate that c1(1) = c1(2) = c1(3) = c1(4) = 0.0001, c2(1) = c2(4) = 0.0002, c2(2) = c2(3) = 0.0004. Select parameters as ε1 = ε4 = 0.6, ε2 = ε3 = 0.45, η = 2. In addition, set τ = τk = 0.1, γk = 0.7, k = 1, 2, 3, 4. Then we can compute that γk2 < 0.5 and 0.1 =

g 

a1h <

h=1 g

0.1 =



a2h <

h=1

0.1 =

g  h=1

a3h <

c1(1) (η − 1) − c2(1) |ηε1 − 1| 2c1(1)

c1(2) (η − 1) − c2(2) |ηε2 − 1| 2c1(2)

c1(3) (η − 1) − c2(3) |ηε1 − 1| 2c1(3)

= 0.3, = 0.3,

= 0.3,

Y. Guo et al. / Journal of the Franklin Institute 355 (2018) 3122–3144

3139

3

2

x(1)

1

y(1)

0

x

(2)

y

(2)

x(3) y(3) x(4)

−1

y(4)

−2

−3

0

2

4

6

8

10

12

14

16

18

20

Fig. 3. The solution of coupled neutral delay oscillators (16) when g = 4.

0.1 =

g 

a4h <

h=1

0.2 =

g 

b1h <

h=1 g

0.1 =



b2h <

h=1

0.1 =

g 

b3h <

h=1 g

0.2 =

 h=1

b4h <

c1(4) (η − 1) − c2(4) |ηε2 − 1| 2c1(4)

= 0.6,

−c1(1) + c2(1) (2ε1 − |ηε1 − 1| ) 2c2(1)

−c1(2) + c2(2) (2ε2 − |ηε2 − 1| ) 2c2(2)

−c1(3) + c2(3) (2ε3 − |ηε3 − 1| ) 2c2(3)

−c1(4) + c2(4) (2ε4 − |ηε4 − 1| ) 2c2(4)

= 0.25, = 0.275,

= 0.275, = 0.25,

which implies condition C1 is satisfied. Clearly, (G, A ) and (G, B) are strongly connected. So, all conditions in Theorem 3 are checked. Therefore, the trivial solution of system (16) is exponentially stable. The corresponding simulation result is shown in Fig. 3 with taking the initial value as (t ) = (3, 2, −2, 1, −2.5, 1.5, 1.5, −1)T , t ∈ [−0.1, 0]. Example 2. Consider the following coupled system of MNDDEs: 5 

 d[xi(k) (t ) − γk xi(k) (t − τk )] = fi(k) x (k) (t ), x (k) (t − τk ), t + Hi(kh) xi(h) (t ) −γh xi(h) (t − τh ) , dt h=1

i = 1, 2, 3, k = 1, 2, 3, 4, 5 , t ≥ 0,

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where xi(k) (t ) ∈ R, τk ≥ 0, γk ≥ 0, Hi(kh) (y) = ui(kh) y which meet condition B1 obviously. Let yi = xi(k) (t ) − γk xi(k) (t − τk ). And set functions as  y13 − ay(k)1 , y1 ≤ 1,

(k) (k) (k) 1 f1 x (t ), x (t − τk ), t = F1 (y1 ) = −b(k) y1 > 1, 1 y1 ,  y3

− y2 − a(k)2 , y2 ≤ 1, (k) (k) (k) f2 x (t ), x (t − τk ), t = F2 (y2 ) = 2 3 −b(k) y2 > 1, 2 y2 ,  (k)

−a3 y3 , y3 ≤ 1, f3(k) x (k) (t ), x (k) (t − τk ), t = F3 (y3 ) = 2 −b(k) y3 > 1, 3 y3 , where ai(k) ≥ 0, b(k) i ≥ 0. Then, we fix attention on the following coupled system of MNDDEs: 5   d[xi(k) (t ) − γk xi(k) (t − τk )] = Fi (yi ) + Hi(kh) xi(h) (t ) − γh xi(h) (t − τh ) , dt (18) h=1 i = 1, 2, 3, k = 1, 2, 3, 4, 5. Choose τ = τk = 0.1, γk = 0.6. Suppose that the weighted matrices ⎛ ⎞ ⎛ 0 1.6 0 0 0 0 ⎜0.5 ⎟ ⎜0.1 0 1 0 0 ⎜ ⎟ ⎜   ⎜0 ⎜0 0.6 0 0.3 0⎟ (kh) (kh) ⎜ ⎟ u1 =⎜ , u2 =⎜ ⎟ ⎜0 0 0 0. 2 0 1 . 5 5×5 5×5 ⎜ ⎟ ⎜ ⎝0 ⎝0 0 0 1.2 0⎠ ⎛

 u3(kh)

5×5

0 ⎜0.1 ⎜ ⎜0.1 =⎜ ⎜0 ⎜ ⎝0

Ui of system (18) are ⎞ 0.4 0 0 0 0 0.1 0.3 0⎟ ⎟ 0.2 0 0.3 0⎟ ⎟, 0.1 0.1 0 0.2⎟ ⎟ 0 0 0.5 0⎠ 0.1 0 0.1 0 0

0.1 0.1 0 0.1 0

0 0 0.1 0 0.1

⎞ 0 0⎟ ⎟ 0⎟ ⎟. 0.1⎟ ⎟ 0⎠

It is clearly that (G, Ui ) is strongly connected (see Fig. 4). For simulation, we assume that the parameters in system (18) are given as follows: a1(1) = a1(2) = a1(3) = 13 , a1(4) = a1(5) = 41 a2(1) = a2(2) = a2(3) = 3, a2(4) = a2(5) = 5 a3(1) = a3(2) = a3(3) = 23 , a3(4) = a3(5) = 56

(2) (3) (4) (5) b(1) 1 = b1 = b1 = 2, b1 = b1 = 3 (1) (2) (3) (4) 7 4 b2 = b2 = b2 = 10 , b2 = b(5) 2 = 5 (1) (2) (3) (4) (5) 1 3 b3 = b3 = b3 = 2 , b3 = b3 = 10

With the above parameters, for k = 1, 2, 3, 4, 5, select  1 (k) (k) (k) (ζ1 )1 = max −b(k) 1 , 1 − (k) , (ζ1 )2 = (ζ1 )3 = 0, a1  1 (ζ2(k) )2 = max −1 + (k) , −b(k) , (ζ2(k) )1 = (ζ2(k) )3 = 0, 2 a2

Y. Guo et al. / Journal of the Franklin Institute 355 (2018) 3122–3144

u(21) 1

u(32) 1

x1

u(43) 1

u(12) 1

u(23) 1

u(21) 2 y2 u(12) 2 u(21) 3 z1

u(23) 2

Group 1

u(45) 1

u(43) 2

u(54) 2 y4 u(45) 2

u(43) 3

u(54) 3

z3

Group 2

z5

z4

u(23) 3

u(13) 3

y5

u(34) 2

u(24) 2

u(32) 3

z2 u(12) 3

u(34) 1

y3

u(31) 3

x5

x4

u(42) 2

u(32) 2

y1

u(54) 1

x3

x2

3141

u(34) 3 Group 3

u(45) 3 Group 4

Group 5

Fig. 4. A structure for system (18) in five groups with three weighted digraphs. 1

x(1) 1 (2)

x1

0.8

x(3) 1 (4)

0.6

x1

(5)

x1

0.4

(1)

x2

x(2)

0.2

2

x(3) 2 (4) 2 x(5) 2 x(1) 3 x(2) 3 (3) x3 (4) x3 x(5) 3

0

x

−0.2 −0.4 −0.6 −0.8 −1

0

2

4

6

8

10

12

14

16

18

20

Fig. 5. Numerical solution of system (18) with initial value (19).

  (ζ3(k) )3 = max −a3(k) , −b(k) , (ζ3(k) )1 = (ζ3(k) )2 = 0. 3 Then we can easily calculate that (ζ1(1) )1 = (ζ1(2) )1 = (ζ1(3) )1 = −2, (ζ1(4) )1 = (ζ1(5) )1 = −3, (ζ2(1) )2 = (ζ2(2) )2 = (ζ2(3) )2 = − 23 , (ζ2(4) )2 = (ζ2(5) )2 = − 45 , (ζ3(1) )3 = (ζ3(2) )3 = (ζ3(3) )3 =  3 − 21 , (ζ3(4) )3 = (ζ3(5) )3 = − 10 . It is not difficult to check that (ζ1(k) )1 < − 5h=1 u1(kh) , (ζ2(k) )2 <   − 5h=1 u2(kh) , (ζ3(k) )3 < − 5h=1 u3(kh) , k = 1, 2, 3, 4, 5. This, together with γk2 = 0.36 < 0.5, shows that condition B3 in Theorem 2 is fulfilled. Therefore, all the conditions in

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Theorem 2 are met. Moreover, for t ∈ [−0.1, 0], we select the following initial value: x1(1) (t ) = − sin 1.01, x1(2) (t ) = cos 0.01, x1(3) (t ) = cos 0.01 − 1.5, x1(4) (t ) = cos 0.49, x1(5) (t ) = sin 0.01 + 0.11, x2(1) (t ) = cos 0.61, x2(2) (t ) = cos 0.01 − 1.7, x2(3) (t ) = 0.5 sin 1.01, x2(4) (t ) = −0.5 cos 0.91, x2(5) (t ) = 0.3 sin 0.005, x3(1) (t ) = − sin 0.29, x3(2) (t ) = − sin 1.01, x3(3) (t ) = sin 0.51, x3(4) (t ) = cos 0.7 − 1.5, x3(5) (t ) = sin 1.405.

(19)

The trivial solution of system (18) is presented in Fig. 5, which clearly demonstrates that the trivial solution of system (18) is exponentially stable. Therefore, the numerical example shows the validity and feasibility of our developed results. 7. Conclusions In this paper, the exponential stability of MNDDEs was studied, based on the method of combining Razumikhin method with graph theory. Two types of sufficient criteria were given to determine exponential stability of MNDDEs, which are Razumikhin-type theorem and the sufficient conditions in the form of coefficients. Then, the coefficient-type criterion was applied to study the stability of coupled neutral delay oscillators which showed the applicability of the theoretical results. Finally, two numerical examples were given to illustrate the effectiveness of our main results. Our method in this paper can be employed to research coupled systems with other delays such as time-varying delay, infinite delay, and mixed delays. Moreover, with the method in this paper, it is also feasible to study exponential stability of stochastic delayed coupled systems. Acknowledgements This work was supported by grants from NNSF of Shandong Province of China (Nos. ZR2017MA008 and ZR2017BA007), National Natural Science Foundation of China (No. 11501150), the Project of Shandong Province Higher Educational Science and Technology Program of China (No. J16LI09). The authors really appreciate the reviewers’ valuable comments. Thank Pengfei Wang for helping us to improve the paper and do the numerical simulations. References [1] J.H. Park, Robust guaranteed cost control for uncertain linear differential systems of neutral type, Appl. Math. Comput. 140 (2003) 523–535. [2] X. Wei, J. He, Research of population with impulsive perturbations based on dynamics of a neutral delay equation and ecological quality system, Saudi. J. Biol. Sci. 23 (2016) S78–S82. [3] G. Fusco, N. Guglielmi, A regularization for discontinuous differential equations with application to state-dependent delay differential equations of neutral type, J. Differ. Equ. 250 (2011) 3230–3279. [4] B. Zhong, Z. Yang, New approach of studying the oscillatory of neutral differential equations, Funkc. Ekvac. 41 (1998) 79–89. [5] W. Wang, Y. Zhang, S. Li, Stability of continuous Runge–Kutta-type methods for nonlinear neutral delay-differential equations, Appl. Math. Model. 33 (2009) 3319–3329.

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