The existence of periodic solutions for coupled systems on networks with time delays

Neurocomputing 152 (2015) 287–293

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The existence of periodic solutions for coupled systems on networks with time delays Xinhong Zhang a,b, Wenxue Li a,n, Ke Wang a a b

Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, PR China College of Science, China University of Petroleum (East China), Qingdao 266555, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 7 May 2014 Received in revised form 13 August 2014 Accepted 29 October 2014 Communicated by H. Jiang Available online 15 November 2014

This paper is concerned with the existence of periodic solutions for coupled systems on networks with time delays (CSND). By the combined method of graph theory, coincidence degree theory and Lyapunov method, a systematic approach for the existence of periodic solutions to CSND is developed. We apply this approach to a coupled system of nonlinear oscillators with time delays and obtain the existence of periodic solutions. Finally, a numerical example is provided to illustrate the effectiveness of the results developed. & 2014 Elsevier B.V. All rights reserved.

Keywords: Periodic solutions Graph theory Coincidence degree theory Coupled systems on networks

1. Introduction The existence of periodic solutions of ordinary and functional differential equations has been studied extensively due to its universal importance. And coincidence degree theory, the upper and lower solutions method, fixed point theorem, bifurcation theory, and Lyapunov method are often used to prove the existence of periodic solutions. In [1–8], by applying continuation theorem of coincidence degree, a large amount of sufficient conditions have been derived to guarantee the existence of periodic solutions of many kinds of systems. On the other hand, coupled systems on networks have attracted great attention in biological systems, artificial neural networks, internet and social networks, complex ecosystems, the spread of infectious diseases, and so on [9–14]. Based on graph theory, coupled systems on networks can be described by a directed graph, in which each vertex represents an individual system called vertex system and the directed arcs stand for the inter-connections and interactions among vertex systems [15]. In recent years, the dynamical behaviors of coupled systems on networks have been widely studied, and many good results have been reported. In [12–14,16–18], the synchronization problems of the complex networks have been considered. By applying graph-theoretic approach, papers [15,19–23] studied the global stability for different kinds of coupled systems on networks, and provided systematic methods for constructing the global Lyapunov functions.

At the same time, the periodicity of coupled systems has also attracted great interests of scientists, and lots of results have been reported, see [24–32] and the references therein. And in these papers, by constructing suitable Lyapunov functionals, applying coincidence degree theory and some analysis techniques, sufficient conditions for the existence of the periodic solutions to different neural networks are obtained. However, few papers combined with the topology of the network. Therefore, in this paper, by using graph-theoretic approach, the existence of periodic solutions of coupled systems on networks with time delays (CSND) is investigated. Given a digraph G with l ðl Z2Þ vertices, assume that each vertex dynamics is described by differential equations with time delays x_ k ðtÞ ¼ f k ðxk ðtÞ; xk ðt  τk Þ; tÞ;

l

x_ k ðtÞ ¼ f k ðxk ðtÞ; xk ðt  τk Þ; tÞ þ ∑ g kh h¼1

ðxk ðt  τk Þ; xh ðt  τh Þ; tÞ; m

n

Corresponding author. Tel.: þ 86 631 5687035; fax: þ86 631 5687572. E-mail addresses: [email protected] (X. Zhang), [email protected] (W. Li). http://dx.doi.org/10.1016/j.neucom.2014.10.067 0925-2312/& 2014 Elsevier B.V. All rights reserved.

1 r k r l;

where xk ðtÞ ¼ ðxk1 ðtÞ; xk2 ðtÞ; …; xkm ðtÞÞT , τk stands for the time delay in the k-th subsystem, f k ¼ ðf k1 ; f k2 ; …; f km ÞT : Rm  Rm  R þ -Rm is a continuous function, and f k ð; ; tÞ ¼ f k ð; ; t þ TÞ for some T 4 0, R þ ¼ ½0; þ 1Þ. If we assume that the influence of the h-th subsystem on the k-th subsystem is described by g kh ðxk ðt  τk Þ; xh ðt  τh Þ; tÞ, g kh ¼ 0 if and only if there exists no influence from h-th subsystem to k-th subsystem, then we can obtain a kind of CSND below

m

1 rk; h r l; 0 r τk ; τh o 1;

m

ð1Þ

where g kh : R  R  R þ -R is continuous, g kh ð; ; tÞ ¼ g kh ð; ; t þ TÞ. The initial value of system (1) is usually given by xðtÞ ¼ ϕðtÞ;

288

X. Zhang et al. / Neurocomputing 152 (2015) 287–293

where ϕðtÞ ¼ ðϕ1 ðtÞ; ϕ2 ðtÞ; …; ϕl ðtÞÞT is a continuous vector function on ½  τ; 0, in which τ ¼ maxfτ1 ; τ2 ; …; τl g. The dynamics of system (1) depends on not only the individual vertex dynamics but also the coupling topology, meanwhile, system (1) exhibits a great of complexity, which are mainly caused by the nonlinearity, periodic coefficients and time delays. Hence, studying such system is meaningful and challenging. In this paper, we mainly focus on the existence of periodic solutions to system (1). By employing graph theory, Lyapunov method and coincidence degree theory, a systematic approach for the existence of periodic solutions of system (1) is developed. The methods of estimating the bound of periodic solutions to the auxiliary equation of system (1) are Lyapunov method and graph theory, which are different from that in the above cited papers. We will show that our approach can be applied to coupled oscillators on a network, a numerical example is given to illustrate the correctness of the developed theory. The contributions and novelties of the current work are as follows: 1. Graph theory, Lyapunov method and coincidence degree theory are combined together to study existence of periodic solutions of coupled systems on networks. 2. A systematic approach for the existence of periodic solutions to general system (1) is obtained. 3. This result is applied to nonlinear coupled oscillators on a network with time delays, and the sufficient conditions for the existence of periodic solutions are derived. The paper is organized as follows: In the following section, we give some useful lemmas. Our main results are presented in Section 3. Then in Section 4, our results are applied to coupled oscillators on a network to demonstrate their applicability and effectiveness. Finally, a numerical example is given to illustrate the correctness of the developed theory.

2. Main lemmas The following process is standard and motivated by [5]. Let X ¼ Z ¼
fx A CðR; Rml Þ : xðt þ TÞ ¼ xðtÞg be equipped with the norm 2 1=2 J x J ¼ ∑lk ¼ 1 maxt A ½0;T ð∑m . Then X and Z are Banach i ¼ 1 jxki ðtÞj Þ spaces. Set !T uk ðtÞ ¼

l

f k ðxk ðtÞ; xk ðt  τk Þ; tÞ þ ∑ g kh ðxk ðt  τk Þ; xh ðt  τh Þ; tÞ h¼1

:

Define an operator L in the following form: L : Dom L  X-X;

Lx ¼ x0

and N : X- X; Nx ¼ ðu1 ; u2 ; …; ul ÞT :

n o It is not difficult to show that Ker L ¼ x A X : x ¼ c A Rlm , RT Im L ¼ fz A Z : 0 zðtÞ dt ¼ 0g is closed in Z, and Dim Ker L ¼ ml ¼ Codim Im L: Thus, the operator L is a Fredholm mapping of index 0. Let project operators P and Q in the following form, respectively: Z Z 1 T 1 T xðtÞ dt; x A X; Qz ¼ zðtÞ dt; z A Z: Px ¼ T 0 T 0 Hence, Im P ¼ Ker L;

Im L ¼ Ker Q ¼ ImðI  Q Þ:

Furthermore, the generalized inverse (of L) K p : Im L-Ker P \ Dom L reads as Z Z Z t 1 T t zðsÞ ds  zðsÞ ds dt: K p ðzÞ ¼ T 0 0 0 We now compute  Z T T Z Z 1 1 T 1 T QNx ¼ u1 ðtÞ dt; u2 ðtÞ dt; …; ul ðtÞ dt T 0 T 0 T 0 and

0Rt

B 0 K p ðI  Q ÞNx ¼ B @R t 0

uT1 ðsÞ ds  T1

RT Rt 0

0

R R 1 T t

uTl ðsÞ ds  T

0

0

uT1 ðsÞ ds dt  ⋮ uTl ðsÞ ds dt 


t

1 T 2


t

RT 0

RT 1

T 2

0

uT1 ðtÞ dt uTl ðtÞ dt

1 C C: A

Clearly, QN and K p ðI  Q ÞN are continuous and QNðΩ Þ is bounded, where

Ω is an open set in X. Then by Arzela–Ascoli theorem, we see

that K p ðI  Q ÞNðΩ Þ is compact. Hence, N is L-compact on Ω . In our proof we will use the continuation theorem of coincidence degree and graph theory. For the sake of convenience, we introduce the results concerning the coincidence degree and graph theory as follows. For more details, see [33,34]. Lemma 1 (Mawhin's continuation theorem). Let L be a Fredholm mapping of index zero and N be L-compact on Ω . Suppose that the following conditions hold. (Y1) (Y2)

for each λ A ð0; 1Þ, every solution x of Lx ¼ λNx is such that x2 = ∂ Ω; for each x A ∂Ω \ Ker L, QNx a0 and degB fJQN; Ω \ KerL; 0g a0;

where B denotes the Brouwer degree. Then the equation Lx¼Nx has at least one solution lying in Dom L \ Ω . The following basic concepts and lemma on graph theory can be found in [33]. A digraph G ¼ ðU; EÞ contains a set U ¼ f1; 2; …; lg of vertices and a set E of arcs (k,h) leading from initial vertex k to terminal vertex h. 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 each arc (h,k) is assigned a positive weight akh. Here akh 4 0 if and only if there exists an arc from vertex h to vertex k in G, and we call A ¼ ðakh Þll as the weight matrix. The weight WðGÞ of G is the product of the weights on all its arcs. A directed path P in G is a subgraph with distinct vertices fi1 ; i2 ; …; is g such that its set of arcs is fðik ; ik þ 1 Þ : k ¼ 1; 2; …; s  1g. If is ¼ i1 , we call P a directed cycle. A connected subgraph T is a tree if it contains no cycles. A tree T is rooted at vertex k, called the root, if k 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 ðG; AÞ. The Laplacian matrix of ðG; AÞ is defined as 0 1 ∑ a1k  a12 ⋯  a1l B ka1 C B a ∑ a2k ⋯  a2l C B C 21 B C ka2 L¼B C: B ⋮ ⋮ ⋱ ⋮ C B C @ A  al2 ⋯ ∑ alk  al1 kal

A weighted digraph ðG; AÞ is said to be balanced if WðCÞ ¼ Wð CÞ 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 ~ be the unicyclic graph unicyclic graph Q with cycle CQ , let Q

X. Zhang et al. / Neurocomputing 152 (2015) 287–293

obtained by replacing CQ with  CQ . Suppose that ðG; AÞ is ~ balanced, then WðQÞ ¼ WðQÞ. Lemma 2 (Li and Shuai [15]). Assume l Z 2. Let ck denote the cofactor of the k-th diagonal element of Laplacian matrix of ðG; AÞ. Then the following identity holds: l

∑ ck akh F kh ðyk ; yh Þ ¼ ∑ WðQ Þ Q AQ

k;h ¼ 1

∑

ðs;rÞ A EðC Q Þ

F rs ðyr ; ys Þ:

where ck denotes the cofactor of the k-th diagonal element of Laplacian matrix of ðG; AÞ, and ck 4 0 since the digraph ðG; AÞ is strongly connected. From (3) it follows that lim VðxÞ ¼ 1:

ð7Þ

jxj-1

By making use of (2), (6) we obtain that l

V 0 ðxðtÞÞ ¼ ∑ ck

Here F rs ðyr ; ys Þ; 1 r r; s rl, are arbitrary functions, Q is the set of all spanning unicyclic graphs of ðG; AÞ; WðQ Þ is the weight of Q, and CQ denotes the directed cycle of Q. In particular, if ðG; AÞ is strongly connected, then ck 4 0 for k ¼ 1; 2; …; l. 3. The existence of periodic solutions of coupled systems on networks with time delays

289

k¼1

dV k ðxk ðtÞÞ dt

l

l

r  λ ∑ ck αk V k ðxk ðtÞÞ þ λ ∑ ck βk V k ðxk ðt  τk ÞÞ k¼1

k¼1

l

l

k¼1

k;h ¼ 1

þ λ ∑ ck γ k þ λ ∑ ck akh F kh ðxk ðt  τk Þ; xh ðt  τh ÞÞ l

l

l

k¼1

k¼1

k¼1

r  λ ∑ ck αk V k ðxk ðtÞÞ þ λ ∑ ck βk V k ðxk ðt  τk ÞÞ þ λ ∑ ck γ k : ð8Þ

Now we are in the position to prove the existence of periodic solutions of system (1). For the sake of simplicity, we use the following notations in this sequel. Let L ¼ f1; 2; …; lg. j  j denotes the Euclidean norm for vectors. Write C 1 ðRn ; R þ Þ for the family of all nonnegative functions V(x) on Rn that are continuously once differentiable in x. Theorem 1. Suppose that the following conditions hold for any k A L.

In (8), the third line follows from condition A2 and Lemma 2. Since x(t) is a T-periodic solution, it follows that VðxðtÞÞ is also a T-periodic function. Hence integrating (8) from 0 to T results in Z T l V k ðxk ðtÞÞ dt 0 r  λ ∑ ck αk k¼1

0

l

þ λ ∑ ck β k

Z

k¼1

There are positive constants αk, βk and γk, mink A L fαk g 4 maxk A L fβ k g, functions V k ðxk Þ A C 1 ðRm ; R þ Þ, F kh ðxk ; xh Þ, and a matrix A ¼ ðakh Þll , akh Z 0 such that

A1.

dV k ðxk ðtÞÞ r  αk V k ðxk ðtÞÞ þ β k V k ðxk ðt  τk ÞÞ dt þ ∑ akh F kh ðxk ðt  τk Þ; xh ðt  τh ÞÞ þ γ k

ð2Þ

h¼1

V k ðxk ðtÞÞ dt

0 T

k¼1

0

Z

kAL

l

V k ðxk ðtÞÞ dt þ λT ∑ ck γ k

T

VðxðtÞÞ dt þ λ maxfβk g kAL

0

Digraph ðG; AÞ is strongly connected and each directed cycle C of weighted digraph ðG; AÞ, there is F kh ðxk ; xh Þ r 0;

ð4Þ

ðh;kÞ A EðCÞ

for all xk ; xh A Rm . Let 0 RT 1 1 l T 0 ½f 1 ðx1 ; x1 ; tÞ þ ∑h ¼ 1 g 1h ðx1 ; xh ; tÞ dt B R C B 1 T ½f ðx ; x ; tÞ þ ∑l C BT 0 2 2 2 h ¼ 1 g 2h ðx2 ; xh ; tÞ dt C ¼ BðxÞx; B C B C ⋮ @ R A 1 T l ½f ðx ; x ; tÞ þ ∑ g ðx ; x ; tÞ dt l l l h l lh T 0 h¼1

VðxðtÞÞ dt 0

which implies that Z T T∑lk ¼ 1 ck γ k ≕M 1 : VðxðtÞÞ dt r mink A L fαk g  maxk A L fβk g 0

ð5Þ

∑lk ¼ 1 ck γ k ≕M 2 : mink A L fαk g  maxk A L fβ k g

Meanwhile, combining (8) yields that for any t A ½ξ; ξ þ T we have Z t VðxðtÞÞ ¼ VðxðξÞÞ þ V 0 ðxðsÞÞ ds ξ # Z " t

r VðxðξÞÞ þ

Proof. We know that system (1) is equivalent to the following operator equation: Lx ¼ Nx:

r VðxðξÞÞ þ

ξ

l

l

∑ ck β k V k ðxk ðs  τk ÞÞ þ ∑ ck γ k ds

k¼1

Z ξþT " ξ

"

l

#

x0k ðtÞ ¼ λ f k ðxk ðtÞ; xk ðt  τk Þ; tÞ þ ∑ g kh ðxk ðt  τk Þ; xh ðt  τ h Þ; tÞ ; h¼1

VðxÞ ¼ ∑ ck V k ðxk Þ;

l

k¼1

¼ VðxðξÞÞ þ maxfβ k g

Z

k¼1

T 0

l

VðxðsÞÞ ds þ T ∑ ck γ k k¼1

l

¼ M 2 þ maxfβ k gM 1 þ T ∑ ck γ k : kAL

#

∑ ck βk V k ðxk ðs  τk ÞÞ þ ∑ ck γ k ds

ð10Þ

k¼1

Therefore, we obtain that

k; h A L:

l

ð6Þ Let

k¼1

l

kAL

Let x(t) be a solution of Lx ¼ λNx for some λ A ð0; 1Þ. Then x(t) must satisfy the following system:

ð9Þ

Inequality (9) yields that there must be a constant ξ A ½0; T such that VðxðξÞÞ r

where BðxÞlmlm is a nonsingular matrix, x A Rlm . Then system (1) has at least one T-periodic solution.

k¼1

T

k¼1

ð3Þ

jxk j-1

l

Z

l

lim V k ðxk Þ ¼ 1:

A3.

r  λ minfαk g

T

þ λT ∑ c k γ k ;

and

∑

l

k¼1

Z

Z

k¼1

l

A2.

l

þ λ ∑ ck β k

l

V k ðxk ðt  τk ÞÞ dt þ λT ∑ ck γ k

0

¼  λ ∑ ck αk k¼1

T

VðxðtÞÞ r M 2 þ maxfβ k gM 1 þ T ∑ ck γ k ; kAL

k¼1

t Z 0:

Hence, by (7), there is a positive number H, which is independent of the choice of λ, such that jxðtÞj o H;

8 t Z 0:

290

X. Zhang et al. / Neurocomputing 152 (2015) 287–293

oscillator system with time delays described by

So it is easy to get J x J oH:

ð11Þ

Consider QNx ¼0, x A Ker L. From condition A3 it follows that the following system of algebraic equations has a unique solution xn ¼ 0 A Rlm : " # 1 0 l R 1 T B T 0 f 1 ðx1 ; x1 ; tÞ þ ∑ g 1h ðx1 ; xh ; tÞ dt C B C h¼1 B C B C ⋮ QNx ¼ B C ¼ BðxÞx ¼ 0: ð12Þ " # B C l B 1RT C @ A T 0 f l ðxl ; xl ; tÞ þ ∑ g lh ðxl ; xh ; tÞ dt h¼1

Set Ω ¼ fx A X : J x J o Hg. From the above discussions we obtain that every solution x of Lx ¼ λNx for λ A ð0; 1Þ belongs to Ω ¼ fx A X : J x J o Hg. Hence the requirement (Y1) of Lemma 1 is satisfied. On the other hand, for x A ∂Ω \ Ker L ¼ ∂Ω \ Rlm , x is a constant vector with J x J ¼ H. From (12) it follows that QNx a 0, x A ∂Ω \ Ker L. Furthermore, in view of condition A3, direct calculation produces degfJQN; Ω \ Ker L; 0g ¼ sgn detðBð0ÞÞ a 0: Here, J can be chosen as the identity map since Im Q ¼Ker L. This implies that (Y2) in Lemma 1 is also satisfied. Hence, Lemma 1 yields that system (1) has at least one T-periodic solution in DomðLÞ \ Ω . The proof is complete. □ Remark 1. Based on the graph theory, coincidence degree theory and Lyapunov method, sufficient criteria to guarantee the existence of periodic solutions of system (1) are obtained. From the proof of Theorem 1, we can observe that the bound of periodic solutions to the auxiliary equation of system (1) can be easily estimated by the method of graph theory and Lyapunov function. Such method is significant for the large-scale dynamical systems, especially the coupled systems. Remark 2. Theorem 1 remains valid if time delays are not positive, that is to say, the conclusion of Theorem 1 is completely suitable for other types of equations of system (1) (e.g., advanced and mixed types). Remark 3. In the study of coupled systems on networks, the method of Lyapunov function is extensively adopted to obtain the boundedness and global stability. However, it is arduous to straightly construct a global Lyapunov function for large-scale coupled systems. Experience has shown that the global Lyapunov function V(x) can be constructed by weighted sum of V k ðxk Þ. Hence, finding a proper Lyapunov function of each vertex system is very important. In many applied fields, Lyapunov functions V k ðxk Þ, k A L can be chosen as the well-known Lyapunov functions for individual vertex systems. Remark 4. From the proof of Theorem 1, we see that the strong connectedness of digraph ðG; AÞ is only used to prove that ck is positive, where ck is the cofactor of the k-th diagonal element of Laplacian matrix of ðG; AÞ. Therefore, the theoretical result of Theorem 1 can be extended to the case where all the cofactors of the diagonal elements of Laplacian matrix of ðG; AÞ are positive. 4. Application to coupled oscillators on a network In this section, we will apply the main result to discuss the existence of periodic solutions for a coupled oscillators system on a network. Given a digraph ðG; AÞ with l ðl Z 2Þ vertices, A ¼ ðaij Þll , a coupled system of nonlinear oscillators on G can be built as follows: assume that in the k-th vertex, it is assigned a periodic

x€ k ðtÞ þ αk ðtÞx_ k ðtÞ þ xk ðtÞ þ x3k ðtÞ þ β k ðtÞxk ðt  τk Þ ¼ pk ðtÞ;

k A L;

where αk ðtÞ, β k ðtÞ and pk(t) are continuous T-periodic functions, and the influence of vertex h to vertex k is provided in the form akh ðxk ðt  τk Þ  xh ðt  τh ÞÞ, in which akh Z0, akh ¼ 0 if there exists no arc from h to k in G. So we arrived at the following coupled systems of second order differential equations: x€ k ðtÞ þ αk ðtÞx_ k ðtÞ þ xk ðtÞ þ x3k ðtÞ þ β k ðtÞxk ðt  τk Þ l

þ ∑ akh ðxk ðt  τk Þ  xh ðt  τh ÞÞ ¼ pk ðtÞ;

k; hA L

h¼1

ð13Þ

Let yk ðtÞ ¼ x_ k ðtÞ þ ηxk ðtÞ, η 4 0, we arrive at a system of first order differential equations ( x_ k ðtÞ ¼ yk ðtÞ  ηxk ðtÞ;

 y_ k ðtÞ ¼ η  αk ðtÞ yk ðtÞ þ ηαk ðtÞ  1  η2 xk ðtÞ x3k ðtÞ  βk ðtÞxk l

ðt  τk Þ þ pk ðtÞ  ∑ akh ðxk ðt  τk Þ xh ðt  τh ÞÞ:

ð14Þ

h¼1

In order to apply Theorem 1 to study the existence of periodic solutions of system (14), the following hypothesis is needed. Assumption 1. Z T pk ðtÞ dt ¼ 0;

k A L:

0

Theorem 2. Assume that all the cofactors of the diagonal elements of Laplacian matrix of ðG; AÞ are positive. Suppose further that there exist constants m1, m2, M1, and M2, such that 0 o m1 o αk ðtÞ o M 1 ; and ðM 1  ηÞη o 1;

0 o m2 o β k ðtÞ o M 2 ;

2η om1 ;

kAL

ð15Þ

l η 4M22  4 4 ∑ akh : η h¼1

ð16Þ

2

Then system (14) has at least one T-periodic solution. Proof. Denote xðkÞ ¼ ðxk ; yk Þ, x ¼ ðxð1Þ ; xð2Þ ; …; xðlÞ ÞT ,  yk ðtÞ  ηxk ðtÞ
  f k ðxðkÞ ðtÞ; xðkÞ ðt  τk Þ; tÞ ¼
η  αk ðtÞ yk ðtÞ þ ηαk ðtÞ  1  η2 xk ðtÞ  x3k  ðtÞ  βk ðtÞxk ðt  τk Þ þ pk ðtÞ ; g kh ðxðhÞ ðt  τh Þ; xðkÞ ðt  τ k Þ; tÞ ¼

0  akh ðxk ðt  τk Þ  xh ðt  τh ÞÞ

! :

The vector spaces X, Z, and mappings L, N, P, Q, Kp are constructed in the same way as in Section 2. Therefore, in order to prove the existence of periodic solutions, it suffice to show that conditions A1, A2 and A3 of Theorem 1 are satisfied. Set V k ðxðkÞ Þ ¼ 12x2k þ 14x4k þ 12y2k , then limjxðkÞ j-1 V k ðxðkÞ Þ ¼ 1. Hence using (15) and (16) we obtain that dV k ðxðkÞ ðtÞÞ ¼  ηx2k ðtÞ  ηx4k ðtÞ þ ½η  αk ðtÞy2k ðtÞ dt
 þ ηαk ðtÞ  η2 xk ðtÞyk ðtÞ  β k ðtÞxk ðt  τk Þyk ðtÞ þ pk ðtÞyk ðtÞ l

 ∑ akh yk ðtÞðxk ðt  τk Þ  xh ðt  τh ÞÞ h¼1

    1 y ðtÞ  r ηαk ðtÞ  η2 2ε1 xk ðtÞ k   ηx2k ðtÞ  2 ε1   ηx4k ðtÞ ½αk ðtÞ  ηy2k ðtÞ     M 2  y ðtÞ 1 y ðtÞ þ 2ε2 xk ðt  τk Þ k  þ 2ε3 pk ðtÞ k  2  ε2  2 ε3  l

l

h¼1

h¼1

þ ∑ akh y2k ðtÞ þ ∑ akh x2k ðt  τk Þ

X. Zhang et al. / Neurocomputing 152 (2015) 287–293

þ

l

1 l ∑ a ðx2 ðt  τh Þ  x2k ðt  τk ÞÞ 2 h ¼ 1 kh h !

η

η

þ pk ðtÞ  ∑ akh ðxk ðt  τk Þ  xh ðt  τ h ÞÞ:

where αk ðtÞ, βk ðtÞ and pk(t) are all T-periodic functions. Now we will illustrate this conclusion. Consider the system (19) with k A f1; 2; 3; 4g, and we choose that

l

η

h¼1

sin t sin t ; α2 ðtÞ ¼ 1:5 þ ; 4 4 sin t sin t ; α4 ðtÞ ¼ 1:8 þ ; α3 ðtÞ ¼ 2 þ 4 4

α1 ðtÞ ¼ 1:75 þ

1 l þ ∑ akh ðx2h ðt  τh Þ  x2k ðt  τk ÞÞ 2h¼1 ! ! l l η 1 η  ∑ akh x2k ðtÞ   ∑ akh x4k ðtÞ r 2 4 h¼1 4 h¼1 ! ! l l η 2M 22 2  ∑ a  þ ∑ akh x2k ðt  τk Þ y ðtÞ þ 4 h ¼ 1 kh k η h¼1

cos t cos t ; β2 ðtÞ ¼ 0:03 þ ; 50 50 cos t cos t ; β 4 ðtÞ ¼ 0:04 þ ; β3 ðtÞ ¼ 0:05 þ 50 50

β1 ðtÞ ¼ 0:06 þ

2 1 l þ p2k ðtÞ þ ∑ akh ðx2h ðt  τh Þ  x2k ðt  τk ÞÞ 2h¼1 η ! ! l l η 4M 22 ðkÞ  2 ∑ akh V k ðx ðtÞÞ þ þ 2 ∑ akh V k r 2 η h¼1 h¼1 ðxðkÞ ðt  τk ÞÞ þ

2

max



η t A ½0;T;k A L

p2k ðtÞ



l

þ ∑ akh F kh ðxk ðt  τk Þ; xh ðt  τh ÞÞ:

ð17Þ

pk ðtÞ ¼ sin t þ cos t;

k A f1; 2; 3; 4g:

Let τk ¼ 0:05k þ 10, k ¼ 1; 2; 3; 4. Furthermore, we assume that the coupling graph of system (19) is as follows: 0 1 0 0:02 0:03 0:01 B 0:02 0 0:01 0:03 C B C ðakh Þ44 ¼ B C: @ 0:03 0:01 0 0:02 A 0:01

h¼1

0:03

0:02

In (17), the sixth line follows from the inequality 2 and ε21 ¼ η, ε22 ¼ 4M2 =η, ε23 ¼ 4=η, 2εab=ε r ε2 a2 þ b =ε2 F kh ðxk ðt  τk Þ; xh ðt  τh ÞÞ ¼ 12ðx2h ðt  τh Þ  x2k ðt  τk ÞÞ. Thus, condition A1 holds. Moreover, it is easy to verify that the function Fkh satisfies A2. Furthermore, the operator equation QNx ¼0, x A Ker L is equivalent to the following system of algebraic equations: 0 RT 1 1 ð1Þ ð1Þ l ð1Þ ðhÞ T 0 ½f 1 ðx ; x ; tÞ þ ∑h ¼ 1 g 1h ðx ; x ; tÞ dt B R C B 1 T ½f ðxð2Þ ; xð2Þ ; tÞ þ ∑l C ð2Þ ðhÞ BT 0 2 h ¼ 1 g 2h ðx ; x ; tÞ dt C ¼ BðxÞx ¼ 0; ð18Þ B C B C ⋮ @ R A 1 T ðlÞ ðlÞ l ðlÞ ðhÞ T 0 ½f l ðx ; x ; tÞ þ ∑h ¼ 1 g lh ðx ; x ; tÞ dt

It is easy to derive that

where the coefficient matrix B(x) of 0 0 0 0 0 η 1 B B a1 b1 a12 0 a13 0 B B 0 0 0 0 η 1 B B a2 b2 a23 0 B a21 0 BðxÞ ¼ B B ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ B B ⋮ ⋮ ⋮ ⋮ ⋮ B ⋮ B B 0 0 0 0 0 0 @

Let η ¼0.6. Then we compute

al1

0

al2

0

al3

0

algebraic equations (18) is 1 ⋯ ⋯ 0 0 C ⋯ ⋯ a1l 0 C C ⋯ ⋯ 0 0C C C ⋯ ⋯ a2l 0 C C; ⋮ ⋮ ⋮ ⋮C C C ⋮ ⋮ ⋮ ⋮C C ⋯ ⋯ η 1 C A ⋯

⋯

ð19Þ

h¼1

 ∑ a r  x2k ðtÞ  ηx4k ðtÞ  y2 ðtÞ 2 4 h ¼ 1 kh k ! l 2M 22 2 þ þ ∑ akh x2k ðt  τk Þ þ p2k ðtÞ

η

291

al

ck ¼ 0:00012 4 0;

0

and 1:25 ¼ m1 o αk ðtÞ o M 1 ¼ 2:25;

In this section, we will give an example to check the feasibility of our results. Theorem 2 shows that if all the cofactors of the elements of Laplacian matrix of ðG; AÞ are positive and inequalities (15) and (16) are satisfied, then the following coupled oscillators on a network with time delays have periodic solutions: ( x_ k ðtÞ ¼ yk ðtÞ  ηxk ðtÞ

 y_ k ðtÞ ¼ η  αk ðtÞ yk ðtÞ þ ηαk ðtÞ  1  η2 xk ðtÞ  x3k ðtÞ  βk ðtÞxk ðt  τ k Þ

0:01 ¼ m2 o βk ðtÞ oM 2 ¼ 0:08;

k ¼ 1; 2; 3; 4:

ðM 1  ηÞη ¼ 0:99 o1;

2η ¼ 1:2 om1 ¼ 1:25; l η 4M22 0:2573 ¼  4 4 ∑ akh ¼ 0:24: 2 η h¼1 Hence all the conditions of Theorem 2 are satisfied. Therefore, from Theorem 2 it follows that (19) has periodic solutions. Our numerical 10 5

in

5. Numerical simulations

k ¼ 1; 2; 3; 4;

where ck is the cofactor of the k-th diagonal element of Laplacian matrix of ðG; AÞ. Direct calculations yield that Z T pk ðtÞ dt ¼ 0; k ¼ 1; 2; 3; 4;

bl

bk ¼ η  α k , which ak ¼ ηα k  1  η2  β k  x2k  ∑h a k akh , RT RT α ¼ ð1=TÞ 0 αk ðtÞ dt, β ¼ ð1=TÞ 0 β k ðtÞ dt and p k ¼ ð1=TÞ k k RT 0 pk ðtÞ dt ¼ 0, k A L. It is not difficult to show that matrix B(x) is nonsingular for any x A Rlm (see the calculation of jBðxÞj in the Appendix). Hence condition A3 holds. An application of Theorem 1 yields that system (14) has at least one T-periodic solution. This completes the proof. □

0

0

y1(t)

−5 −10 −15 −20 −25 −4

−2

0

2

4

6

x1(t)

Fig. 1. The dynamical behavior of the subsystem xð1Þ of system (19) with initial value ðx1 ð0Þ; y1 ð0ÞÞ ¼ ð6; 2Þ.

292

X. Zhang et al. / Neurocomputing 152 (2015) 287–293 3.5

simulations verify the effectiveness of the proposed results (see Figs. 1–4).

3 2.5

6. Conclusion and future directions

2

y4(t)

2

In this paper, we study a class of coupled systems on networks with time delays. Using the combined method of coincidence degree, graph theory and Lyapunov function, we develop a systematic approach that allows one to obtain the existence of periodic solutions for coupled systems on networks. We apply this approach to a coupled system of nonlinear oscillators and derive the sufficient conditions for the existence of periodic solutions. We can see that in the proof of our main result, the methods used to estimate the bound of periodic solutions are graph theory and Lyapunov function. Compared with the existing papers studying the existence of periodic solutions, which were cited in the Introduction, the methods of graph theory and Lyapunov function are more effective, which are extensively used in dealing with the boundedness and global stability of coupled systems. Hence, the combined method of coincidence degree, graph theory and Lyapunov function is a decided advantage in studying the existence of periodic solutions for large-scale coupled systems.

1.5 1 0.5 0

−0.5 −1 −2

−1.5

−1

−0.5

0

0.5

1

1.5

x4(t)

Fig. 4. The dynamical behavior of the subsystem xð4Þ of system (19) with initial value ðx4 ð0Þ; y4 ð0ÞÞ ¼ ð  2; 3Þ.

We would like to mention that results reported here are not exhaustive. We will continue to do some researches about the existence of coupled systems on networks with time-varying delays, neutral functional differential coupled system and so on.

0

Acknowledgments

y2(t)

−2

The authors are thankful to the editor and the reviewers for their valuable comments and suggestions that helped to improve the quality of the paper. This work was supported by the NNSF of China (Nos. 11301112, 11171081 and 11171056), the NNSF of Shandong Province (No. ZR2013AQ003), China Postdoctoral Science Foundation funded project (Nos. 2013M541352, 2014T70313), HIT.IBRSEM.A.2014014 and the Key Project of Science and Technology of Weihai (No. 2013DXGJ04).

−4

−6

−8

−10 −2

−1

0

1

2

3

4

x2(t)

Appendix

Fig. 2. The dynamical behavior of the subsystem xð2Þ of system (19) with initial value ðx2 ð0Þ; y2 ð0ÞÞ ¼ ð4; 2Þ.

  η    a1   0    a21 jBðxÞj ¼   ⋮   ⋮   0    al1

2

1.5

y3(t)

1

0.5

0

−0.5

−1 −1

The calculation of determinant jBðxÞj in Theorem 2. Let ak ¼ ηα k  1  η2  β k  x2k  ∑h a k akh , bk ¼ η  α k , ρk ¼  1  β k  x2k  ∑h a k akh . Then

−0.5

0

0.5

1

x3(t)

Fig. 3. The dynamical behavior of the subsystem xð3Þ of system (19) with initial value ðx3 ð0Þ; y3 ð0ÞÞ ¼ ð 1; 2Þ.

  0    ρ1   0    a21 ¼   ⋮   ⋮   0    al1

1

0

0

0

0

⋯

⋯

0

b1

a12

0

a13

0

⋯

⋯

a1l

0

η

1

0

0

⋯

⋯

0

0

a2

b2

a23

0

⋯

⋯

a2l

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮ 0

⋮ 0

⋮ 0

⋮ 0

⋮ 0

⋮ ⋯

⋮ ⋯

⋮ η

0

al2

0

al3

0

⋯

⋯

al

1

0

0

0

0

⋯

⋯

0

b1

a12

0

a13

0

⋯

⋯

a1l

0

η

1

0

0

⋯

⋯

0

0

a2

b2

a23

0

⋯

⋯

a2l

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

0

0

0

0

0

⋯

⋯

η

0

al2

0

al3

0

⋯

⋯

al

 0   0  0   0  ⋮   ⋮  1   bl   0   0  0   0  ⋮   ⋮  1   bl 

X. Zhang et al. / Neurocomputing 152 (2015) 287–293

   ρ1   0    a21   ¼ ⋮   ⋮   0    al1     ρ1   0    a21   ¼ ⋮   ⋮   0    al1 

   ρ1   a21    ⋮ ¼   ⋮   0  a  l1

a12

0

a13

0

⋯

⋯

η

1

0

0

⋯

⋯

a2 ⋮

b2 ⋮

a23 ⋮

0 ⋮

⋯ ⋮

⋯ ⋮

⋮

⋮

⋮

⋮

⋮

⋮

0

0

0

0

⋯

⋯

al2

0

al3

0

⋯

⋯

a12 0

0 1

a13 0

0 0

⋯ ⋯

⋯ ⋯

ρ2

b2

a23

0

⋯

⋯

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

0

0

0

0

⋯

⋯

al2

0

al3

0

⋯

⋯

a12

a13

0

⋯

a1l

ρ2

a23

0

⋯

a2l

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

⋮

0

0

0

⋯

η

  0  0   ⋮  ⋮   1  bl 

al2 al3 0 ⋯ al    ρ1 a12 ⋯ a1;l  1   a ρ ⋯ a2;l  1  21 2  l ⋮ ⋮ ⋱ ⋮ ¼ ð  1Þ  a a ⋯ ρ l  1;2  l  1;1 l1   al1 al2 ⋯ al;l  1 

  0  0 0   a2l 0  ⋮ ⋮   ⋮ ⋮   η 1   al bl    a1l 0   0 0   a2l 0  ⋮ ⋮   ⋮ ⋮   η 1   al bl  a1l

       ⋮ :  al  1;l   ρl  a1l a2l

Since ρk ¼  1  β k  x2k  ∑h a k akh , it follows that for any x A Rlm that jBðxÞj is a strictly diagonally dominant determinant, so B(x) is nonsingular. References [1] K. Wang, S. Lu, The existence, uniqueness and global attractivity of periodic solution for a type of neutral functional differential system with delays, J. Math. Anal. Appl. 335 (2007) 808–818. [2] H. Gao, B. Liu, Existence and uniqueness of periodic solutions for forced Rayleigh-type equations, Appl. Math. Comput. 211 (2009) 148–154. [3] A.L.A. Araujo, Periodic solutions for a nonautonomous ordinary differential equation, Nonlinear Anal. 75 (2012) 2897–2903. [4] Z. Zhang, Z. Hou, Existence of four positive periodic solutions for a ratiodependent predator–prey system with multiple exploited (or harvesting) terms, Nonlinear Anal. 11 (2010) 1560–1571. [5] M. Fan, K. Wang, Periodicity in a delayed ratio-dependent predator–prey system, J. Math. Anal. Appl. 262 (2001) 179–190. [6] S. Lu, W. Ge, On the existence of periodic solutions for neutral functional differential equation, Nonlinear Anal. Theor. Methods Appl. 54 (2003) 1285–1306. [7] L. Wang, W. Li, Existence and global stability of positive periodic solutions of a predator–prey system with delays, Appl. Math. Comput. 146 (2003) 167–185. [8] G. Liu, J. Yan, Existence of positive periodic solutions for neutral delay Gausetype predator–prey system, Appl. Math. Model. 35 (2011) 5741–5750. [9] C. Bishop, Neural Networks for Pattern Recognition, Oxford University Press, Oxford, 1995. [10] R.M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 2001. [11] R.V. Sole, J. Bascompte, Self-Organization in Complex Ecosystems, Princeton University Press, Princeton, 2006. [12] Y. Dai, Y. Cai, X.M. Xu, Synchronization criteria for complex dynamical networks with neutral-type coupling delay, Physica A 387 (2008) 4673–4682. [13] H. Liu, J. Chen, J. Lu, M. Cao, Generalized synchronization in complex dynamical networks via adaptive couplings, Physica A 389 (2010) 1759–1770. [14] B. Shen, Z. Wang, X. Liu, Bounded H-infinity synchronization and state estimation for discrete time-varying stochastic complex networks over a finite-horizon, IEEE Trans. Neural Netw. 22 (2011) 145–157. [15] M.Y. Li, Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equ. 248 (2010) 1–20.

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[16] Z. Wang, S. Cao, Z. Duan, X. Liu, Synchronization of coupled Duffing-type oscillator dynamical networks, Neurocomputing 136 (2014) 162–169. [17] G. Liu, W. Li, H. Yang, K. Gareth, The control gain region for synchronization in non-diffusively coupled complex networks, Physica A 405 (2014) 17–24. [18] Y. Sun, D. Zhao, Effects of noise on the outer synchronization of two unidirectionally coupled complex dynamical networks, Chaos 22 (2012) 023131. [19] H. Su, W. Li, K. Wang, Global stability analysis of discrete-time coupled systems on networks and its applications, Chaos 22 (2012) 033135. [20] W. Li, H. Su, K. Wang, Global stability analysis for stochastic coupled systems on networks, Automatica 47 (2011) 215–220. [21] H. Chen, J. Sun, Stability analysis for coupled systems with time delay on networks, Physica A 391 (2012) 528–534. [22] J. Suo, J. Sun, Y. Zhang, Stability analysis for impulsive coupled systems on networks, Neurocomputing 99 (2013) 172–177. [23] H. Guo, M.Y. Li, Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Am. Math. Soc. 136 (2008) 2793–2802. [24] J. Wei, M.Y. Li, Global existence of periodic solutions in a tri-neuron network model with delays, Physica D 198 (2004) 106–119. [25] J. Cao, On exponential stability and periodic solutions of CNNs with delays, Phys. Lett. A 267 (2000) 312–318. [26] Y. Li, Existence and stability of periodic solutions for Cohen–Grossberg neural networks with multiple delays, Chaos Solitons Fractals 20 (2004) 459–466. [27] Z. Liu, A. Chen, J. Cao, L. Huang, Existence and global exponential stability of periodic solution for BAM neural networks with periodic coefficients and time-varying delays, IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50 (2003) 1162–1173. [28] H. Huang, J. Cao, J. Wang, Global exponential stability and periodic solutions of recurrent neural networks with delays, Phys. Lett. A 298 (2002) 393–404. [29] A. Chen, J. Cao, Periodic bi-directional Cohen–Grossberg neural networks with distributed delays, Nonlinear Anal. 66 (2007) 2947–2961. [30] A. Chen, F. Chen, Periodic solution to BAM neural network with delays on time scales, Neurocomputing 73 (2009) 274–282. [31] M. Dong, Global exponential stability and existence of periodic solutions, Phys. Lett. A 300 (2002) 49–57. [32] Z. Zhang, K. Liu, Existence and global exponential stability of a periodic solution to interval general bidirectional associative memory (BAM) neural networks with multiple delays on time scales, Neural Netw. 24 (2011) 427–439. [33] D.B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996. [34] R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, 1977. Xinhong Zhang was born in 1979. She received her M.S. degree from China University of Petroleum. She is currently a Ph.D. student in Harbin Institute of Technology, China. Her current research interests include stochastic differential equations with or without Poisson jumps, dynamic systems.

Wenxue Li was born in 1981. He received his Ph.D. degree from the Harbin Institute of Technology, China, in 2009. He is currently a instructor in the Harbin Institute of Technology at Weihai. His current research interests include stability theory for stochastic differential and integral equations.

Ke Wang received his M.S. and Ph.D. degrees from the Northeast Normal University and Jilin University, China, in 1984 and 1995, respectively. Now he is a professor in the Harbin Institute of Technology at Weihai. His main research interests are ordinary and functional differential equations, stochastic differential equations, random dynamical systems.