Global Exponential Stability of Periodic Oscillation for Nonautonomous BAM Neural Networks with Distributed Delay

Journal of Inequalities and Applications Volume 2009, Article ID 385298,20pages doi:10.1155/2009/385298

Research Article

Global Exponential Stability of Periodic

Oscillation for Nonautonomous BAM Neural

Networks with Distributed Delay

Yi Wang,

_{Zhongjun Ma,}

_{and Hongli Liu}

1_{School of Mathematics and Statistics, Zhejiang University of Finance & Economical,}

Hangzhou 310012, China

2_{School of Mathematics and Computing Science, Guilin University of Electronic Technology,}

Guilin 541004, China

3_{School of Mathematical Science and Computing Technology, Central South University,}

Changsha 410083, China

Correspondence should be addressed to Yi Wang,[email protected]

Received 22 March 2009; Revised 7 July 2009; Accepted 2 October 2009 Recommended by Alexander I. Domoshnitsky

We derive a new criterion for checking the global stability of periodic oscillation of bidirectional associative memoryBAM neural networks with periodic coefficients and distributed delay, and find that the criterion relies on the Lipschitz constants of the signal transmission functions, weights of the neural network, and delay kernels. The proposed model transforms the original interacting network into matrix analysis problem which is easy to check, thereby significantly reducing the computational complexity and making analysis of periodic oscillation for even large-scale networks.

Copyrightq2009 Yi Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The bidirectional associative memoryBAMneural network which was first introduced by Kosko in 19871,2is formed by neurons arranged in two layers. The neurons in one layer are fully interconnected to the neurons in the other layer, while there are no interconnections among neurons in the same layer. Through iterations of forward and backward information flows between the two layers, it performs a two-way associative search for stored bipolar vector pairs and generalizes the single-layer autoassociative hebbian correlation to a two-layer pattern matched heteroassociative.

to some practical problems, the properties of equilibrium points play important roles. An equilibrium point can be looked as a special periodic solution of neural networks with arbitrary period. In this sense, the analysis of periodic solutions of neural networks could be more general than that of equilibrium points. There are some results on the existence and stability of periodic solution of BAM neural networks. Liu et al.20, 21obtained several sufficient conditions which ensure existence and stability of periodic solution for BAM neural networks with periodic coefficients and time-varying delays. Subsequently, Guo et al.22

obtained some sufficient conditions ensuring the existence, uniqueness, and stability of the periodic solution for BAM neural networks with periodic variable coefficients and variable delays, and they also estimated the exponentially convergent rate. Song et al. obtained several sufficient conditions which ensure existence and stability of periodic solution for BAM neural networks with periodic coefficients and periodic time-varying delays23. Moreover, neural networks usually has a spatial extent due to the presence of an amount of parallel pathways with a variety of axon sizes and lengths. Thus, the delays in neural networks are usually continuously distributed. Recently, there are some authors studied the BAM neural networks with distributed delays and constants coefficients24–26.

Until recently, few studies have considered periodic solution for the BAM neural networks with periodic coefficients and distributed delays. Zhou et al. considered the periodic solution for the BAM neural networks with period coefficients and continuously distributed delays 27. However, the result in Zhou et al. contains two limitations, one made for periodicT and the other is min1i,jn{ai−m_j1djiNji, cj−_i1n bijMij} > 0,

which also in the Wang et al.28. This limitation is being removed by this paper. Based on the continuation theorem of Mawhin’s coincidence degree theory, the nonsingularM-matrix and Lyapunov functionals, we derive a new global exponential stability criterion in matrix form for periodic oscillation of BAM neural networks with period coefficients and distributed delay. Moreover, our criterion is easy to check out.

The paper is organized as follows. Our model and some preliminaries are given in

Section 2. The existence of periodic solution is proved inSection 3. The exponential stability of periodic oscillator is considered inSection 4. An example is shown in Section 5. Several summary remarks are finally given inSection 6.

2. Preliminaries

In this paper, we study the BAM neural networks with periodic coefficients and continuously distributed delays modeled by the following system:

˙

uit −aituit

m

j1

bijthij

tij

0

fijsvjt−sds

Iit,

˙

vjt −cjtvjt

n

i1

djiteji

τji

0

gjisuit−sds

Ljt,

2.1

wherei1,2, . . . , n;j 1,2, . . . , m;ait>0 andcjt>0 denote the rate with which the cells

iandjreset their potential to the resting state when isolated from the other cells and inputs;

bijtanddjitare connection weights of the neural network;Iit,Ljtdenote theith and

cell andjth cell at timet, respectively. Moreover, thejth cell has an impact on theith cell in the time oftijand thejth cell has an impact on theith cell in the time ofτji.

Ifuitandvjtsatisfy system2.1anduitT uit,vjtT vjt, then they

areT-periodic solutions of system2.1. The initial conditions associated with system2.1

are given as follows:

uis φis, vjs ψjs, s∈−∞,0, 2.2

whereφis, ψjsare continuous functioni1,2, . . . , n;j1,2, . . . , m.

Throughout this paper, we make the following assumptions.

Assumption 2.1. ait,bijt,cjt,djit,IitandJjtare continuousT-periodic functions on

R. In addition,a_i sup_t∈R|ait| <∞,a−_i inft∈R|ait|> 0,b_ij sup_t∈R|bijt|< ∞,c_i

sup_t∈R|cjt|<∞,c_i−inft∈R|cjt|>0,d_ji sup_t∈R|djit|<∞,I_isup_t∈R|Iit|<∞, and

L_j sup_t∈R|Ljt|<∞.

Assumption 2.2. Signal transmission functionshiju,ejivare bounded onR, and there exist

numberMij>0 andNji>0 such that

_{hiju−hijv Mij|u−v|, ejiu−ejiv Nji|u−v| 2.3}

for eachu, v∈R, i1,2, . . . , nandj1,2, . . . , m.

Assumption 2.3. The delay kernelsfijs, gjis :0,∞ → 0,∞ i 1,2, . . . , n;j 1,2,

. . . , mare continuous and integrable and satisfy

_∞

0

fijsds1,

_∞

0

gjisds1. 2.4

Assumption 2.4. The delay kernelsfijs, gjis :0,∞ → 0,∞ i 1,2, . . . , n;j 1,2,

. . . , msatisfy

_∞

0

eαsfijsds1,

_∞

0

eαsgjisds1, 2.5

whereαis a bounded positive real number.

Now, we give some useful notations, definitions, and lemmas as follows: u

T

0|us|2ds

1/2_{andv 1/T}T

0vsds, whereus∈CR,R, vsisT-periodic function.

Assume thatTn×n_{{A a}

ijn×n:aij0, i /j}, then we have the following.

Lemma 2.5see17,29,30. LetA∈Tn×n_{. Then, each of the following conditions is equivalent to}

the statement ‘Ais a nonsingular M -matrix’:

aall of the principal minors ofAare positive;

cAis inverse-positive; that is,A−1_{exists andA}−1_0;

dthere is a vectorx(ory), whose elements are all positive, such that the elements ofAx(or

AT_{y) are all positive;}

eAhas all positive diagonal elements and there exists a positive diagonal matrixDsuch that

ADis strictly diagonally dominant; that is,

aiidi>

i /j

aij dj, i1,2, . . . , n. 2.6

In the following, we introduce some concepts and results from the book by Gaines and Mawhin31.

Let X and Z be two Banach spaces,L : DomL ⊂ X → Z a linear mapping, and

N :X → Za continuous mapping. The mappingLwill be called a Fredholm mapping of index zero if dim KerLcodimImL <∞andImLis closed inZ. IfLis a Fredholm mapping of index zero and there exist continuous projectorsP :X → XandQ:Z → Zsuch thatImP

KerL, KerQImLImI−Q, it follows that mappingL|DomLKerP :I−PX → ImLis

invertible. We denote the inverse of that mapping byKP. IfΩis an open bounded subset ofX,

the mappingNwill be calledL-compact on ifQNΩis bounded andKPI−QN:Ω → X

is compact. SinceImQis isomorphic to KerL, there exists an isomorphismJ :ImQ → KerL.

Lemma 2.6Mawhin’s continuation theorem. Let X and Zbe two Banach spaces andL be a

Fredholm mapping of index zero. Assume thatΩ ⊂Xis an open bounded set andN :X → Zis a continuous operator which isL-compact onΩ. ThenLxNxhas at least one solution inDomLΩ, if the following conditions are satisfied:

afor eachλ∈0,1, x∈∂ΩDomL, Lx /λNx;

bfor eachx∈∂ΩKerL,QNx /0;

cdeg{JQNx,ΩKerL,0}/0,

whereJ:ImQ → KerLis an isomorphism.

3. Existence of Periodic Solutions

Now we give the following sufficient conditions on the existence of periodic solutions.

Theorem 3.1. Assume that Assumptions2.1–2.3hold. Then, system2.1has at least one T-periodic

solution, if

P

In P12

P21 In

3.1

is a nonsingularM-matrix, whereIn is unite matrix andP12 pijn×m, pij −b_ijMij/a−_i;P21

Proof. LetX Z {xt u1t, . . . , unt, v1t, . . . , vmt ∈ CR,Rnm | uit uit

T, vjt vjtT, i1,2, . . . , n; j 1,2, . . . , m}, thenXis a Banach space with the norm

x₁n_i1maxt∈0,T|uit|

_m

j1maxt∈0,T|vjt|.

LetL: DomL⊂X → Z,P :XDomL → KerL,Q:X → X/ImL, andN:X → Z

be given by the following:

Lx u˙1t,u˙2t, . . . ,u˙nt,v˙1t,v˙2t, . . . ,v˙mt,

P xQx u1t, u2t, . . . , unt, v1t, v2t, . . . , vmt,

Nx ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

−a1tu1t m

j1

b1jth1j

t1j

0

f1jsvjt−sds

I1t

· · ·

−antunt m

j1

bnjthnj

tnj

0

fnjsvjt−sds

Int

−c1tv1t n

i1

d1ite1i

τ1i

0

g1isuit−sds

L1t · · ·

−cmtvmt n

i1

dmitemi

τmi

0

gmisuit−sds

Lmt

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . 3.2

It is easy to see thatL is a linear operator with KerL {xt | xt κ ∈ Rnm}.

ImL {xt | xt ∈ Z,T₀xtdt 0}is closed inZ, and dim KerL codimImL nm. Therefore,Lis a Fredholm mapping of index zero. It is easy to prove thatP andQare two projectors, andImP KerL,ImLKerQImI−Q. By using the Arzel´a-Ascoli theorem, it is easy to prove that for every bounded subsetΩ∈X, KpI−QNare relatively compact

onΩinX, that is,NisL-compact onΩ. Consider the operator equation

LxλNx, λ∈0,1, 3.3

that is

˙

uit −λaituit λ

m

j1

bijthij

tij

0

fijsvjt−sds

λIit,

˙

vjt −λcjtvjt λ

n

i1

djiteji

τji

0

gjisuit−sds

λLjt,

3.4

wherei 1,2, . . . , nandj 1,2, . . . , m.Denoteμi λ

m

j1bijsups∈R|hijs|Ii. By using

3.4, we obtain

Multiplying both sides of3.5byeλt₀aisds_{, we have}

−μieλ

t

0aisds

uiteλ

t

0aisds

μieλ

t

0aisds. 3.6

Integrating the inequality above from 0 toν ν0 27, we obtain

−μi

_ν

0

eλt₀aisds_dtu

iνeλ

ν

0aisds−u_i0μ_i

_ν

0

eλt₀aisds_{dt. 3.7}

Hence,

−μi

ν

0

e−λνtaisds_dtu

i0e−λ

ν

0aisds_u

iνμi

ν

0

e−λνtaisds_dtu

i0e−λ

ν

0aisds _3.8

forνt0. So we have

−

|ui0| −

μi

λa−_i

e−λa−iν− μi

λa−_i uiν

|ui0| −

μi

λa−_i

e−λa−iν μi

λa−_i , 3.9

that is,

|uiν|

|ui0| −

μi

λa−_i

μi

λa−_i , 3.10

which implies thatuiis bounded and similarlyvj. By theAssumption 2.3, we know that

_tij

0

fijsvjt−sds 3.11

is uniformly convergent. Therefore, the following iterated integral:

_T

0

uit

_tij

0

fijsvjt−sds dt 3.12

can be changed integrating order.

Suppose thatu1t, . . . , unt, v1t, . . . , vmt∈Xis any periodic solution of system

2.1for a certainλ∈0,1. Multiplying both sides of

˙

uit −λaituit λ

m

j1

bijthij

tij

0

fijsvjt−sds

byuitand integrating from 0 toT, we obtain

T

0

λaitu2itdt

T

0

λ

m

j1

bijtuithij

tij

0

fijsvjt−sds

dt

T

0

λIituitdt. 3.14

From Assumptions 2.1, 2.2, and 2.3 and noting that T₀|vjt − s|2dt1/2

T

0|vjt|2dt

1/2_v

j, we have

a−_i

T

0

u2_itdt

T

0

aitu2itdt

_T

0 m

j1

bijtuit

hij

tij

0

fijsvjt−sds

−hij0

dt

_T

0 m

j1

bijthij0uitdt

_T

0

Iituitdt

m

j1

b_ijMij

_∞

0

fijs

_T

0

vjt−s |uit|dt ds

√T

⎛

⎝m

j1

b_ij hij0 Ii

⎞

⎠_ui

m

j1

b_ijMijvjui

√

T

⎛

⎝m

j1

b_ij hij0 Ii

⎞ ⎠ui,

3.15

that is,

ui

1

a−_i

m

j1

b_ijMijvj

√

T a−_i

⎛

⎝m

j1

b_ij hij0 Ii

⎞

⎠_{. 3.16}

By similar argument, we have

vj

1

c−_j

n

i1

d_jiNjiui

√

T c−_j

_n

i1

d_ji eji0 Lj

. 3.17

It follows3.16and3.17that

whereP In P12

P21 In

, In is a unite matrix andP12 pijn×m, pij −b_ijMij/a−_i;P21 qjim×n,

qji−d_jiNji/c_j−, y u1,u2, . . . ,un,v1,v2, . . . ,vm,s s1, s2, . . . , snm, si

√

Tm_j1b_ij|hij0|I_i/a−_i, snj√Tni1dji|eji0|Lj/c−j, 1in, 1j m.

Application ofLemma 2.5yields

yP−1sr1, r2, . . . , rnm, 3.19

which implies that

uiri, vjrnj. 3.20

It is not difficult to check that there existt∗_i, t∗_nj∈0, Tsuch that

ui

t∗_i √ri

T, vj

t∗_njr√nj

T 3.21

for 1 inand 1 j m. Multiplying both sides of3.13by ˙uit i 1,2, . . . , nand

integrating from 0 toT, we obtain

u˙i2λ m

j1

T

0

bijtu˙ithij

tij

0

fijsvjt−sds

dt

−λ

T

0

aitu˙ituitdtλ

T

0

Iitu˙itdt

λ

m

j1

_T

0

bijtu˙it

hij

tij

0

fijsvjt−sds

dt−hij0

λ

m

j1

_T

0

bijthij0u˙itdt−λ

_T

0

aituitu˙itdtλ

_T

0

Iitu˙itdt

m

j1

b_ijMij

_∞

0

fijs

T

0

|u˙it|2dt

1/2_T

0

|vjt−s|2dt

1/2

ds

√T

m

j1

b_ij hij0 u˙iai

T

0

|u˙it|2dt

1/2_T

0

|uit|2dt

1/2

√TI_iu˙i

m

j1

b_ijMiju˙ivjaiu˙iui

√

T

m

j1

b_ij hij0 u˙i √

TI_iu˙i.

3.22

Therefore,u˙im_j1b_ijMijvja_iui √

It is easy check that

uit uit∗

_T

t∗u˙itdt

ri √

T

√

Tu˙i

_√ri

T

√

T

⎡

⎣m

j1

b_ijMijvjaiui

√

T

⎛

⎝m

j1

b_ij hij0 Ii

⎞ ⎠ ⎤ ⎦

_√ri

T

√

T

⎡

⎣m

j1

b_ijMijrnjairi

√

T

⎛

⎝m

j1

b_ij hij0 Ii

⎞ ⎠ ⎤

⎦_√ri

T ζi,

vjt

r_nj

√

T

√

T

_n

i1

d_jiNjiricjrnj

√

T

_n

i1

d_ji eji0 Lj

r√nj

T ζnj.

3.23

Letξiri/ √

Tζiandξnjrnj/

√

Tζ_nj 0< 1. We take

Ω x u1t, . . . , unt, v1t, . . . , vmt∈X| |uit|< ξi, vjt < ξnj

. 3.24

Obviously, conditionaofLemma 2.6is satisfied. Whenx∈∂ΩKerL∂ΩRnm,

xis a constant vector inRnmwith|uit|ξi,|vjt|ξnj. Then, we have

uiQNxi−aiu2_i ui

m

j1

bijhij

vj

uiIi,

vjQNx_nj−cjv_j2vj

n

i1

djiejiui vjLj.

3.25

We claim that there exists someiorj 1in, 1jmsuch that

uiQNxi<0 or vjQNxnj<0. 3.26

IfuiQNxi 0 andvjQNxnj0, then, we obtain

aiu2i ui m

j1

bijhij

vj

uiIi

m

j1

uibij

hij

vj

−hij0 ui m

j1

bijhij0 uiIi

|ui|

m

j1

b_ijMij vj |ui| m

j1

b_ij hij0 |ui|Ii.

Which implies

ξi

1

ai m

j1

b_ijMijξnj

1

ai

⎛

⎝m

j1

b_ij hij0 Ii

⎞

⎠ 1

a−_i

m

j1

b_ijMijξnj

si √

T. 3.28

By a similar argument, we obtain

ξnj_c1

j n

i1

d_jiNjiξi 1

cj

_n

i1

d_ji eij0 Lj

1

c−_j

n

i1

d_jiNjiξi

snj

√

T. 3.29

On the other hand,

ξ >√r

T ζ >

r

√

T

P−1_s √

T , 3.30

whereξ ξ1, ξ2, . . . , ξnm,ζ ζ1, ζ2, . . . , ζnm,andr r1, r2, . . . , rnm. It follows that there

exists someiorjsuch that

ξi−

1

a−_i

m

j1

b_ijMij

ξ_nj> √si

T 3.31

or

ξnj− 1

c−_j

n

i1

d_jiNjiξi>

snj

√

T, 3.32

which is contradiction with 3.28 and 3.29. Then, there exists some ior j 1 i n, 1j msuch thatuiQNxi<0 orvjQNxnj<0. Therefore,

QNx₁

n

i1

|QNx_i|

m

j1

_QNx

nj >0. 3.33

This indicates that conditionbofLemma 2.6is satisfied.

Define Hx, θ −θx 1−θQNx, θ ∈ 0,1, where x u1t, . . . , unt, v1t,

. . . , vmt∈Rnm. Whenx∈KerL

∂Ω, we have

Hx, θ₁

n

i1

|Hui, θ| m

j1

_H

vj, θ

n

i1

|−θui 1−θQNxi| m

j1

_−θv

j 1−θQNxj >0.

According to the invariant of homology, we have

degJQNx,Ω!KerL,0/0, 3.35

where J : ImQ → KerL is an isomorphism. Therefore, according to the continuation theorem of Gaines and Mawhin, system2.1has at least oneT-periodic solution. The proof is completed.

Remark 3.2. In 27, the period T was assumed to be T < min{1/a_i,1/c_j}. In 28, the inequations min1i,jn{ai−m_j1djiNji, cj−n_i1bijMij}>0 must hold and also in27.

Here, the limitation is being removed.

4. Global Exponential Stability of Periodic Solution

In this section, we discuss the global exponential stability of the periodic solution of system

2.1. Under the assumptions ofTheorem 3.1, system2.1has at least oneT-periodic solution

x∗t u∗₁t, . . . , u∗_nt, v₁∗t, . . . , v_m∗t. 4.1

Now, we give the following definition about global exponential of periodic solution:

Definition 4.1. The periodic solutionx∗tof model2.1is said to be globally exponentially stable, if there exist positive constantsβ, Msuch that

⎛

⎝n

i1

|ut−u∗t|2

m

j1

|vt−v∗t|2

⎞ ⎠

1/2

≤Mx0−x₀∗₂e−βt 4.2

for all t > 0, wherex0, x∗₀ represent the history of x and x∗ on −∞,0,respectively, and x0−x∗₀₂ {n_i1sup_−∞<t0|φit−u∗_it|2m_j1sup_−∞<t0|ψjt−v∗_jt|2}1/2.

Definition 4.2. A matrix M is said to be diagonally dominant if|mii| _{j /}i|mij|for alli, |mii|>_{j /}i|mij|for at least onei, wheremijdenotes the entry in theith row andjth column.

Theorem 4.3. Assume that Assumptions2.1,2.2,2.4hold andPis a nonsingularM-matrix, where

P is the same as inTheorem 3.1. System2.1has a unique T-periodic solution, which is globally exponentially stable, ifQ−αIT ∈Tand is a weakly diagonally dominant matrix as

Q

Q11 Q12 Q21 Q22

, 4.3

where Q11 diag2a−1 −

_m

j1b1jM1j,2a−2 −

_m

j1b2jM2j, . . . ,2a−n −

_m

j1bnjMnj; Q12

mijn×m,mij −b_ijMij;Q21 njim×n,nji −d_jiNji;Q22 diag2c−₁ −

_n

i1d1iN1i,2c−2 −

_n

i1d2iN2i, . . . ,2c−m−

_n

Proof. Letzt u1t−u₁∗t, . . . , unt−u∗nt, v1t−v∗₁t, . . . , vmt−vm∗t. Then,

zitz˙it −ait|zit|2 m

j1

bijtzithij

"tij

0

fijs

#

znjt−s v∗jt−s

$

ds

%

−m j1

bijtzithij

tij

0

fijsv∗jt−sds

−a−_i|zit|2

m

j1

b_ijMij

_tij

0

fijszitznjt−sds

−a−_i|zit|2

m

j1

b_ijMij

_∞

0

fijs

|zit|2 znjt−s 2

2 ds

−a−_i|zit|2

1 2

m

j1

b_ijMij

|zit|2

_∞

0

fijs znjt−s 2ds

,

z_njtz˙_njt−c−_j z_njt 2 1

2

n

i1

d_jiNji znjt 2

_∞

0

gjis|zit−s|2ds

4.4

fori1,2, . . . , nandj1,2, . . . , m.

We define the following Lyapunov functionals:

&

Vit

eαt

α |zit|

2 1

α

m

j1

b_ijMij

_∞

0

t

t−s

fijs znjγ 2eαγsdγds,

&

Vnjt e

αt

α znjt

2 1

α

n

i1

d_jiNji

_∞

0

_t

t−s

gjis zi

γ 2eαγsdγds.

4.5

Calculating the Dini upper right derivative ofV&jtandV&njtalong the solution of

2.1, and estimating it via the assumptions32, we have

DV&iteαt|zit|2e αt

α

⎧ ⎨

⎩−2a−i|zit| 2m

j1

b_ijMij

*

|zit|2

_∞

0

fijs znjt−s 2ds

+⎫_⎬ ⎭ eαt α ⎧ ⎨ ⎩ m j1

b_ijMij

*_∞

0

fijs znjt 2eαsds−

_∞

0

fijs znjt−s 2ds

+⎫_⎬ ⎭

eαt

α

⎛

⎝_αm

j1

b_ijMij−2a−_i

⎞

⎠_|zit|2 eαt

α

m

j1

b_ijMij znjt 2

_∞

0

fijseαsds

eαt α ⎧ ⎨ ⎩ ⎛ ⎝α m j1

b_ijMij−2a−i

⎞

⎠_|zit|2 m

j1

b_ijMij znjt 2

⎫ ⎬

DV&njt e αt α " α n i1

d_jiNji−2c−_j

_znjt 2n i1

d_jiNji|zit|2

%

. 4.6

Let

Q

Q11 Q12 Q21 Q22

, 4.7

where Q11 diag2a−1 −

_m

j1b1jM1j,2a−2 −

_m

j1b2jM2j, . . . ,2a−n −

_m

j1bnjMnj; Q12

mijn×m,mij −b_ijMij;Q21 njim×n,nji −d_jiNji;Q22 diag2c−₁ −

_n

i1d1iN1i,2c−2 −

_n

i1d2iN2i, . . . ,2c−m−

_n

i1dmi Nmi.

Consider the Lyapunov functional

Vt

nm

k1

&

Vkt. 4.8

WhenQ−αIT∈Tand is a weakly diagonally dominant matrix, calculating the Dini upper right derivative ofValong the solution of2.1, we have

DVt D

nm

k1

&

Vkt

eαt α ⎧ ⎨ ⎩ ⎛ ⎝_−2a−

i m

j1

b_ijMijα

⎞

⎠_|zit|2m j1

b_ijMij znjt 2

⎫ ⎬ ⎭ eαt α " −2c−_j

n

i1

d_jiNjiα

_znjt 2n i1

d_jiNji|zit|2

%

eαt

α

⎧ ⎨ ⎩−2a−i

m

j1

b_ijMij

n

i1

d_jiNjiα

⎫ ⎬ ⎭|zit|2

eαt

α

⎧ ⎨ ⎩−2c−j

n

i1

d_jiNji

m

j1

b_ijMijα

⎫ ⎬

⎭|znit|2<0.

4.9

Therefore, forVtV0,

V0 1

α ⎧ ⎨ ⎩ nm k1

z2_i0

m

j1

b_ijMij

_∞

0

₀

−s

fijs znjγ 2eαγsdγds

n

i1

d_jiNji

_∞

0

0

−s

gjis zi

γ 2eαγsdγds

1

α

⎧ ⎨

⎩nm

m

j1

b_ijMij

_∞

0

₀

−s

fijseαγsdγds

n

i1

d_jiNji

_∞

0

₀

−s

gjiseαγsdγds

% max

k∈{1,...,mn}sup_γ0z 2 k

γ

1

αnmk∈{max1,...,mn}sup_γ0z 2 k

γ

1

αnm

⎧ ⎨ ⎩

n

i1

sup

−∞<t0

_φit−u∗ it

2

m

j1

sup

−∞<t0

_ψ

jt−vj∗t

2⎫_⎬

⎭.

4.10

Thus,

eαt

⎛

⎝n

i1

|ut−u∗t|2

m

j1

|vt−v∗t|2

⎞

⎠_{< αVt< αV0}

nm

⎧ ⎨ ⎩

n

i1

sup

−∞<t0

_φit−u∗ it

2

m

j1

sup

−∞<t0

_ψ

jt−v∗jt

2⎫_⎬

⎭,

4.11

that is,

⎛

⎝n

i1

|ut−u∗t|2

m

j1

|vt−v∗t|2

⎞ ⎠

1/2

<√nmx0−x∗02e−

αt/2_{. 4.12}

This means that periodic solution of system2.1is globally exponentially stable. The proof is completed.

Remark 4.4. For system2.1, when delay kernelsfijs,gjisareδ-functions, that is, we take

fijs δs−iandgjis δs−σj, then system2.1can be reduced to a fixed time delay

system and all above theorems are hold.

5. Example

50 40

30 20

10 0

T −1.5

−1

−0.5 0 0.5 1 1.5 2

u1

Figure 1:Time evolution ofu1of system5.1.

Consider the following BAM neural networks:

˙

uit −aituit

3

j1

bijthij

tij

0

fijsvjt−sds

Iit,

˙

vjt −cjtvjt

3

i1

djiteji

τji

0

gjisuit−sds

Ljt.

5.1

We select a set of parameters asα4/5,ait 20.1 sint,cjt 20.1 cost,fijs

gjis e−s,Iit −0.20.4 costandLjt 0.30.3 sint,hijs tanhs,ejis 1/1s,

tijτji20,

bijt

3×3

⎛ ⎜ ⎜ ⎝

0.10.1 sint 0.20.1 sint 0.30.1 sint

0.50.1 sint 0.40.1 sint 0.30.1 sint

0.30.1 sint 0.20.1 sint −0.50.1 sint

⎞ ⎟ ⎟

⎠,

djit

3×3

⎛ ⎜ ⎜ ⎝

0.20.1 cost 0.40.1 cost 0.10.1 cost

−0.20.1 cost 0.30.1 cost 0.50.1 cost

0.50.1 cost 0.20.1 cost 0.30.1 cost

⎞ ⎟ ⎟

⎠.

5.2

50 40

30 20

10 0

T −1.5

−1

−0.5 0 0.5 1 1.5 2

u2

Figure 2:Time evolution ofu2of system5.1.

50 40

30 20

10 0

T −2

−1.5

−1

−0.5 0 0.5 1 1.5 2

u3

Figure 3:Time evolution ofu3of system5.1.

It is easy to check that maxh_ij 1 and maxe_ji 1. LetMij 1 andNji1. By some

computations, we obtain

P

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 0 0 −0.1053 −0.1579 −0.2105

0 1 0 −0.3158 −0.2632 −0.2105

0 0 1 −0.3158 −0.1579 −0.2105

−0.1579 −0.2632 −0.1053 1 0 0

−0.1579 −0.2105 −0.3158 0 1 0

−0.3158 −0.1579 −0.2105 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

50 40

30 20

10 0

T −2

−1.5

−1

−0.5 0 0.5 1 1.5

v1

Figure 4:Time evolution ofv1of system5.1.

50 40

30 20

10 0

T −2

−1.5

−1

−0.5 0 0.5 1 1.5 2

v2

Figure 5:Time evolution ofv2of system5.1.

Q

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

2.9 0 0 −0.2 −0.3 −0.4

0 2.3 0 −0.6 −0.5 −0.4

0 0 2.5 −0.6 −0.3 −0.4

−0.3 −0.5 −0.2 2.8 0 0

−0.3 −0.4 −0.6 0 2.5 0

−0.6 −0.3 −0.4 0 0 2.5 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

.

5.3

50 40

30 20

10 0

T −2

−1.5

−1

−0.5 0 0.5 1 1.5

v3

Figure 6:Time evolution ofv3of system5.1.

is globally exponentially stable. In Figures1,2,3,4,5, and6, we plot the trajectories ofuiand

vj, respectively.

6. Conclusion

In this paper, we derive a new criterion for checking the global stability of periodic oscillation of BAM neural networks with distributed delay and periodic external input sources and find that the criterion rely on the Lipschitz constants of the signal transmission functions, weights of the neural network and delay kernels by using the continuation theorem of Mawhin’s coincidence degree theory, the nonsingularM-matrix and Lyapunov function. The proposed model transforms the original interacting network into matrix analysis, thereby significantly reducing the computational complexity and making analysis of periodic oscillation for even large-scale networks. Most importantly, our result is very practical in the design of BAM neural networks.

Acknowledgments

The authors thank the anonymous reviewers for the insightful and constructive comments, and also thank for helpful discussion Professor Zengrong Liu. This work is supported by the NNSFno. 60964006.

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