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Volume 2008, Article ID 648148,14pages doi:10.1155/2008/648148

Research Article

New Results of a Class of Two-Neuron Networks

with Time-Varying Delays

Chuangxia Huang,1Yigang He,2and Lihong Huang3

1College of Mathematics and Computers, Changsha University of Science and Technology,

Changsha, Hunan 410076, China

2College of Electrical and Information Engineering, Hunan University, Changsha, Hunan 410082, China

3College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

Correspondence should be addressed to Chuangxia Huang,cxiahuang@126.com

Received 19 September 2008; Accepted 4 December 2008

Recommended by Marta Garcia-Huidobro

With the help of the continuation theorem of the coincidence degree, a priori estimates, and differential inequalities, we make a further investigation of a class of planar systems, which is generalization of some existing neural networks under a time-varying environment. Without assuming the smoothness, monotonicity, and boundedness of the activation functions, a set of sufficient conditions is given for checking the existence of periodic solution and global exponential stability for such neural networks. The obtained results extend and improve some earlier publications.

Copyrightq2008 Chuangxia Huang 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

Neural networks are complex and large-scale nonlinear dynamics, while the dynamics of the delayed neural network are even richer and more complicated 1. To obtain a deep and clear understanding of the dynamics of neural networks, one of the usual ways is to investigate the delayed neural network models with two neurons, which can be described by differential systemssee2–8. It is hoped that, through discussing the dynamics of two-neuron networks, we can get some light for our understanding about the large networks. In

7, T´aboas considered the system of delay differential equations

˙

x1t −x1t αf1

x1t−τ, x2t−τ

,

˙

x2t −x2t αf2

x1t−τ, x2t−τ

, 1.1

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∂f1

∂x20,0/0,

∂f2

∂x10,0/0, 1.2

and the negative feedback conditions:x2f1x1, x2>0,x2 /0;x1f2x1, x2<0,x1 /0. T´aboas

showed that there is anα0>0 such that forα > α0, there exists a nonconstant periodic solution

with period greater than 4. Further study on the global existence of periodic solutions to system1.1can be found in2,3. All together there is only one delay appearing in both equations. Ruan and Wei6investigated the existence of nonconstant periodic solutions of the following planar system with two delays

˙

x1t −a0x1t a1f1

x1t−τ1, x2t−τ2

,

˙

x2t −b0x2t b1f2

x1t−τ1, x2t−τ2

, 1.3

wherea0 > 0, b0 > 0,a1 andb1 are constants, the functionsf1 andf2 satisfyfjC3R2, fj0,0 0,∂fj/∂xj0,0 0,j 1,2;x2f1x1, x2/0 forx2 /0;x1f2x1, x2/0 forx1 /0; ∂f1/∂x20,0/0,∂f2/∂x10,0/0.

Recently, Chen and Wu9considered the following system:

˙

xt μ1xt F

ytτ1

I1,

˙

yt μ2ytG

xtτ2

I2,

1.4

whereμ1andμ2are positive constants,τ1andτ2are nonnegative constants withτ :τ1 τ2>

0,I1andI2are constants, andFandGare boundedC1-functions with ˙Ft>0, and ˙Gt>0

fortR. By discussing a two-dimensional unstable manifold of the transformation of the system1.4, they got some results about slowly oscillating periodic solutions.

However, delays considered in all above systems1.1–1.4are constants. It is well known that the delays in artificial neural networks are usually time-varying, and they sometime vary violently with time due to the finite switching speed of amplifiers and faults in the electrical circuit. They slow down the transmission rate and tend to introduce some degree of instability in circuits. Therefore, fast response must be required in practical artificial neural-network designs. As pointed out by Gopalsamy and Sariyasa10,11, it would be of great interest to study the neural networks in periodic environments. On the other hand, drop the assumptions of continuous first derivative, monotonicity, and boundedness for the activation might be better candidates for some purposes, see 12. Motivated by above mentioned, in this paper, we continue to consider the following planar system:

˙

x1t −a1tx1t b1tf1

x1

tτ11t

, x2

tτ12t

I1t,

˙

x2t −a2tx2t b2tf2

x1

tτ21t

, x2

tτ22t

I2t,

1.5

whereait∈CR,0,∞,bit, Iit∈CR, R, i1,2, are periodic with a common period ω>0,fi·,·∈CR2, R,τijt∈CR,0,∞, i,j1,2 beingω-periodic.

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functions, a family of sufficient conditions are given for checking the existence of periodic solution and global exponential stability for such neural networks. Our results extend and improve some earlier publications. The remainder of this article is organized as follows. In

Section 2, the basic notations and assumptions are introduced. After giving the criteria for checking the existence of periodic solution and global exponential stability for the neuron networks in Section 3, one illustrative example and simulations are given inSection 4. We also conclude this paper inSection 5.

2. Preliminaries

In this section, we state some notations, definitions, and lemmas. Assume that nonlinear system1.5is supplemented with initial values of the type

uit φit, −τt≤0, τ max 1≤i,j≤2

τijt

, 2.1

in whichφit, i1,2 are continuous functions. Setφ φ1, φ2, φ3T, ifx∗t x1∗t, x∗2tT

is a solution of system1.5, then we denote

φx2

i1

sup

τt≤0

φitx

it. 2.2

Definition 2.1. The solutionx∗t x∗1t, x2∗tT is said to be globally exponentially stable, if there existλ >0 andm≥1 such that for any solutionxt x1t, x2tT of1.5, one has

xitx

it≤xeλt fort≥0, 2.3

whereλis called to be globally exponentially convergent rate.

To establish the main results of the model1.5, some of the following assumptions will be applied:

H1|fix1, x2| ≤ αi|x1| βi|x2| Mi for allx1, x2TR2, i 1,2, whereαi ≥ 0, βi

0, Mi>0 are constants;

H2there exist constantsαi ≥0, βi ≥ 0,such that|fjx1, x2−fjy1, y2| ≤αi|x1−y1| βi|x2−y2|for anyx1, x2TR2, y1, y2TR2.

ForVCa, ∞, R, one denotes

DVt lim sup

h→0−

Vt hVt

h ,

DVt lim inf

h→0−

Vt hVt

h .

2.4

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Consider an abstract equation in a Branch space X In this section, we use the coincidence degree theory to obtain the existence of anω-periodic solution to1.5. For the sake of convenience, we briefly summarize the theory as below.

LetXandZbe normed spaces,L: DomLXZbe a linear mapping, andN:XZbe a continuous mapping. The mappingLwill be called a Fredholm mapping of index zero if dim KerLcodim ImL <∞and ImLis closed inZ. IfLis a Fredholm mapping of index zero, then there exist continuous projectorsP:XXandQ:ZZsuch that ImP KerL

and ImLKerQImI−Q. It follows thatL|DomL∩KerP:I−PX →ImLis invertible. We denote the inverse of this map byKp. IfΩis a bounded open subset ofX, the mappingN

is calledL-compact onΩifQNΩis bounded andKpI−QN:Ω →Xis compact. Because

ImQis isomorphic to KerL, there exists an isomorphismJ: ImQ →KerL.

LetΩ⊂Rnbe open and bounded,f C1Ω, RnCΩ, RnandyRn\f∂ΩS f,

that is,yis a regular value off. Here,Sf {x∈Ω:Jfx 0}, the critical set off, andJf is

the Jacobian offatx. Then, the degree deg{f,Ω, y}is defined by

deg{f,Ω, y} xf−1y

sgnJfx, 2.5

with the agreement that the above sum is zero iff−1y . For more details about the

degree theory, one refers to the book of Deimling14.

Now, with the above notation, we are ready to state the continuation theorem.

Lemma 2.2Continuation theorem13, page 40. LetLbe a Fredholm mapping of index zero and

letNbeL-compact onΩ. Suppose that

afor eachλ∈0,1, every solutionxofLxλNxis such thatx/∂Ω; bQNx /0for eachx∂Ω∩KerLand

degJQN, Ω∩KerL, 0/0. 2.6

Then, the equationLx Nx has at least one solution lying inDomL∩Ω. For more details about degree theory, we refer to the book by Deimling [14].

For the simplicity of presentation, in the remaining part of this paper, we also introduce the following notation:

aimin

ait

, bijmaxbijt, IimaxIit,

Da1−b1α1 −b1β1 −b2α2 a2−b2β2

,

D1

I1 b1M1 −b1β1 I2 b2M2 a2−b2β2

, D2

a1−b1α1 I1 b1M1 −b2α2 I2 b2M2 .

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3. Main results

Theorem 3.1. SupposeH1holds andD1/D >0,D2/D > 0, then system1.5has aω-periodic

solution.

Proof. TakeX{ut x1t, x2tTCR, R2:ut ut ωfortR}and denote

|xi| max

t∈0,ω|xit|, i1,2; u0maxi1,2|xi|. 3.1

Equipped with the norm·0,Xis a Banach space. For anyutX, because of the periodicity, it is easy to check that

t −→ −a1txt b1tf1

xtτ11t

, ytτ12t

I1t −a2tyt b2tf2

xtτ21t

, ytτ22t

I2t

X. 3.2

Let

L: DomLuX:uC1R, R2u −→uX,

P :Xu −→uX, Q:X x −→xX, 3.3

where for anyγ γ1, γ2TR2, we identify it as the constant function inXwith the value

vectorγ γ1, γ2T. DefineN:XXgiven by

Nut −a1tx1t b1tf1

x1

tτ11t

, x2

tτ12t

I1t −a2tx2t b2tf2

x1

tτ21t

, x2

tτ22t

I2t

X. 3.4

Then, system1.5can be reduced to the operator equationLuNu. It is easy to see that

KerLR2,

ImL{xX :x0}, which is closed inX,

dim KerLcodim ImL2<,

3.5

andP,Qare continuous projectors such that

ImP KerL, KerQImLImI−Q. 3.6

It follows thatLis a Fredholm mapping of index zero. Furthermore, the generalized inverse

toLKp: ImL →KerP∩DomLis given by

Kpu

t

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

t

0

x1sds− 1 ω

ω

0

s

0

x1vdvds

t

0

x2sds

1

ω

ω

0

s

0

x2vdvds ⎞ ⎟ ⎟ ⎟ ⎟

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Thus, QNut ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 ω ω 0

a1sx1s b1sf1

x1

sτ11s

, x2

sτ12s

I1s ds 1 ω ω 0

a2sx2s b2sf2

x1

sτ21s

, x2

sτ22s

I2s ds ⎞ ⎟ ⎟ ⎟ ⎟ ⎠,

KpI−QNu t ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ t 0

a1sx1s b1sf1

x1

sτ11s

, x2

sτ12s

I1s

ds

t

0

a2sx2s b2sf2

x1

sτ21s

, x2

sτ22s

I2s ds ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ − ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 ω ω 0 s 0

a1vx1v b1vf1

x1

vτ11v

, x2

vτ12v

I1v

dvds

1 ω ω 0 s 0

a2vx2v b2vf2

x1

vτ21v

, x2

vτ22v

I2v

dvds

⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2− t ω ω 0

a1sx1s b1sf1

x1

sτ11s

, x2

sτ12s

I1s ds 1 2− t ω ω 0

a2sx2s b2sf2

x1

sτ21s

, x2

sτ22s

I2s ds ⎞ ⎟ ⎟ ⎟ ⎟ ⎠. 3.8

Clearly,QNandKpI−QNare continuous. For any bounded open subsetΩ⊂X,QNΩ

is obviously bounded. Moreover, applying the Arzela-Ascoli theorem, one can easily show thatKpI−QNΩis compact. Note thatKpI−QNis a compact operator and QNΩ

is bounded, therefore,N isL-compact onΩ with any bounded open subsetΩ ⊂ X. Since ImQ KerL, we take the isomorphismJ of ImQ onto KerL to be the identity mapping. Corresponding to equationLuλNu, λ∈0,1, we have

˙

x1t λ

a1tx1t b1tf1

x1t−τ11t

, x2t−τ12t

I1t

,

˙

x2t λ

a2tx2t b2tf2

x1

tτ21t

, x2

tτ22t

I2t

. 3.9

Now we reach the position to search for an appropriate open bounded subset Ω for the application of the Lemma 2.2. Assume that u utX is a solution of system 3.9 for someλ ∈0,1. Then, the componentsxit i 1,2ofutare continuously differentiable.

Thus, there existsti∈0, ωsuch that|xiti|maxt∈0,ω|xit|. Hence, ˙xiti 0. This implies

a1 t1 x1

t1b1 t1 f1 x1

t1−τ1

t1

, x2

t1−τ2

t1

I1

t1,

a2

t2

x2

t2b2 t2 f2 x1

t2−τ3

t2

, x2

t2−τ4

t2

I1

t2.

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Since

fi

x1, x2≤αix1 βix2 Mi fori1,2, 3.11

we get

x1

t1≤

α1b1x1

t1−τ11

t1 a1

β1b1x2

t1−τ12

t1 a1

b1M1 I1 a1

,

x2

t2≤

α2b2x1

t2−τ21

t2 a2

β2b2x2

t2−τ22

t2 a2

b2M2 I2 a2

.

3.12

Setk1D1/D,k2D2/D, we find that

k1 α1b1

a1 k1 β1b1

a1 k2

b1M1 I1 a1 ,

k2 α2b2

a2 k1

β2b2 a2

k2

b2M2 I2 a2

.

3.13

Now, we choose a constant numberd >1 and take

Ω x1, x2

T

R2; xi< dki fori1,2

, 3.14

wherek1D1/D >0,k2D2/D >0. We will show thatΩsatisfies all the requirements given

inLemma 2.2. In fact, we will prove that ifxit−τijt∈Ωthenxit∈Ωfori, j 1,2, t≥0.

Therefore, it means that u utis uniformly bounded with respect toλ when the initial value function belongs toΩ. It follows from3.12and3.13that

xi αibi ai

x1

tiτi1t βibi

ai

x1

tiτi2t biMi Ii ai

αibi ai dk1

βibi ai dk2

biMi Ii ai < d

α

ibi ai k1

βibi ai k2

biMi Ii ai

,

3.15

i1,2. Therefore,

xi

0< dki fori1,2. 3.16

Clearly,dki,i 1,2 are independent of λ. It is easy to see that there are noλ ∈ 0,1and u∂Ωsuch thatLu λNu. Ifu x1, x2T∂Ω∩KerL ∂ΩR2, thenuis a constant

vector inR2with|xi|dk

ifori1,2. Note thatQNuJQNu, we have

QNu ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

x11 ω

ω

0

a1tdt f1

x1, x2

1

ω

ω

0

b1tdt 1 ω

ω

0 I1tdt

x2

1

ω

ω

0

a2tdt f2

x1, x2

1

ω

ω

0 b2tdt

1

ω

ω

0 I2tdt

⎞ ⎟ ⎟ ⎟ ⎟

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We claim that

QNu

i>0 u∂Ω∩KerL∂ΩR3, i1,2. 3.18

Contrarily, suppose that there exists someisuch that|QNui|0, that is,

xi

1

ω

ω

0

aitdtfi

x1, x2

1

ω

ω

0 bitdt

1

ω

ω

0

Iitdt. 3.19

So, we have

dki|xi| ≤

1

ai

fi

x1, x2

1

ω

ω

0 bitdt

1

ω

ω

0 Iitdt

αibi ai

dk1 βibi

ai dk2

biMi Ii ai

< d

αibi ai

k1 βibi

ai k2

biMi Ii ai

dki,

3.20

this is a contradiction. Therefore,3.17holds, and hence,

QNu /0, for any u∂Ω∩KerL∂ΩR2. 3.21

Now, consider the homotopyF:Ω∩KerL×0,1 → Ω∩KerL, defined by

Fu, μ μdiag

1

ω

ω

0

a1tdt,

1

ω

ω

0 a2tdt

u 1−μQNu, 3.22

whereu∈Ω∩KerL Ω∩R2andμ0,1. Whenu∂ΩKerL∂ΩR2andμ0,1, u x1, x2Tis a constant vector inR2with|xi|dkifori1,2. Thus

Fu, μ

0maxi1,2

xi1

ω

ω

0

aitdt 1−μfi

x1, x2

1

ω

ω

0 bitdt

. 3.23

We claim that

Fu, μ/ 0 u, μ∈∂Ω∩KerL×0,1. 3.24

Contrarily, suppose thatFu, μ00, then,

xi

1

ω

ω

0

aitdt 1−μfi

x1, x2

1

ω

ω

0

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Thus,

dki|xi| ≤

1−μ

ai bifi

x1, x2

< d

αibi ai

k1 βibi

ai

k2 biMi Ii ai

dki. 3.26

This is impossible. Thus,3.22holds. From the property of invariance under a homotopy, it follows that

degJQN,Ω∩KerL,0

degF·,0,Ω∩KerL,0

degF·,1,Ω∩KerL,0

sgn

−1 ω

ω

0

a1tdt 0

0 −1

ω

ω

0 a2tdt

sgn

1

ω

ω

0 a1tdt

1

ω

ω

0 a2tdt

/ 0.

3.27

We have shown thatΩsatisfies all the assumptions of Lemma 2.2. Hence, Lu Nuhas at least oneω-periodic solution on DomL∩Ω. This completes the proof.

Corollary 3.2. Suppose that there exist positive constantsMisuch that|fix1, x2| ≤Mifori1,2.

Then system1.5has at least anω-periodic solution.

Proof. As|fix1, x2| ≤Mii1,2,implies thatαi, βi 0, hence, the conditions in Theorem

2.1 are all satisfied.

Remark 3.3. From Corollary 3.2, we can find that the conditionai > bi in4, Theorem 2.1

can be dropped out. Therefore, Theorem 3.1is greatly generalized results to 4, Theorem 2.1. Furthermore, we should point out that theTheorem 3.1is different from the the existing work in5when without assuming the boundedness, monotonicity, and differentiability of activation functions. In fact, the explicit presence ofωin Theorem 2.1see5may impose a very strict constraint on the coefficients of1.5 e.g., whenωis very large or small. Since our results are presented independent ofω, it is more convenient to design a neural network with delays.

Theorem 3.4. Suppose thatfi, i1,2satisfy the hypothesesH2. IfD1/D >0,D2/D >0, then

system1.5has exactly one periodic solutionxt . Moreover, it is globally exponentially stable.

Proof. From H2, we can conclude H1 is true. It is obvious that all the hypotheses in

Theorem 2.1 hold withMi |fi0,0|, i 1,2.Thus, system1.5has at least one periodic

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calculation of the upper left derivative of|xit−xit|along the solutions of system1.5

leads to

D xit−xit

D−sgnxit−xit

xit−xit

≤ −aitx1t−x1t bit ×fi

x1

tτi1t

,x2

tτi2t

fi

x1

tτi1t

, x2

tτi2t ≤ −aitx1t−x1t bit

×αix1

tτi1t

x1

tτi1t βix2

tτi2t

x2

tτi2t,

3.28

whereD−denotes the upper left derivative,i1,2. Letzit |xit−xit|. Then,3.27can

be transformed into

Dzit≤ −aizit biαi sup tτst

z1s biβi sup tτst

z2s. 3.29

FromD1/D >0,D2/D >0 and3.13, we have

k1> b1α1

a1 k1 b1β1

a1 k2,

k2> b2α2

a2 k1

b2β2 a2

k2.

3.30

Then, there exists a constantσ >0 such that

k1a1 k1b1α1 k2b1β1< σ,k2a2 k1b2α2 k2b2β2< σ.

3.31

Thus, we can choose a constant 0< λ1 such that

λki

kiai

k1biαi k2biβi

eλτ<0, i1,2. 3.32

Now, we choose a constantN1 such that

Nkieλt>1,t∈−τ,0, i1,2. 3.33

Set

Yit Nki

sup

τs≤0

z1s sup

τs≤0

z2s ε

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for∀ε >0. From3.32and3.34, we obtain

DYit>

kiai

k1biαi k2biβi

eλτN

2

j1

sup

τs≤0

zjs ε

eλt

aiYit biαi sup tτst

Y1s biβi sup tτst

Y2s.

3.35

In view of3.33and3.34, we have

Yit > 2

j1

sup

τs≤0

zjs ε > zit, ∀t∈−τ,0. 3.36

We claim that

zit< Yit, ∀t >0, i1,2. 3.37

Contrarily, there must existi∈ {1,2}andti>0 such that

ziti Yiti, zit< Yit ∀t∈−τ, ti, i1,2. 3.38

It follows that

0≤Dzi

ti

Yi

ti

≤lim sup

h→0−

zi

ti h

zi

ti

h −lim infh→0−

Yi

ti h

Yi

ti

h

Dzi

ti

DYi

ti

.

3.39

From3.29,3.35, and3.38, we obtain

Dzi

ti

≤ −aizi

ti

biαi sup tiτsti

z1s biβi sup tiτsti

z2s< DYi

ti

, 3.40

which contradicts3.39. Hence,3.37holds. Lettingε → 0 andm max1≤i≤2{Nki 1},

from3.34and3.37, we have

xit−xitzit≤Nki

2

j1

sup

τs≤0

zjs ε

eλt

xeλt, fort≥0, i1,2.

3.41

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0 50 100 150 Timet

−0.6

−0.4

−0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Solution

x1

,x

2

Figure 1:Numerical solutionx1t, x2tof system4.1, whereτ11t 5, τ12t 2, τ21t 3, τ22t 6,

x1s −0.2, x2s 1.4 fors∈−6,0.

Remark 3.5. From Theorem 3.4, it is easy to see that our results independent ofω and τ, we have dropped out the condition b of Theorem 3.1 in 4 and the condition: ai > αi βi|bi|ea

of Theorem 3.1 in5. Therefore, it is generalized the corresponding results

in4,5.

4. Example and numerical simulation

In this section, we will give a example to illustrate our results. Using the method of numerical simulation in15, we will find that the theoretical conclusions are in excellent agreement with the numerically observed behavior.

Example 4.1. Consider a two-neuron networks with delays as follows:

˙

x1t −a1tx1t b1tf1

x1t−τ11t

, x2

tτ12t

I1t,

˙

x2t −a2tx2t b2tf2

x1

tτ21t

, x2

tτ22t

I2t,

4.1

wherea1t 49/8 cos t,a2t 15/2 sint,b1t 1 2 cost,b2t 1−sint,I1t sint, I2t 2 cost,f1x1, x2 cos1/4x1 1/8|x1| sinx2−1/2|x2| 1,f2x1, x2 sinx1− 1/4x1 cos1/2x2 1/4|x2| 2, and 0 ≤ τijt < ∞ are any 2π-periodic continuous

functions fori, j1,2.

Obviously, the requirements of smoothness, monotonicity, and boundedness on the activation functions are relaxed in our model. By some simple calculations, we obtainα1

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−0.5 0 0.5 1

x1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x2

Figure 2:Phase trajectories of system4.1.

5. Conclusion

In this paper, we have derived sufficient algebraic conditions in terms of the parameters of the connection and activation functions for periodic solutions and global exponential stability of a two-neuron networks with time-varying delays. The obtained results extend and improve some earlier publications, which are all independent of the delays and the periodωand may be highly important significance in some applied fields.

Acknowledgments

This work was supported in part by the High-Tech Research and Development Program of China under Grant no. 2006AA04A104, China Postdoctoral Science Foundation under Grants no. 20070410300 and no. 200801336, the Foundation of Chinese Society for Electrical Engineering, and the Hunan Provincial Natural Science Foundation of China under Grant no. 07JJ4001.

References

1 J. Wu,Introduction to Neural Dynamics and Signal Transmission Delay, vol. 6 ofde Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 2001.

2 M. Z. Baptistini and P. Z. T´aboas, “On the existence and global bifurcation of periodic solutions to planar differential delay equations,”Journal of Differential Equations, vol. 127, no. 2, pp. 391–425, 1996.

3 S. M. S. Godoy and J. G. dos Reis, “Stability and existence of periodic solutions of a functional differential equation,”Journal of Mathematical Analysis and Applications, vol. 198, no. 2, pp. 381–398, 1996.

4 C. Huang and L. Huang, “Existence and global exponential stability of periodic solution of two-neuron networks with time-varying delays,”Applied Mathematics Letters, vol. 19, no. 2, pp. 126–134, 2006.

5 C. Huang, L. Huang, and T. Yi, “Dynamics analysis of a class of planar systems with time-varying delays,”Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1233–1242, 2006.

(14)

7 P. T´aboas, “Periodic solutions of a planar delay equation,”Proceedings of the Royal Society of Edinburgh. Section A, vol. 116, no. 1-2, pp. 85–101, 1990.

8 K. Liu and G. Yang, “Strict stability criteria for impulsive functional differential systems,”Journal of Inequalities and Applications, vol. 2008, Article ID 243863, 8 pages, 2008.

9 Y. Chen and J. Wu, “Slowly oscillating periodic solutions for a delayed frustrated network of two neurons,”Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 188–208, 2001.

10 K. Gopalsamy and S. Sariyasa, “Time delays and stimulus-dependent pattern formation in periodic environments in isolated neurons,”IEEE Transactions on Neural Networks, vol. 13, no. 3, pp. 551–563, 2002.

11 K. Gopalsamy and S. Sariyasa, “Time delays and stimulus dependent pattern formation in periodic environments in isolated neurons—II,”Dynamics of Continuous, Discrete & Impulsive Systems. Series B, vol. 9, no. 1, pp. 39–58, 2002.

12 M. Forti and A. Tesi, “New conditions for global stability of neural networks with application to linear and quadratic programming problems,”IEEE Transactions on Circuits and Systems. I, vol. 42, no. 7, pp. 354–366, 1995.

13 R. E. Gaines and J. L. Mawhin,Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977.

14 K. Deimling,Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.

Figure

Figure 1: Numerical solution x1�t�, x2�t� of system �4.1�, where τ11�t� � 5, τ12�t� � 2, τ21�t� � 3, τ22�t� � 6,x1�s� � −0.2, x2�s� � 1.4 for s ∈ �−6, 0�.

Figure 1.

Numerical solution x1 t x2 t of system 4 1 where 11 t 5 12 t 2 21 t 3 22 t 6 x1 s 0 2 x2 s 1 4 for s 6 0 . View in document p.12
Figure 2: Phase trajectories of system �4.1�.

Figure 2.

Phase trajectories of system 4 1 . View in document p.13

References

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