dynamics in bam fuzzy neural networks with delays and ......international mathematical forum, 3,...

International Mathematical Forum, 3, 2008, no. 20, 979 - 1000

Dynamics in BAM Fuzzy Neural Networks

with Delays and Reaction-Diffusion Terms1

Zuoan Li

Department of Mathematics

Sichuan University of Science & Engineering

Sichuan 643000, China

[email protected]

Abstract

In this paper, employing some analysis techniques and Lyapunovfunctional, the existence, uniqueness and global exponential stabilitiesof both the equilibrium point and the periodic solution are investigatedfor a class of bi-directional associative memory (BAM) fuzzy neural net-works with delays and reaction-diffusion terms. We obtain two concisesufficient conditions ensuring the existence, uniqueness and global expo-nential stability of both the equilibrium point and the periodic solution.Moreover, an illustrate example is given to show the effectiveness of ob-tained results.

Mathematics Subject Classification: 92B20, 34K20, 34K13

Keywords: Bi-directional associative memory; fuzzy neural networks;

reaction-diffusion; delays; global exponential stability; periodic solution

1 Introduction

The bi-directional associative memory (BAM) neural networks was first in-

troduced by Kosto [1]. It is important model with the ability of information

memory and information association, which is crucial for application in pattern

recognition, solving optimization problems and automatic control engineering

[2]-[4]. In such applications, the stability of networks plays an important role,

it is of significance and necessary to investigate the stability. In both biological

1This work was jointly supported by grant 2006A109 and 07ZA047 from the ScientificResearch Fund of Sichuan Provincial Education Department.

980 Zuoan Li

and man-made neural networks, the delays arise because of the processing of

information [5]. Time delays may lead to oscillation, divergence, or instability

which may be harmful to a system [5, 6]. Therefore, study of neural dynamics

with consideration of the delayed problem becomes extremely important to

manufacture high quality neural networks. Recently, BAM neural networks

have been extensively studied both in theory and applications, for example,

see [3]-[17] and references therein.

On the other hand, the fuzzy cellular neural networks (FCNN) was intro-

duced by Yang in 1996 [18], it combines fuzzy logic with traditional cellular

neural networks (CNN). Studies have shown the potential of FCNN in image

producing and pattern recognition. Many stability conditions have been given

for FCNN [19]-[21]. In [19], the authors have obtained some conditions for the

existence and the global stability of the equilibrium point of FCNN without

delay. In [20], Liu and Tang have considered FCNN with either constant de-

lays or time-varying delays, several sufficient conditions have been obtained

to ensure the existence and uniqueness of the equilibrium point and its global

exponential stability. In [21], Yuan, Cao and Deng have given several criteria

of dynamics for FCNN with time-varying delays. However, as pointed out in

[16, 22, 23], diffusion effect cannot be avoided in the neural networks when

electrons are moving in asymmetric electromagnetic fields, so we must con-

sider the activations vary in space as well as in time. Recently, Huang has

considered the stability of FCNN with diffusion terms and time-varying delay,

the model is expressed by partial differential equations [23]. To the best of our

knowledge, few authors have considered BAM fuzzy neural networks model

with delays and reaction-diffusion terms.

Moreover, studies on neural dynamical systems not only involve the discus-

sion of stability properties, but also involve many dynamic behaviors such as

periodic oscillatory behavior, bifurcation, and chaos. In many applications, the

properties of periodic oscillatory solutions and global exponential stability are

of great interest. For example, the human brain has been in periodic oscillatory

or chaos state, hence it is of prime importance to study periodic oscillation,

global exponential stability and chaos phenomenon of neural networks.

Motivated by the above discussions, the objective of this paper is to study

the existence, uniqueness and global exponential stability of both the equi-

librium point and the periodic solution for BAM fuzzy neural networks with

delays and reaction-diffusion terms. The model is described by the following

Dynamics in BAM fuzzy neural networks 981

functional differential equation⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂ui(t,x)∂t

=l∑

k=1

∂∂xk

(Dik∂ui(t,x)∂xk

) − aiui(t, x) +m∑j=1

ajigj(vj(t, x))

+m∑j=1

ajiwj(t) + Ii(t) +m∧j=1

αjigj(vj(t − τji, x))ds

+m∨j=1

αjigj(vj(t − τji, x)) +m∧j=1

Tjiwj(t) +m∨j=1

Hjiwj(t),

∂vj(t,x)

∂t=

l∑k=1

∂∂xk

(Djk∂vj(t,x)

∂xk) − bjvj(t, x) +

n∑i=1

bijfi(ui(t, x))

+n∑i=1

bijwi(t) + Jj(t) +n∧i=1

βijfi(ui(t − σij , x))ds

+n∨i=1

βijfi(ui(t − σij , x)) +n∧i=1

Tijwi(t) +n∨i=1

Hijwi(t),

(1)

for i ∈ I := {1, 2, · · · , n}, j ∈ J := {1, 2, · · · , m}, t > 0, where x = (x1, x2, · · · , xl, )T ∈Ω ⊂ Rl, Ω is a compact set with smooth boundary and mesΩ > 0 in space

Rl; u = (u1, u2, · · · , un)T ∈ Rn, v = (v1, v2, · · · , vm)T ∈ Rm. ui(t, x) and vj(t, x)

are the state of the ith neuron and the jth neuron at time t and in space

x, respectively; fi and gj denote the signal functions of the ith neuron and

the jth neuron at time t and in space x, respectively; wi(t) and wj(t) denote

inputs of the ith neuron and the jth neuron at the time t, respectively; and

Ii(t) and Jj(t) denote bias of the ith neuron and the jth neuron at the time

t, respectively; ai > 0, bj > 0, aji, aji, αji, αji, bij , bij, βij , βij are constants, aiand bj represent the rate with which the ith neuron and the jth neuron will

reset their potential to the resting state in isolation when disconnected from

the networks and external inputs, respectively; aji, bij and aji, bij denote con-

nection weights of feedback template and feedforward template, respectively;

αji, βij and αji, βij denote connection weights of the delays fuzzy feedback MIN

template and the delays fuzzy feedback MAX template, respectively; Tji, Tijand Hji, Hij are elements of fuzzy feedforward MIN template and fuzzy feed-

forward MAX template, respectively;∧

and∨

denote the fuzzy AND and

fuzzy OR operation, respectively; smooth functions Dik = Dik(t, x, u) ≥ 0 and

Djk = Djk(t, x, v) ≥ 0 correspond to the transmission diffusion operators along

the ith neuron and the jth neuron, respectively.

The boundary conditions and initial conditions are given by{∂ui

∂n:= ( ∂ui

∂x1, ∂ui

∂x2, · · · , ∂ui

∂xl)T = 0, i ∈ I,

∂vj

∂n:= (

∂vj

∂x1,∂vj

∂x2, · · · , ∂vj

∂xl)T = 0, j ∈ J,

(2)

and{ui(t, x) = φui(t, x), −τ ≤ t ≤ o, τ = max1≤i≤n,1≤j≤m{τji},vj(t, x) = φvj(t, x), −σ ≤ t ≤ 0, σ = max1≤i≤n,1≤j≤m{σij}, (3)

982 Zuoan Li

where φui(t, x), φvj(t, x) (i ∈ I, j ∈ J) are bounded and continuous on [−τ, 0]×Ω, [−σ, 0] × Ω, respectively.

Throughout this paper, we make the following assumptions:

(H1) The neurons activation functions fi and gj (i ∈ I, j ∈ J) are Lipschitz-

continuous, that is, there exist constants Fi > 0 and Gj > 0 such that

|fi(ξ1) − fi(ξ2)| ≤ Fi|ξ1 − ξ2|, |gj(ξ1) − gj(ξ2)| ≤ Gj|ξ1 − ξ2|for all ξ1, ξ2 ∈ R.

(H2) There exist constants λi > 0, λn+j > 0(i ∈ I, j ∈ J) such that⎧⎪⎪⎨⎪⎪⎩

−2λiai + λim∑j=1

(|aji| + |αji| + |αji|) + F 2i

m∑j=1

λn+j(|bij | + |βij | + |βij |) < 0,

−2λn+jbj + λn+j

n∑i=1

(|bij | + |βij | + |βij |) +G2j

n∑i=1

λi(|aji| + |αji| + |αji|) < 0

for i ∈ I, j ∈ J.To prove our main results, we need the following lemma:

Lemma 1 [19] Suppose u and u∗ are two state of model (1), then we have∣∣∣ n∧j=1

αijfj(uj) −n∧j=1

αijfj(u∗j )∣∣∣ ≤ n∑

j=1

∣∣∣αij∣∣∣ · ∣∣∣fj(uj) − fj(u∗j )∣∣∣,

∣∣∣ n∨j=1

βijfj(uj) −n∨j=1

βijfj(u∗j )∣∣∣ ≤ n∑

j=1

∣∣∣βij∣∣∣ · ∣∣∣fj(uj) − fj(u∗j )∣∣∣.

2 Global exponential stability of the equilib-

rium point

In this section, we will discuss the existence, uniqueness and global exponentialstability of the equilibrium point of BAM fuzzy neural networks with delays anddiffusion terms, and give their proofs. Consider the case of model (1) as wi(t) = wi,wj(t) = wj, Ii(t) = Ii and Jj(t) = Jj (i ∈ I, j ∈ J), i.e.,⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂ui(t,x)∂t =

l∑k=1

∂∂xk

(Dik∂ui(t,x)∂xk

) − aiui(t, x) +m∑j=1

ajigj(vj(t, x))

+m∑j=1

ajiwj + Ii +m∧j=1

αjigj(vj(t− τji, x))

+m∨j=1

αjigj(vj(t− τji, x)) +m∧j=1

Tjiwj +m∨j=1

Hjiwj ,

∂vj(t,x)∂t =

l∑k=1

∂∂xk

(Djk∂vj(t,x)∂xk

) − bjvj(t, x) +n∑i=1

bijfi(ui(t, x))

+n∑i=1

bijwi + Jj +n∧i=1

βijfi(ui(t− σij, x))

+n∨i=1

βijfi(ui(t− σij , x)) +n∧i=1

Tijwi +n∨i=1

Hijwi

(1′)


for i ∈ I, j ∈ J.We introduce a notation before giving the following definition. For u(t, x) =

(u1(t, x), u2(t, x), · · · , uk(t, x))T ∈ Rk, denote

‖ui(t, x)‖2 =[ ∫

Ω|ui(t, x)|2dx

] 12, i = 1, 2, · · · , k.

Definition 1 The equilibrium point (u∗, v∗)T of model (1′) is said to be globallyexponentially stable, if there exist constants ε > 0 and M ≥ 1 such that

n∑i=1

‖ ui(t, x) − u∗i ‖22 +

m∑j=1

‖ vj(t, x) − v∗j ‖22≤Me−εt(‖ φu − u∗ ‖ + ‖ φv − v∗ ‖)

for all t ≥ 0, where (u1(t, x), · · · , un(t, x), v1(t, x), · · · , vm(t, x))T is any solution ofmodel (1′), u∗ = (u∗1, · · · , u∗n)T , v∗ = (v∗1 , · · · , v∗m)T , φu = (φu1, φu2, · · · , φun)T , φv =

(φv1, φv2, · · · , φvm)T , and ‖φu − u∗‖ = sup−τ≤t≤0

n∑i=1

‖φui(t, x) − u∗i ‖22, ‖φv − v∗‖ =

sup−σ≤t≤0

m∑j=1

‖φvj(t, x) − v∗j ‖22.

Theorem 1 Under assumptions (H1) and (H2), then there is exactly one equilib-rium point of model (1′).

Proof.For the sake of simplification, let⎧⎪⎪⎨⎪⎪⎩

Ii =m∑j=1

ajiwj + Ii +m∧j=1

Tjiwj +m∨j=1

Hjiwj , i ∈ I,

Jj =n∑i=1

bijwi + Jj +n∧i=1

Tijwj +n∨i=1

Hijwi, j ∈ J,

then model (1′) is reduced to⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂ui(t,x)∂t =

l∑k=1

∂∂xk

(Dik∂ui(t,x)∂xk

) − aiui(t, x) +m∑j=1

ajigj(vj(t, x))

+m∧j=1

αjigj(vj(t− τji, x)) +m∨j=1

αjigj(vj(t− τji, x)) + Ii,

∂vj(t,x)∂t =

l∑k=1

∂∂xk

(Djk∂vj(t,x)∂xk

) − bjvj(t, x) +n∑i=1

bijfi(ui(t, x))

+n∧i=1

βijfi(ui(t− σij , x)) +n∨i=1

βijfi(ui(t− σij, x)) + Jj

(4)

for i ∈ I, j ∈ J.It is evident that the dynamical characteristics of model (1′) is the same as of

model (4).We denote

984 Zuoan Li

h(u1, · · · , un, v1, · · · , vm) = (h1, · · · , hn, h1, · · · , hm)T , where⎧⎪⎪⎨⎪⎪⎩

hi = aiui −m∑j=1

ajigj(vj) −m∧j=1

αjigj(vj) −m∨j=1

αjigj(vj) − Ii, i ∈ I,

hj = bjvj −n∑i=1

bijfi(ui) −n∧i=1

βijfi(ui) −n∨i=1

βijfi(ui) − Jj , j ∈ J.

Obviously, the equilibrium points of model (4) are the solutions of system of equa-tions {

hi = 0, i ∈ I,hj = 0, j ∈ J.

(5)

Define the following homotopic mapping:

H(u1, · · · , un, v1, · · · , vm) = λh(u1, · · · , un, v1, · · · , vm)

+(1 − λ)(u1, · · · , un, v1, · · · , vm)T ,

where λ ∈ [0, 1]. Let Hk(k = 1, 2, · · · , n+m) denote the kth component of H(u1, · · · ,un, v1, · · · , vm), then from Lemma 1, we can get the following inequalities⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

|Hi| ≥ λai|ui| − λm∑j=1

(|aji| + |αji| + |αji|)|gj(vj) − gj(0)|

−λm∑j=1

(|aji| + |αji| + |αji|)|gj(0)| − λ|Ii|,

|Hn+j | ≥ λbj |vj | − λn∑i=1

(|bij | + |βij | + |βij |)|fi(ui) − fi(0)|

−λn∑i=1

(|bij | + |βij | + |βij |)|fi(0)| − λ|Jj |

(6)

for i ∈ I, j ∈ J. In fact, for i ∈ I, we have

|Hi| ≥ (1 − λ)|ui| + λai|ui| − λ

n∑j=1

(|aji| + |αji| + |αji|)|gj(vj)| − λ|Ii|

≥ λai|ui| − λn∑j=1

(|aji| + |αji| + |αji|)|gj(vj) − gj(0) + gj(0)| − λ|Ii|

≥ λai|ui| − λ

n∑j=1

(|aji| + |αji| + |αji|)|gj(vj) − gj(0)|

−λn∑j=1

(|aji| + |αji| + |αji|)|gj(0)| − λ|Ii|,

also, for j ∈ J, we have

|Hn+j | ≥ λbj |vj | − λ

n∑i=1

(|bij | + |βij | + |βij |)|fi(ui) − fi(0)|

−λn∑i=1

(|bij | + |βij | + |βij |)|fi(0)| − λ|Jj |.


It follows from the inequality 2ab ≤ a2 + b2 and (H1) that

n∑i=1

λi|ui||Hi| ≥ λ

n∑i=1

[λiai|ui|2 − 1

2λi|ui|2

m∑j=1

(|aji| + |αji| + |αji|)

−12λi

m∑j=1

(|aji| + |αji| + |αji|)G2j |vj |2

]

−λn∑i=1

λi|ui|[|Ii| +

m∑j=1

(|aji| + |αji| + |αji|)|gj(0)|]

(7)

and

m∑j=1

λn+j |vj ||Hn+j | ≥ λ

m∑j=1

[λn+jbj|vj |2 − 1

2λn+j |vj|2

n∑i=1

(|bij | + |βij | + |βij |)

−12λn+j

n∑i=1

(|bij | + |βij | + |βij |)F 2i |ui|2

]

−λm∑j=1

λn+j |vj|[|Ji| +

n∑i=1

(|bij | + |βij | + |βij |)|fi(0)|]. (8)

We obtain from (7) and (8) that

n∑i=1

λi|ui||Hi| +m∑j=1

λn+j |vj ||Hn+j | ≥ λρ‖w‖2 − λμ√n+m‖w‖, (9)

where

ρ = min1≤k≤n+m

{ρk},

ρi = λiai − 12λi

∑mj=1(|aji| + |αji| + |αji|) − 1

2F2i

m∑j=1

λn+j(|bij | + |βij | + |βij |),

ρn+j = λn+jbj− 12λn+j

∑ni=1(|bij |+|βij |+|βij |)− 1

2G2j

n∑i=1

λi(|aji|+|αji|+|αji|),

μ = max1≤k≤n

{μk},

μi = λi

[|Ii| +

m∑j=1

(|aji| + |αji| + |αji|)|gj(0)|],

μn+j = λn+j

[|Jj | +

n∑i=1

(|bij | + |βij | + |βij |)|fi(0)|],

‖w‖ =√w2

1 + · · · + w2n + v2

1 + · · · + v2m.

986 Zuoan Li

Define

Γ ={w = (u1, · · · , un, v1, · · · , vm)T | ‖w‖ ≤

√n+m(μ+ 1)

ρ

},

then, it follows that for any w = (u1, · · · , un, v1, · · · , vm)T ∈ ∂Γ, we have ‖w‖ =√n+m(μ+1)

ρ , hence

n∑i=1

λi|ui||Hi| +m∑j=1

λn+j |vj ||Hn+j | ≥ λρ‖w‖2 − λμ√n+m‖w‖

= λρ(n+m)(μ+ 1)2

ρ2

−λμ√n+m

√n+m(μ+ 1)

ρ

> 0, λ ∈ (0, 1].

this means H(u1, · · · , un, v1, · · · , vm) �= 0, for any (u1, · · · , un, v1, · · · , vm)T ∈ ∂Γ, λ ∈(0, 1]. Also, as λ = 0, H(u1, · · · , un, v1, · · · , vm) = (u1, · · · , un, v1, · · · , vm) �= 0, forany (u1, · · · , un, v1, · · · , vm)T ∈ ∂Γ. Hence, we have H(u1, · · · , un, v1, · · · , vm) �= 0,for any (u1, · · · , un, v1, · · · , vm)T ∈ ∂Γ, λ ∈ [0, 1].

From homotopy invariance theorem [24], we get

deg(h,Γ, 0) = deg(H,Γ, 0) = 1,

by topological degree theory, we know that (4) has at least one solution in Γ. Thatis, model (1′) has at least an equilibrium point.

If (u1, · · · , un, v1, · · · , vm)T and (u∗1, · · · , u∗n, v∗1 , · · · , v∗m)T are two equilibrium pointsof the model (4), by using of Lemma1, (H1) and the inequality 2ab ≤ a2 + b2, weeasily obtain the following inequalities

λiai|ui − u∗i |2 ≤ 12λi|ui − u∗i |2

m∑j=1

(|aji| + |αji| + |αji|)

+12λi

m∑j=1

(|aji| + |αji| + |αji|)G2j |vj − v∗j |2, (10)

and

λn+jbj|vj − v∗j |2 ≤ 12λn+j |vj − v∗j |2

n∑i=1

(|bij | + |βij | + |βij |)

+12λn+j

n∑i=1

(|bij | + |βij | + |βij |)F 2i |ui − u∗i |2. (11)

It follows from (10) and (11) thatn∑i=1

[λiai − 1

2λi

m∑j=1

(|aji| + |αji| + |αji|)


−12F 2i

m∑j=1

λn+j(|bij | + |βij | + |βij |)]|ui − u∗i |2

+m∑j=1

[λn+jbj − 1

2λn+j

n∑i=1

(|bij | + |βij | + |βij |)

−12G2j

n∑i=1

λi(|aji| + |αji| + |αji|)]|vj − v∗j |2 ≤ 0.

This implies from (H2) that ui = u∗i , vj = v∗j , i ∈ I, j ∈ J. Therefore, the system (4)has one unique equilibrium point. The proof is completed.

Theorem 2 Under assumptions (H1) and (H2), the equilibrium point of model (1′)is globally exponentially stable.

Proof. Since (H2) holds, we can choose a small ε > 0 such that

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2λi(ε− ai) + λim∑j=1

(|aji| + |αji| + |αji|) + F 2i

×m∑j=1

λn+j

[|bij | + (|βij | + |βij |)e2εσ

]< 0,

2λn+j(ε− bj) + λn+j

n∑i=1

(|bij | + |βij | + |βij |) +G2j

×n∑i=1

λi

[|aji| + (|αji| + |αji|)e2ετ

]< 0

(12)

for i ∈ I, j ∈ J. Let (u∗, v∗) be the unique equilibrium point of the system (1′),rewrite model (1′) as

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂(ui−u∗i )∂t =

l∑k=1

∂∂xk

(Dik∂(ui−u∗i )∂xk

) − ai(ui − u∗i ) +m∑j=1

aji[gj(vj) − gj(vj∗)]

+m∧j=1

αjigj(vj(t− τji, x)) −m∧j=1

αjigj(v∗j )

+m∨j=1

αjigj(vj(t− τji, x)) −m∨j=1

αjigj(v∗j ),

∂(vj−v∗j )

∂t =l∑

k=1

∂∂xk

(Djk∂(vj−v∗j )

∂xk) − bj(vj − v∗j ) +

n∑i=1

bij [fi(ui) − fi(u∗i )]

+n∧i=1

βijfi(ui(t− σij, x)) −n∧i=1

βijfi(u∗i )

+n∨i=1

βijfi(ui(t− σij, x)) −n∨i=1

βijfi(u∗i )

(13)

for i ∈ I, j ∈ J. By using of Lemma 1 and hypotheses (H1), we can obtain as

988 Zuoan Li

following inequalities⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(ui − u∗i )∂(ui−u∗i )

∂t ≤ (ui − u∗i )l∑

k=1

∂∂xk

(Dik∂(ui−u∗i )∂xk

) − ai(ui − u∗i )2

+|ui − u∗i |m∑j=1

|aji|Gj |vj − v∗j |

+|ui − u∗i |m∑j=1

(|αji| + |αji|)Gj |vj(t− τji, x) − v∗j |,

(vj − v∗j )∂(vj−v∗j )

∂t ≤ (vj − v∗j )l∑

k=1

∂∂xk

(Djk∂(vj−v∗j )

∂xk) − bj(vj − v∗j )

2

+|vj − v∗j |n∑i=1

|bij |Fi|ui − u∗i |

+|vj − v∗j |n∑i=1

(|βij | + |βij |)Fi|ui(t− σij , x) − u∗i |.

(14)

Now we construct the Lyapunov functional as follows

V (t) =∫

Ω

n∑i=1

λi

[|ui − u∗i |2e2εt +

m∑j=1

(|αji| + |αji|)G2j

×∫ t

t−τji

|vj(s, x) − v∗j |2e2ε(s+τji)ds]dx

+∫

Ω

m∑j=1

λn+j

[|vj − v∗j |2e2εt +

n∑i=1

(|βij | + |βij |)F 2i

×∫ t

t−σij

|ui(s, x) − u∗i |2e2ε(s+σij )ds]dx.

Calculating the upper right Dini derivative D+V (t) of V (t) along the solution ofmodel (1′), and by using of some analysis techniques, we have

D+V (t) =∫

Ω

n∑i=1

λi

[2(ui − u∗i )

∂(ui − u∗i )∂t

e2εt + 2εe2εt(ui − u∗i )2

+m∑j=1

(|αji| + |αji|)G2j (vj(t, x) − v∗j )

2e2ε(t+τji)

−m∑j=1

(|αji| + |αji|)G2j (vj(t− τji, x) − v∗j )

2e2εt]dx

+∫

Ω

m∑j=1

λn+j

[2(vj − v∗j )

∂(vj − v∗j )∂t

e2εt + 2εe2εt(vj − v∗j )2

+n∑i=1

(|βij | + |βij |)Fi(ui(t, x) − u∗i )2e2ε(t+σij )

−n∑i=1

(|βij | + |βij |)F 2i (ui(t− σij , x) − u∗i )

2e2εt]dx.


≤∫

Ω

n∑i=1

λi

[2e2εt(ui − u∗i )

l∑k=1

∂

∂xk

(Dik

∂(ui − u∗i )∂xk

)

+2e2εt(ε− ai)(ui − u∗i )2 + 2e2εt|ui − u∗i |

m∑j=1

|aji|Gj |vj − v∗j |

+2e2εt|ui − u∗i |m∑j=1

(|αji| + |αji|)Gj |vj(t− τji, x) − v∗j |

+m∑j=1

(|αji| + |αji|)G2j (vj(t, x) − v∗j )

2e2ε(t+τji)

−e2εtm∑j=1


2]dx

+∫

Ω

m∑j=1

λn+j

[2e2εt(vj − v∗j )

l∑k=1

∂

∂xk

(Djk

∂(vj − v∗j )∂xk

)

+2e2εt(ε− bj)(vj − v∗j )2 + 2e2εt|vj − v∗j |

n∑i=1

|bij |Fi|ui − u∗i |

+2e2εt|vj − v∗j |n∑i=1

(|βij | + |βij |)Fi|ui(t− σij, x) − u∗i |

+e2εtn∑i=1

(|βij | + |βij |)F 2i (ui − u∗i )

2e2εσij

−e2εtn∑i=1


2]dx

≤ e2εt∫

Ω

n∑i=1

λi

[2(ui − u∗i )

l∑k=1

∂

∂xk

(Dik


)

+2(ε− ai)(ui − u∗i )2 +

m∑j=1

|aji|(ui − u∗i )2

+m∑j=1

|aji|G2j |vj − v∗j |2 +

m∑j=1

(|αji| + |αji|)|ui − u∗i |2

+m∑j=1

(|αji| + |αji|)G2j |vj(t− τji, x) − v∗j |2

+m∑j=1

(|αji| + |αji|)G2j (vj − v∗j )

2e2ετ

−m∑j=1


2]dx

990 Zuoan Li

+e2εt∫

Ω

m∑j=1

λn+j

[2(vj − v∗j )

l∑k=1

∂

∂xk

(Djk


)

+2(ε− bj)(vj − v∗j )2 +

n∑i=1

|bij |(vj − v∗j )2

+n∑i=1

|bij |F 2i |ui − u∗i |2 +

n∑i=1

(|βij | + |βij |)|vj − v∗j |2

+n∑i=1

(|βij | + |βij |)F 2i |ui(t− σij , x) − u∗i |2

+n∑i=1

(|βij | + |βij |)F 2i (ui − u∗i )

2e2εσ

−n∑i=1


2]dx

= 2e2εt∫

Ω

n∑i=1

λi(ui − u∗i )l∑

k=1

∂

∂xk

(Dik


)dx

+2e2εt∫

Ω

m∑j=1

λn+j(vj − v∗j )l∑

k=1

∂

∂xk

(Djk


)dx

+e2εt∫

Ω

n∑i=1

[(2λi(ε− ai) + λi

m∑j=1

(|aji| + |αji| + |αji|)

+F 2i

m∑j=1

λn+j [|bij | + (|βij | + |βij |)e2εσ])|ui − u∗i |2

]dx

+e2εt∫

Ω

m∑j=1

[(2λn+j(ε− bj) + λn+j

n∑i=1

(|bij | + |βij | + |βij |)

+G2j

n∑i=1

λi[|aji| + (|αji| + |αji|)e2ετ ])|vj − v∗j |2

]dx. (15)

From the boundary condition (2) and the proof in [16], we can get

∫Ω

n∑i=1

λi(ui − u∗i )l∑

k=1

∂

∂xk

(Dik


)dx

= −n∑i=1

λi

l∑k=1

∫ΩDik

(∂(ui − u∗i )∂xk

)2dx (16)

and∫

Ω

m∑j=1

λn+j(vj − v∗j )l∑

k=1

∂

∂xk

(Djk


)dx


= −m∑j=1

λn+j

l∑k=1

∫ΩDik

(∂(vj − v∗j )∂xk

)2dx. (17)

Since Dik ≥ 0, Djk ≥ 0(i ∈ I, j ∈ J, k = 1, 2, · · · , l), from (12), (15), (16) and (17),we obtain that D+V (t) ≤ 0 for t > 0. So V (t) ≤ V (0) for t ≥ 0. Moreover, we have

V (0) =∫

Ω

n∑i=1

λi

[|ui − u∗i |2 +

m∑j=1

(|αji| + |αji|)G2j

×∫ 0

−τji

|vj(s, x) − v∗j |2e2ε(s+τji)ds]dx

+∫

Ω

m∑j=1

λn+j

[|vj − v∗j |2 +

n∑i=1

(|βij | + |βij |)F 2i

×∫ 0

−σij

|ui(s, x) − u∗i |2e2ε(s+σij)ds]dx.

≤∫

Ω

[max1≤i≤n

{λi}n∑i=1

|φui(s, x) − u∗i |2 + e2εσ max1≤i≤n

{F 2i }

×m∑j=1

(λn+j

n∑i=1

(|βij | + |βij |)∫ 0

−σij

|φui(s, x) − u∗i |2e2εsds)]dx

+∫

Ω

[max

1≤j≤m{λn+j}

m∑j=1

|φvj(s, x) − v∗j |2 + e2ετ max1≤j≤m

{G2j}

×n∑i=1

(λi

m∑j=1

(|αji| + |αji|)∫ 0

−τji

|φvj(s, x) − v∗j |2e2εsds)]dx

≤∫

Ω

[max1≤i≤n

{λi}n∑i=1

|φui(s, x) − u∗i |2 + e2εσ max1≤i≤n

{F 2i }

×m∑j=1

(λn+j max

1≤i≤n{|βij | + |βij |}

n∑i=1

∫ 0

−σ|φui(s, x) − u∗i |2e2εsds

)]dx

+∫

Ω

[max

1≤j≤m{λn+j}

m∑j=1

|φvj(s, x) − v∗j |2 + e2ετ max1≤j≤m

{G2j}

×n∑i=1

(λi max

1≤j≤m{|αji| + |αji|}

m∑j=1

∫ 0

−τ|φvj(s, x) − v∗j |2e2εsds

)]dx

≤[

max1≤i≤n

{λi} + σe2εσ max1≤i≤n

{F 2i }

×m∑j=1

(λn+j max

1≤i≤n{|βij | + |βij |}

)]‖φu − u∗i ‖

+[

max1≤j≤m

{λn+j} + τe2ετ max1≤j≤m

{G2j}

992 Zuoan Li

×n∑i=1

(λi max


)]‖φv − v∗j ‖,

and

V (t) ≥∫

Ωe2εt

n∑i=1

λi|ui − u∗i |2dx+∫

Ωe2εt

m∑j=1

λn+j |vj − v∗j |2dx.

≥ e2εt min1≤i≤n+m

{λi}( n∑i=1

‖ui(t, x) − u∗i ‖22 +

m∑j=1

‖vj(t, x) − v∗j ‖22

)

for t > 0. So

e2εt min1≤i≤n+m

{λi}( n∑i=1

‖ui(t, x) − u∗i ‖22 +

m∑j=1

‖vj(t, x) − v∗j ‖22

)

≤[

max1≤i≤n


{F 2i }

m∑j=1

(λn+j max

1≤i≤n{|βij | + |βij |}

)]‖φu − u∗i ‖

+[

max1≤j≤m


{G2j}

n∑i=1

(λi max


)]‖φv − v∗j ‖.

Let

M1 = max1≤i≤n


{F 2i }

m∑j=1

(λn+j max

1≤i≤n{|βij | + |βij |}

),

M2 = max1≤j≤m


{G2j}

n∑i=1

(λi max


),

M =max{M1,M2}

min1≤i≤n+m{λi} ,

then M ≥ 1, and we easily getn∑i=1

‖ui(t, x) − u∗i ‖22 +

m∑j=1

‖vj(t, x) − v∗j ‖22 ≤Me−2εt(‖φu − u∗i ‖ + ‖φu − u∗i ‖)

for all t ≥ 0, it implies that the equilibrium point (u∗, v∗)T is globally exponentiallystable. The proof is completed.

3 Periodic oscillatory solution

In this section, we will discuss the periodic oscillatory solutions of model (1). Letwi : R+ → R, wj : R+ → R, Ii : R+ → R and Jj : R+ → R be continuously periodicfunctions with period ω, i.e.,

wi(t+ ω) = wi(t), wj(t+ ω) = wj(t), Ii(t+ ω) = Ii(t), Jj(t+ ω) = Jj(t)

for i ∈ I, j ∈ J.


Theorem 3 Under assumptions (H1) and (H2), then there exists exactly one ω-periodic solution of model (1), and all other solutions of model (1) converge expo-nentially to it as t→ +∞.

Proof. Let φ =(φuφv

)= (φu1, · · · , φun, φv1, · · · , φvm)T , denote

C ={φ

∣∣∣φ :(

[−τ, 0] ×Rl

[−σ, 0] ×Rl

)→ Rn+m

}.

For φ ∈ C, we define

‖φ‖ =∥∥∥∥(φuφv

)∥∥∥∥ = ‖φu‖ + ‖φv‖,

then C is a Banach space of continuous functions which maps(

[−τ, 0] ×Rl

[−σ, 0] ×Rl

)into

Rn+m with the topology of uniform convergence.

For any(φuφv

),

(ψuψv

)∈ C, we denote the solutions of model (1) through((

00

),

(φuφv

))and

((00

),

(ψuψv

))as

u(t, φu, x) = (u1(t, φu, x), u2(t, φu, x), · · · , un(t, φu, x))T ,

v(t, φv , x) = (v1(t, φv , x), v2(t, φv , x), · · · , vm(t, φv , x))T

and

u(t, ψu, x) = (u1(t, ψu, x), u2(t, ψu, x), · · · , un(t, ψu, x))T ,

v(t, ψv , x) = (v1(t, ψv , x), v2(t, ψv , x), · · · , vm(t, ψv , x))T ,

respectively. Define

ut(φu, x) = u(t+ θ, φu, x), θ ∈ [−τ, 0], t ≥ 0,

vt(φv , x) = v(t+ θ, φv, x), θ ∈ [−σ, 0], t ≥ 0,

994 Zuoan Li

then(ut(φu, x)vt(φv , x)

)∈ C for t ≥ 0. Also, from model (1), we have

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂(ui(t,φu,x)−ui(t,ψu,x))∂t =

l∑k=1

∂∂xk

(Dik∂(ui(t,φu,x)−ui(t,ψu,x))

∂xk)

−ai(ui(t, φu, x) − ui(t, ψu, x))

+m∑j=1

aji[gj(vj(t, φv, x)) − gj(vj(t, ψv , x))]

+m∧j=1

αjigj(vj(t− τji, φv, x)) −m∧j=1

αjigj(vj(t− τji, ψv , x))

+m∨j=1

αjigj(vj(t− τji, φv, x)) −m∨j=1

αjigj(vj(t− τji, ψv , x)),

∂(vj(t,φv,x)−vj(t,ψv ,x))∂t =

l∑k=1

∂∂xk

(Djk∂(vj (t,φv,x)−vj(t,ψv ,x)

∂xk)

−bj(vj(t, φv, x) − vj(t, ψv, x))

+n∑i=1

bij[fi(ui(t, φu, x)) − fi(ui(t, ψu, x))]

+n∧i=1

βijfi(ui(t− σij , φu, x)) −n∧i=1

βijfi(ui(t− σij , ψu, x))

+n∨i=1

βijfi(ui(t− σij , φu, x)) −n∨i=1

βijfi(ui(t− σij , ψu, x))

(18)for i ∈ I, j ∈ J.

By using of Lemma 1 and assumption (H1), we can obtain as following inequal-ities

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(ui(t, φu, x) − ui(t, ψu, x))∂(ui(t,φu,x)−ui(t,ψu,x))

∂t

≤ (ui(t, φu, x) − ui(t, ψu, x))l∑

k=1

∂∂xk

(Dik∂(ui(t,φu,x)−ui(t,ψu,x))

∂xk)

−ai(ui(t, φu, x) − ui(t, ψu, x))2

+|ui(t, φu, x) − ui(t, ψu, x)|m∑j=1

|aji|Gj |vj(t, φv, x) − vj(t, ψv, x)|

+|ui(t, φu, x) − ui(t, ψu, x)|m∑j=1

(|αji| + |αji|)×Gj |vj(t− τji, φv , x) − vj(t− τji, ψv, x)|,

(vj(t, φv , x) − vj(t, ψv , x))∂(vj (t,φv,x)−vj(t,ψv,x))

∂t

≤ (vj(t, φv , x) − vj(t, ψv , x))l∑

k=1

∂∂xk

(Djk∂(vj (t,φv,x)−vj(t,ψv ,x))

∂xk)

−bj(vj(t, φv , x) − vj(t, ψv , x))2

+|vj(t, φv, x) − vj(t, ψv, x)|n∑i=1

|bij |Fi|ui(t, φu, x) − ui(t, ψu, x)|

+|vj(t, φv, x) − vj(t, ψv, x)|n∑i=1

(|βij | + |βij |)×Fi|ui(t− σij , φu, x) − ui(t− σij, ψu, x)|

(19)


for i ∈ I, j ∈ J and t ≥ 0. Since (H2) holds, we can choose a small positive numberε > 0 such that ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2λi(ε− ai) + λim∑j=1

(|aji| + |αji| + |αji|)

+F 2i

m∑j=1

λn+j

[|bij | + (|βij | + |βij |)e2εσ

]< 0,

2λn+j(ε− bj) + λn+j

n∑i=1

(|bij | + |βij | + |βij |)

+G2j

n∑i=1

λi

[|aji| + (|αji| + |αji|)e2ετ

]< 0

(20)

for i ∈ I, j ∈ J. We consider the another Lyapunov functional

V (t) =∫

Ω

n∑i=1

λi|ui(t, φu, x) − ui(t, ψu, x)|2e2εtdx

+∫

Ω

m∑j=1

λn+j |vj(t, φv, x) − vj(t, ψv , x))|2e2εtdx

+∫

Ω

n∑i=1

λi

[ m∑j=1

(|αji| + |αji|)

×G2j

∫ t

t−τji

|vj(s, φv , x) − vj(s, ψv , x))|2e2ε(s+τji)ds]dx

+∫

Ω

m∑j=1

λn+j

[ n∑i=1

(|βij | + |βij |)

×F 2i

∫ t

t−σij

|ui(s, φu, x) − ui(s, ψu, x)|2e2ε(s+σij)ds]dx.

By a minor modification of the proof of Theorem 2, we can easily deriven∑i=1

‖ui(t, φu, x) − ui(t, ψu, x)‖22 +

m∑j=1

‖vj(t, φv , x) − vj(t, ψv , x)‖22

≤Me−2εt(‖φu − ψu‖ + ‖φv − ψv‖)for all t ≥ 0, where M ≥ 1 is a constant. Hence we have

n∑i=1

‖ui(t, φu, x) − ui(t, ψu, x)‖22 ≤Me−2εt(‖φu − ψu‖ + ‖φv − ψv‖)

andm∑j=1

‖vj(t, φv, x) − vj(t, ψv, x)‖22 ≤Me−2εt(‖φu − ψu‖ + ‖φv − ψv‖)

for all t ≥ 0. We can choose a positive integer N such that

Me−2ε(Nω−τ) ≤ 16, Me−2ε(Nω−σ) ≤ 1

6.

996 Zuoan Li

Now, we define a Poincare mapping C → C by

P

(φuφv

)=

(uω(φu, x)vω(φv , x)

),

then

PN(φuφv

)=

(uNω(φu, x)vNω(φv, x)

).

Let t = Nω, then∥∥∥∥PN(φuφv

)− PN

(ψuψv

)∥∥∥∥ ≤ 13

∥∥∥∥(φuφv

)−

(ψuψv

)∥∥∥∥ .This implies that PN is a contraction mapping, hence there exist a unique fixed

point(φ∗uφ∗v

)∈ C such that

PN(φ∗uφ∗v

)=

(φ∗uφ∗v

),

since

PN(P

(φ∗uφ∗v

))= P

(PN

(φ∗uφ∗v

))= P

(φ∗uφ∗v

),

then P(φ∗uφ∗v

)∈ C is a fixed point of PN , and so

P

(φ∗uφ∗v

)=

(φ∗uφ∗v

), i.e.,

(uω(φ∗u)vω(φ∗v)

)=

(φ∗uφ∗v

).

Let(u(t, φ∗u, x)v(t, φ∗v , x)

)be the solution of model (1) through

((00

)(φ∗uφ∗v

)), then(

u(t+ ω, φ∗u, x)v(t+ ω, φ∗v, x)

)is also a solution of model (1). Obviously, we have

(ut+ω(φ∗u, x)vt+ω(φ∗v, x)

)=

(ut(uω(φ∗u, x))vt(vω(φ∗v , x))

)=

(ut(φ∗u, x)vt(φ∗v, x)

)

for all t ≥ 0. Hence (u(t+ ω, φ∗u, x)v(t+ ω, φ∗v, x)

)=

(u(t, φ∗u, x)v(t, φ∗v , x)

).

This shows that(u(t, φ∗u, x)v(t, φ∗v , x)

)is exactly one ω-periodic solution of model (1) and

all other solutions of model (1) converge exponentially to it as t→ +∞. The proofis completed.


4 An illustrate example

Consider the following BAM fuzzy neural networks with delays and diffusion terms:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂ui(t,x)∂t = ∂

∂x(Di∂ui(t,x)∂x ) − aiui(t, x) +

2∑j=1

ajigj(vj(t, x)) +2∑j=1

ajiwj(t) + Ii(t)

+2∧j=1

αjigj(vj(t− τji, x))ds+2∨j=1

αjigj(vj(t− τji, x))

+2∧j=1

Tjiwj(t) +2∨j=1

Hjiwj(t),

∂vj(t,x)∂t = ∂

∂x(Dj∂vj(t,x)∂x ) − bjvj(t, x) +

2∑i=1

bijfi(ui(t, x)) +2∑i=1

bijwi(t) + Jj(t)

+2∧i=1

βijfi(ui(t− σij, x))ds +2∨i=1

βijfi(ui(t− σij , x))

+2∧i=1

Tijwi(t) +2∨i=1

Hijwi(t),

(21)for i = 1, 2, j = 1, 2, t > 0, where

D1 = t2u21, D2 = t4u6

2, D1 = t4v21 , D2 = t2v8

2 , a1 = a2 = 1,

a11 = 0.25, a21 = −0.25, a12 = 0.25, a22 = 0.25, α11 = 0.125,

α21 = −0.125, α12 = 0.125, α22 = 0.125, α11 = 0.1, α21 = 0.1,

α12 = −0.1, α22 = 0.1, b1 = b2 = 2, b11 = 0.2, b21 = 0.2,

b12 = −0.2, b22 = 0.2, β11 = 0.1, β21 = −0.1, β12 = −0.1,

β22 = 0.1, β11 = 0.125, β21 = −0.125, β12 = 0.125, β22 = 0.125.

Let fi(y) = 11+e−y , i = 1, 2; gj(y) = |y+1|−|y−1|

2 , j = 1, 2, then we have F1 = F2 =1,G1 = G2 = 1, and let I1 = I2 = J1 = J2 = 2, w1 = w2 = w1 = w2 = 1,0 ≤ τji ≤ π

2 , i = 1, 2, j = 1, 2, 0 ≤ σij ≤ π2 , i = 1, 2, j = 1, 2, the matrices T =

(Tji), T = (Tij),H = (Hji) and H = (Hij) are the identity matrix.

998 Zuoan Li

Take constants λ1 = λ2 = λ3 = λ4 = 1, we have

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−2λ1a1 + λ1

2∑j=1

(|aj1| + |αj1| + |αj1|

)

+F 21

2∑j=1

λ2+j(|b1j | + |β1j | + |β1j |) ≤ −0.2 < 0,

−2λ2a2 + λ2

2∑j=1

(|aj2| + |αj2| + |αj2|

)

+F 22

2∑j=1

λ2+j(|b2j | + |β2j | + |β2j |) ≤ −0.2 < 0,

−2λ3b1 + λ3

2∑i=1

(|bi1| + |βi1| + |βi1|

)+G2

1

2∑i=1

λi(|a1i| + |α1i| + |α1i|) ≤ −0.2 < 0,

−2λ4b2 + λ4

2∑i=1

(|bi2| + |βi2| + |βi2|

)+G2

2

2∑i=1

λi(|a2i| + |α2i| + |α2i|) ≤ −0.2 < 0.

It follows that the assumption (H2) is satisfied. From Theorem 1 and Theorem2, the system (21) has exactly one equilibrium point, and the equilibrium point isglobally exponentially stable.

Furthermore, as the inputs wi(t), wj(t) , Ii(t) and Jj(t) are periodic functionswith common period, for example, wi(t) = cos t, i = 1, 2, wj(t) = sin t, Ii(t) = 3 sin tand Jj(t) = −2 cos t, here, tkae λ1 = λ2 = λ3 = λ4 = 1, by Theorem 3, we concludethat there exists exactly one 2π-periodic solution of the system (21), and all othersolutions of (21) converge exponentially to it as t→ +∞.

5 Conclusions

We have dealt with the problem of global exponential stability analysis and the ex-istence of both the equilibrium point and the periodic solution for model (1). Thegeneral sufficient conditions have been obtained to ensure the existence, uniquenessand global exponential stability of both the equilibrium point and the periodic so-lution for BAM fuzzy neural networks with delays and reaction-diffusion terms. Inparticular, an illustrate example is given to show the effectiveness of obtained re-sults. In addition, the sufficient conditions what we obtained are delay-independent,and are easily verified. This has practical benefits, since easily verifiable conditionsfor the global exponential stability are important in the design and applications ofneural networks.


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Received: November 27, 2007

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