Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 385298, 20 pages
doi:10.1155/2009/385298
Research Article
Global Exponential Stability of Periodic
Oscillation for Nonautonomous BAM Neural
Networks with Distributed Delay
Yi Wang,1 Zhongjun Ma,2, 3 and Hongli Liu1
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, wangyihzh@hotmail.com
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 memory BAM 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 largescale networks.
Copyright q 2009 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 memory BAM neural network which was first introduced by
Kosko in 1987 1, 2 is 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 twolayer pattern matched heteroassociative.
As it is well known, research on neural dynamical systems not only involves a
discussion of stability properties, but also involves many dynamic behavior such as periodic
oscillatory behavior, bifurcation, and chaos 3–19. In the application of neural networks
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Journal of Inequalities and Applications
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, 21 obtained 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 delays 23. 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 coefficients 24–26.
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
n
made for periodic T and the other is min1i,jn {ai − m
j1 dji Nji , cj −
i1 bij Mij } > 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 nonsingular M-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 in Section 3. The exponential stability
of periodic oscillator is considered in Section 4. An example is shown in Section 5. Several
summary remarks are finally given in Section 6.
2. Preliminaries
In this paper, we study the BAM neural networks with periodic coefficients and continuously
distributed delays modeled by the following system:
m
u̇i t −ai tui t bij thij
v̇j t −cj tvj t t
ij
j1
0
n
τji
i1
dji teji
fij svj t − sds
Ii t,
gji sui t − sds Lj t,
2.1
0
where i 1, 2, . . . , n; j 1, 2, . . . , m; ai t > 0 and cj t > 0 denote the rate with which the cells
i and j reset their potential to the resting state when isolated from the other cells and inputs;
bij t and dji t are connection weights of the neural network; Ii t, Lj t denote the ith and
the jth component of an external input source introduced from outside the network to the ith
Journal of Inequalities and Applications
3
cell and jth cell at time t, respectively. Moreover, the jth cell has an impact on the ith cell in
the time of tij and the jth cell has an impact on the ith cell in the time of τji .
If ui t and vj t satisfy system 2.1 and ui t T ui t, vj t T vj t, then they
are T -periodic solutions of system 2.1. The initial conditions associated with system 2.1
are given as follows:
ui s φi s,
vj s ψj s,
s ∈ −∞, 0,
2.2
where φi s, ψj s are continuous function i 1, 2, . . . , n; j 1, 2, . . . , m.
Throughout this paper, we make the following assumptions.
Assumption 2.1. ai t, bij t, cj t, dji t, Ii t and Jj t are continuous T -periodic functions on
R. In addition, ai supt∈R |ai t| < ∞, a−i inft∈R |ai t| > 0, bij supt∈R |bij t| < ∞, ci supt∈R |cj t| < ∞, ci− inft∈R |cj t| > 0, dji supt∈R |dji t| < ∞, Ii supt∈R |Ii t| < ∞, and
Lj supt∈R |Lj t| < ∞.
Assumption 2.2. Signal transmission functions hij u, eji v are bounded on R, and there exist
number Mij > 0 and Nji > 0 such that
hij u − hij v Mij |u − v|,
eji u − eji v Nji |u − v|
2.3
for each u, v ∈ R, i 1, 2, . . . , n and j 1, 2, . . . , m.
Assumption 2.3. The delay kernels fij s, gji s : 0, ∞ → 0, ∞ i 1, 2, . . . , n; j 1, 2,
. . . , m are continuous and integrable and satisfy
∞
fij sds 1,
∞
0
gji sds 1.
2.4
0
Assumption 2.4. The delay kernels fij s, gji s : 0, ∞ → 0, ∞ i 1, 2, . . . , n; j 1, 2,
. . . , m satisfy
∞
eαs fij sds 1,
∞
0
eαs gji sds 1,
2.5
0
where α is a bounded positive real number.
Now, we give some useful notations, definitions, and lemmas as follows: u T
0 |us|2 ds1/2 and v 1/T 0 vsds, where us ∈ CR, R, vs is T -periodic function.
j}, then we have the following.
Assume that Tn×n {A aij n×n : aij 0, i /
T
Lemma 2.5 see 17, 29, 30. Let A ∈ Tn×n . Then, each of the following conditions is equivalent to
the statement ‘A is a nonsingular M -matrix’:
a all of the principal minors of A are positive;
b the real parts of all the eigenvalue of A are positive;
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Journal of Inequalities and Applications
c A is inverse-positive; that is, A−1 exists and A−1 0;
d there is a vector x (or y), whose elements are all positive, such that the elements of Ax (or
AT y) are all positive;
e A has all positive diagonal elements and there exists a positive diagonal matrix D such that
AD is strictly diagonally dominant; that is,
aii di >
aij dj ,
i 1, 2, . . . , n.
2.6
i/
j
In the following, we introduce some concepts and results from the book by Gaines and
Mawhin 31.
Let X and Z be two Banach spaces, L : Dom L ⊂ X → Z a linear mapping, and
N : X → Z a continuous mapping. The mapping L will be called a Fredholm mapping of
index zero if dim Ker L codimImL < ∞ and ImL is closed in Z. If L is a Fredholm mapping
of index zero and there exist continuous projectors P : X → X and Q : Z → Z such that ImP
Ker L, Ker Q ImL ImI − Q, it follows that mapping L|Dom L Ker P : I − P X → ImL is
invertible. We denote the inverse of that mapping by KP . If Ω is an open bounded subset of X,
the mapping N will be called L-compact on if QNΩ is bounded and KP I − QN : Ω → X
is compact. Since ImQ is isomorphic to Ker L, there exists an isomorphism J : ImQ → Ker L.
Lemma 2.6 Mawhin’s continuation theorem. Let X and Z be two Banach spaces and L be a
Fredholm mapping of index zero. Assume that Ω ⊂ X is an open bounded set and N : X → Z is a
continuous operator which is L-compact on Ω. Then Lx Nx has at least one solution in Dom L Ω,
if the following conditions are satisfied:
a for each λ ∈ 0, 1, x ∈ ∂Ω Dom L, Lx /
λNx;
b for each x ∈ ∂Ω Ker L, QNx /
0;
c deg{JQNx, Ω Ker L, 0} / 0,
where J : ImQ → Ker L is 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 Assumptions 2.1–2.3 hold. Then, system 2.1 has at least one T-periodic
solution, if
P
In P12
P21 In
3.1
is a nonsingular M-matrix, where In is unite matrix and P12 pij n×m , pij −bij Mij /a−i ; P21 qji m×n , qji −dji Nji /cj− .
Journal of Inequalities and Applications
5
Proof. Let X Z {xt u1 t, . . . , un t, v1 t, . . . , vm t ∈ CR, Rnm | ui t ui t T , vj t vj t T , i 1, 2, . . . , n; j 1, 2, . . . , m}, then X is a Banach space with the norm
|v t|.
x1 ni1 maxt∈0,T |ui t| m
j1 max
t∈0,T j
Let L : Dom L ⊂ X → Z, P : X Dom L → Ker L, Q : X → X/ImL, and N : X → Z
be given by the following:
Lx u̇1 t, u̇2 t, . . . , u̇n t, v̇1 t, v̇2 t, . . . , v̇m t ,
P x Qx u1 t, u2 t, . . . , un t, v1 t, v2 t, . . . , vm t ,
⎛
m
t
1j
⎞
f1j svj t − sds I1 t ⎟
⎜ −a1 tu1 t b1j th1j
⎜
⎟
0
j1
⎜
⎟
⎜
⎟
···
⎜
⎟
⎜
⎟
⎜
⎟
t
m
nj
⎜
⎟
⎜ −an tun t bnj thnj
⎟
f
−
sds
I
sv
t
t
nj
j
n
⎜
⎟
0
⎟.
j1
Nx ⎜
⎜
⎟
τ
n
1i
⎜
⎟
⎜ −c1 tv1 t d1i te1i
⎟
g
−
sds
L
su
t
t
1i
i
1
⎜
⎟
0
⎜
⎟
i1
⎜
⎟
···
⎜
⎟
⎜
⎟
τmi
⎜
⎟
n
⎝
⎠
−cm tvm t dmi temi
gmi sui t − sds Lm t
i1
3.2
0
It is easy to see that L is a linear operator with Ker L {xt | xt κ ∈ Rnm }.
T
ImL {xt | xt ∈ Z, 0 xtdt 0} is closed in Z, and dim Ker L codimImL n m.
Therefore, L is a Fredholm mapping of index zero. It is easy to prove that P and Q are two
projectors, and ImP Ker L, ImL Ker Q ImI − Q. By using the Arzelá-Ascoli theorem,
it is easy to prove that for every bounded subset Ω ∈ X, Kp I − QN are relatively compact
on Ω in X, that is, N is L-compact on Ω.
Consider the operator equation
Lx λNx,
λ ∈ 0, 1,
3.3
that is
m
u̇i t −λai tui t λ bij thij
j1
t
ij
fij svj t − sds
λIi t,
0
τji
n
v̇j t −λcj tvj t λ dji teji
gji sui t − sds λLj t,
i1
3.4
0
m
j1
bij sups∈R |hij s| Ii . By using
−μi u̇i t λai tui t μi .
3.5
where i 1, 2, . . . , n and j 1, 2, . . . , m. Denote μi λ
3.4, we obtain
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Journal of Inequalities and Applications
Multiplying both sides of 3.5 by eλ
−μi eλ
t
0 ai sds
t
0 ai sds
, we have
t
t
ui teλ 0 ai sds μi eλ 0 ai sds .
3.6
Integrating the inequality above from 0 to ν ν 0 27, we obtain
−μi
ν
e
t
λ 0 ai sds
dt ui νe
ν
λ 0 ai sds
− ui 0 μi
ν
0
eλ
t
0 ai sds
3.7
dt.
0
Hence,
−μi
ν
e−λ
ν
t
ai sds
dt ui 0e−λ
ν
0 ai sds
ui ν μi
0
ν
e−λ
ν
t
ai sds
dt ui 0e−λ
ν
0 ai sds
3.8
μi
,
λa−i
3.9
0
for ν t 0. So we have
μi
− |ui 0| − −
λai
e
−λa−i ν
μi
− − ui ν λai
μi
|ui 0| − −
λai
−
e−λai ν that is,
μi
μi |ui ν| |ui 0| − − − ,
λai
λai
3.10
which implies that ui is bounded and similarly vj . By the Assumption 2.3, we know that
tij
3.11
fij svj t − sds
0
is uniformly convergent. Therefore, the following iterated integral:
T
ui t
tij
0
3.12
fij svj t − sds dt
0
can be changed integrating order.
Suppose that u1 t, . . . , un t, v1 t, . . . , vm t ∈ X is any periodic solution of system
2.1 for a certain λ ∈ 0, 1. Multiplying both sides of
m
u̇i t −λai tui t λ bij thij
j1
t
ij
0
fij svj t − sds
λIi t
3.13
Journal of Inequalities and Applications
7
by ui t and integrating from 0 to T , we obtain
T
0
λai tu2i tdt
T
t
T T
m
ij
bij tui thij
fij svj t − sds dt λIi tui tdt.
λ
0
3.14
0
0
j1
T
From Assumptions 2.1, 2.2, and 2.3 and noting that 0 |vj t − s|2 dt1/2
1/2
0 |vj t| dt
2
a−i
vj , we have
T
0
u2i tdt
T
0
ai tu2i tdt
T
m
bij tui t hij
t
ij
0 j1
T
m
fij svj t − sds
− hij 0 dt
0
bij thij 0ui tdt 0 j1
T
Ii tui tdt
0
∞
T
m
bij Mij
fij s vj t − s|ui t|dt ds
0
j1
√
3.15
0
⎞
⎛
m
T ⎝ b hij 0 I ⎠ui ij
i
j1
⎛
⎞
m
m
√
bij Mij vj ui T ⎝ bij hij 0 Ii ⎠ui ,
j1
j1
that is,
⎞
√ ⎛m
m
T
1
ui − bij Mij vj − ⎝ bij hij 0 Ii ⎠.
ai j1
ai
j1
3.16
By similar argument, we have
√ n
n
1
T
vj d Nji ui −
dji eji 0 Lj .
cj− i1 ji
cj
i1
3.17
It follows 3.16 and 3.17 that
Py s,
3.18
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Journal of Inequalities and Applications
In P12
, In is a unite matrix and P12 pij n×m , pij −bij Mij /a−i ; P21 qji m×n ,
P21 In
qji −d Nji /cj− , y u1 , u2 , . . . , un , v1 , v2 , . . . , vm , s s1 , s2 , . . . , snm , si √ √ m ji T j1 bij |hij 0| Ii /a−i , snj T ni1 dji |eji 0| Lj /cj− , 1 i n, 1 j m.
where P Application of Lemma 2.5 yields
y P−1 s r1 , r2 , . . . , rnm ,
3.19
which implies that
ui ri ,
vj rnj .
3.20
It is not difficult to check that there exist t∗i , t∗nj ∈ 0, T such that
rnj
vj t∗nj √
T
ri
ui t∗i √ ,
T
3.21
for 1 i n and 1 j m. Multiplying both sides of 3.13 by u̇i t i 1, 2, . . . , n and
integrating from 0 to T , we obtain
2
u̇i λ
m T
bij tu̇i thij
0
j1
T
−λ
t
ij
fij svj t − sds dt
0
T
ai tu̇i tui tdt λ
0
Ii tu̇i tdt
0
t
m T
ij
λ
bij tu̇i t hij
fij svj t − sds dt − hij 0
j1
0
0
T
T
m T
λ
bij thij 0u̇i tdt − λ ai tui tu̇i tdt λ Ii tu̇i tdt
j1
0
0
0
T
1/2 T
1/2
∞
m
2
2
bij Mij
fij s
|vj t − s| dt
ds
|u̇i t| dt
0
j1
0
0
T
1/2 T
1/2
m
√ √
2
2
T bij hij 0 u̇i ai
T Ii u̇i |u̇i t| dt
|ui t| dt
0
j1
0
m
m
√ √
bij Mij u̇i vj ai u̇i ui T bij hij 0u̇i T Ii u̇i .
j1
Therefore, u̇i j1
m
j1
bij Mij vj ai ui √
T
m
j1
bij |hij 0| Ii .
3.22
Journal of Inequalities and Applications
9
It is easy check that
T
√
ri
u̇i tdt √ T u̇i T
t∗
⎛
⎞⎤
⎡
m
m
√ √
ri
√ T ⎣ bij Mij vj ai ui T ⎝ bij hij 0 Ii ⎠⎦
T
j1
j1
ui t ui t∗ ⎞⎤
⎡
⎛
m
m
√
√ ri
ri
√ T ⎣ bij Mij rnj ai ri T ⎝ bij hij 0 Ii ⎠⎦ √ ζi ,
T
T
j1
j1
3.23
n
n
√
rnj
rnj √ √ ζnj .
dji Nji ri cj rnj T
dji eji 0 Lj
vj t √ T
T
T
i1
i1
√
√
Let ξi ri / T ζi and ξnj rnj / T ζnj 0 < 1. We take
Ω x u1 t, . . . , un t, v1 t, . . . , vm t ∈ X | |ui t| < ξi , vj t < ξnj .
3.24
Obviously, condition a of Lemma 2.6 is satisfied. When x ∈ ∂Ω Ker L ∂Ω Rnm ,
x is a constant vector in Rnm with |ui t| ξi , |vj t| ξnj . Then, we have
ui QNxi −ai u2i ui
m
bij hij vj ui I i ,
j1
vj QNxnj −cj vj2
vj
n
3.25
dji eji ui vj Lj .
i1
We claim that there exists some i or j 1 i n, 1 j m such that
ui QNxi < 0 or
vj QNxnj < 0.
3.26
If ui QNxi 0 and vj QNxnj 0, then, we obtain
ai u2i ui
m
bij hij vj ui Ii
j1
m
m
ui bij hij vj − hij 0 ui bij hij 0 ui Ii
j1
|ui |
j1
m
m
bij Mij vj |ui | bij hij 0 |ui |Ii .
j1
j1
3.27
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Journal of Inequalities and Applications
Which implies
⎛
⎞
m
m
m
1
1
1
si
bij Mij ξnj ⎝ bij hij 0 Ii ⎠ − bij Mij ξnj √ .
ξi ai j1
ai j1
ai j1
T
3.28
By a similar argument, we obtain
ξnj
n
1
1
d Nji ξi cj i1 ji
cj
n
n
snj
1
dji eij 0 Lj − dji Nji ξi √ .
cj i1
T
i1
3.29
On the other hand,
r
P−1 s
r
ξ > √ ζ > √ √ ,
T
T
T
3.30
where ξ ξ1 , ξ2 , . . . , ξnm , ζ ζ1 , ζ2 , . . . , ζnm , and r r1 , r2 , . . . , rnm . It follows that there
exists some i or j such that
ξi −
m
1
si
bij Mij ξnj > √
−
ai j1
T
3.31
or
ξnj −
n
snj
1
d Nji ξi > √ ,
−
cj i1 ji
T
3.32
which is contradiction with 3.28 and 3.29. Then, there exists some i or j 1 i n,
1 j m such that ui QNxi < 0 or vj QNxnj < 0. Therefore,
QNx1 n
|QNxi | i1
m QNxnj > 0.
3.33
j1
This indicates that condition b of Lemma 2.6 is satisfied.
Define Hx, θ −θx 1 − θQNx, θ ∈ 0, 1, where x u1 t, . . . , un t, v1 t,
. . . , vm t ∈ Rnm . When x ∈ Ker L ∂Ω, we have
Hx, θ1 n
i1
n
i1
|Hui , θ| m H vj , θ j1
m |−θui 1 − θQNxi | −θvj 1 − θQNxj > 0.
j1
3.34
Journal of Inequalities and Applications
11
According to the invariant of homology, we have
!
deg JQNx, Ω
Ker L, 0 / 0,
3.35
where J : ImQ → Ker L is an isomorphism. Therefore, according to the continuation
theorem of Gaines and Mawhin, system 2.1 has at least one T -periodic solution. The proof
is completed.
Remark 3.2. In 27, the period T was assumed to be T < min{1/ai , 1/cj }. In 28, the
n
inequations min1i,jn {ai − m
j1 dji Nji , cj −
i1 bij Mij } > 0 must hold and also in 27.
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 of Theorem 3.1, system 2.1 has at least one T -periodic solution
∗
t .
x∗ t u∗1 t, . . . , u∗n t, v1∗ t, . . . , vm
4.1
Now, we give the following definition about global exponential of periodic solution:
Definition 4.1. The periodic solution x∗ t of model 2.1 is said to be globally exponentially
stable, if there exist positive constants β, M such that
⎞1/2
n
m
2
2
∗
∗
⎝ |ut − u t| |vt − v t| ⎠ ≤ Mx0 − x∗ e−βt
0 2
⎛
i1
4.2
j1
for all t > 0, where x0 , x0∗ represent the history of x and x∗ on −∞, 0, respectively, and
2 1/2
∗
.
x0 − x0∗ 2 { ni1 sup−∞<t0 |φi t − u∗i t|2 m
j1 sup−∞<t0 |ψj t − vj t| }
Definition 4.2. A matrix M is said to be diagonally dominant if |mii | j / i |mij | for all i,
|mii | > j / i |mij | for at least one i, where mij denotes the entry in the ith row and jth column.
Theorem 4.3. Assume that Assumptions 2.1, 2.2, 2.4 hold and P is a nonsingular M-matrix, where
P is the same as in Theorem 3.1. System 2.1 has a unique T -periodic solution, which is globally
exponentially stable, if Q − αIT ∈ T and is a weakly diagonally dominant matrix as
Q
Q11 Q12
Q21 Q22
,
4.3
m m −
−
where Q11 diag2a−1 − m
j1 b1j M1j , 2a2 −
j1 b2j M2j , . . . , 2an −
j1 bnj Mnj ; Q12 −
mij n×m , mij −bij Mij ; Q21 nji m×n , nji −dji Nji ; Q22 diag2c1 − ni1 d1i
N1i , 2c2− −
n n
−
i1 d2i N2i , . . . , 2cm −
i1 dmi Nmi .
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Journal of Inequalities and Applications
∗
t . Then,
Proof. Let zt u1 t − u∗1 t, . . . , un t − u∗n t, v1 t − v1∗ t, . . . , vm t − vm
2
zi tżi t −ai t|zi t| m
bij tzi thij
" t
ij
0
j1
m
− bij tzi thij
ij
0
j1
−a−i |zi t|2 −a−i |zi t|2 m
−a−i |zi t|2
bij Mij
fij svj∗ t
tij
0
m
∞
bij Mij
− sds
fij szi tznj t − sds
j1
j1
t
%
#
$
∗
fij s znj t − s vj t − s ds
0
2
|zi t|2 znj t − s
ds
fij s
2
4.4
∞
m
2
1
2
b Mij |zi t| fij s znj t − s ds ,
2 j1 ij
0
∞
n
2 1 2
znj tżnj t −cj− znj t dji Nji znj t gji s|zi t − s|2 ds
2 i1
0
for i 1, 2, . . . , n and j 1, 2, . . . , m.
We define the following Lyapunov functionals:
m
1
eαt
b Mij
V&i t |zi t|2 α
α j1 ij
∞ t
0
n
2 1 eαt V&nj t znj t d Nji
α
α i1 ji
2
fij sznj γ eαγs dγds,
t−s
∞ t
0
2
gji szi γ eαγs dγds.
4.5
t−s
Calculating the Dini upper right derivative of V&j t and V&nj t along the solution of
2.1, and estimating it via the assumptions 32, we have
⎧
⎫
∞
+⎬
*
m
αt ⎨
2
e
2
2
−
−2ai |zi t| bij Mij |zi t| fij sznj t − s ds
D V&i t e |zi t| ⎭
α ⎩
0
j1
2
αt
⎧
⎫
∞
+⎬
* ∞
m
eαt ⎨
2
2
b Mij
fij sznj t eαs ds −
fij sznj t − s ds
⎭
α ⎩ j1 ij
0
0
⎞
⎛
m
m
2 ∞
eαt eαt ⎝
2
−⎠
α bij Mij − 2ai |zi t| b Mij znj t
fij seαs ds
α
α j1 ij
0
j1
⎧⎛
⎫
⎬
e
⎝α b Mij − 2a− ⎠|zi t|2 b Mij znj t2 ,
ij
ij
i
⎭
α ⎩
j1
j1
αt ⎨
m
⎞
m
Journal of Inequalities and Applications
%
"
n
n
2 eαt
2
&
− znj t dji Nji |zi t| .
α dji Nji − 2cj
D Vnj t α
i1
i1
13
4.6
Let
Q
Q11 Q12
Q21 Q22
4.7
,
m m −
−
where Q11 diag2a−1 − m
j1 b1j M1j , 2a2 −
j1 b2j M2j , . . . , 2an −
j1 bnj Mnj ; Q12 mij n×m , mij −bij Mij ; Q21 nji m×n , nji −dji Nji ; Q22 diag2c1− − ni1 d1i
N1i , 2c2− −
n n
−
i1 d2i N2i , . . . , 2cm −
i1 dmi Nmi .
Consider the Lyapunov functional
Vt nm
V&k t.
4.8
k1
When Q − αIT ∈ T and is a weakly diagonally dominant matrix, calculating the Dini
upper right derivative of V along the solution of 2.1, we have
D Vt D
nm
V&k t
k1
⎫
⎧⎛
⎞
m
m
⎬
eαt ⎨⎝
2
2
−2a−i bij Mij α⎠|zi t| bij Mij znj t
⎭
α ⎩
j1
j1
eαt
α
"
−2cj−
%
n
n
2 2
dji Nji α znj t dji Nji |zi t|
i1
⎧
i1
⎫
⎬
e
−2a−i bij Mij dji Nji α |zi t|2
⎭
α ⎩
j1
i1
αt ⎨
m
n
⎫
⎧
n
m
⎬
eαt ⎨
−2cj− dji Nji bij Mij α |zni t|2 < 0.
⎭
α ⎩
i1
j1
Therefore, for Vt V0,
⎧
∞ 0
m
2
1 ⎨nm
2
zi 0 bij Mij
fij sznj γ eαγs dγds
V0 ⎩
α k1
0
−s
j1
%
∞ 0
n
2 αγs
dji Nji
gji s zi γ e
dγds
i1
0
−s
4.9
14
Journal of Inequalities and Applications
⎧
1⎨
nm
α⎩
∞ 0
m
bij Mij
fij seαγs dγds
0
j1
−s
%
∞ 0
n
αγs
dji Nji
gji se
dγds
0
i1
−s
max
supz2k γ
k∈{1,...,mn} γ0
1
n m max supz2k γ
α
k∈{1,...,mn} γ0
⎧ ⎫
2
n
m
⎨
2⎬
1
∗
∗
sup φ t − ui t
sup ψj t − vj t
n m
.
⎭
⎩ i1 −∞<t0 i
α
j1 −∞<t0
4.10
Thus,
⎛
⎞
n
m
eαt ⎝ |ut − u∗ t|2 |vt − v∗ t|2 ⎠ < αVt < αV0
i1
j1
⎧ ⎫
2
n
m
⎨
2⎬
∗
∗
n m
,
sup φ t − ui t
sup ψj t − vj t
⎩ i1 −∞<t0 i
⎭
j1 −∞<t0
4.11
that is,
⎛
⎞1/2
n
m
√
⎝ |ut − u∗ t|2 |vt − v∗ t|2 ⎠ < n mx0 − x∗ e−αt/2 .
0 2
i1
4.12
j1
This means that periodic solution of system 2.1 is globally exponentially stable. The proof
is completed.
Remark 4.4. For system 2.1, when delay kernels fij s, gji s are δ-functions, that is, we take
fij s δs − i and gji s δs − σj , then system 2.1 can be reduced to a fixed time delay
system and all above theorems are hold.
5. Example
In this section, we present an example to show the effectiveness and correctness of our
theoretical results.
Journal of Inequalities and Applications
15
2
1.5
1
u1
0.5
0
−0.5
−1
−1.5
0
10
20
30
40
50
T
Figure 1: Time evolution of u1 of system 5.1.
Consider the following BAM neural networks:
3
u̇i t −ai tui t bij thij
v̇j t −cj tvj t t
ij
j1
0
3
τji
i1
dji teji
fij svj t − sds
Ii t,
gji sui t − sds Lj t.
5.1
0
We select a set of parameters as α 4/5, ai t 2 0.1 sin t, cj t 2 0.1 cos t, fij s gji s e−s , Ii t −0.2 0.4 cos t and Lj t 0.3 0.3 sin t, hij s tanhs, eji s 1/1 s,
tij τji 20,
⎛
bij t
3×3
dji t
3×3
⎞
⎟
⎜
⎟
⎜
⎝0.5 0.1 sin t 0.4 0.1 sin t 0.3 0.1 sin t ⎠,
0.3 0.1 sin t 0.2 0.1 sin t −0.5 0.1 sin t
⎞
0.2 0.1 cos t 0.4 0.1 cos t 0.1 0.1 cos t
⎟
⎜
⎟
⎜
⎝−0.2 0.1 cos t 0.3 0.1 cos t 0.5 0.1 cos t⎠.
0.5 0.1 cos t 0.2 0.1 cos t 0.3 0.1 cos t
⎛
0.1 0.1 sin t 0.2 0.1 sin t 0.3 0.1 sin t
5.2
Remark 5.1. A frequently used model for distributed time delays in biological, neural
networks applications is to choose for the Gamma kernel due to mathematical difficulties.
In this paper, we set a weak delay kernel, that is, an exponential kernel.
16
Journal of Inequalities and Applications
2
1.5
1
u2
0.5
0
−0.5
−1
−1.5
0
10
20
30
40
50
T
Figure 2: Time evolution of u2 of system 5.1.
2
1.5
1
u3
0.5
0
−0.5
−1
−1.5
−2
0
10
20
30
40
50
T
Figure 3: Time evolution of u3 of system 5.1.
It is easy to check that max hij 1 and max eji 1. Let Mij 1 and Nji 1. By some
computations, we obtain
⎛
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⎟
⎟
⎜
P⎜
⎟,
⎟
⎜−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
Journal of Inequalities and Applications
17
1.5
1
0.5
v1
0
−0.5
−1
−1.5
−2
0
10
20
30
40
50
T
Figure 4: Time evolution of v1 of system 5.1.
2
1.5
1
v2
0.5
0
−0.5
−1
−1.5
−2
0
10
20
30
40
50
T
Figure 5: Time evolution of v2 of system 5.1.
⎛
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⎟
⎟
⎜
Q⎜
⎟.
⎜−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
It is easy to check that P is a nonsingular M-matrix and Q − 0.8IT is a weakly
diagonally dominant matrix. From Theorem 4.3 system 5.1 has T -periodic oscillation, which
18
Journal of Inequalities and Applications
1.5
1
0.5
v3
0
−0.5
−1
−1.5
−2
0
10
20
30
40
50
T
Figure 6: Time evolution of v3 of system 5.1.
is globally exponentially stable. In Figures 1, 2, 3, 4, 5, and 6, we plot the trajectories of ui and
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 nonsingular M-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
NNSF no. 60964006.
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