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Existence and stability of almost periodic solution for BAM neural networks with delays
Applied Mathematics and Computation 137 (2003) 177–193 www.elsevier.com/locate/amc
Existence and stability of almost periodic solution for BAM neural networks with delays Anping Chen a, Lihong Huang b, Jinde Cao a
c,*
Department of Mathematics, Chenzhou Teacher’s College, Chenzhou, Hunan 423000, China b College of Mathematics and Econometrics, Hunan University, Changsha 410082, China c Department of Applied Mathematics, Southeast University, Nanjing 210096, China
Abstract By using the Banach fixed point theorem and constructing suitable Lyapunov function, some sufficient conditions are obtained ensuring existence, uniqueness and global stability of almost periodic solution of the BAM neural networks with variable coefficients and delays. These results are helpful to design global exponential stable BAM networks and almost periodic oscillatory BAM networks. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Almost periodic solution; Global stability; BAM neural networks
1. Introduction Recently, a class of two-layer heteroassociative networks called bidirectional associative memory (BAM) networks [5,12–15] with and without axonal signal transmission delays has been proposed and used in many fields such as pattern recognition and automatic control. At the same time, it has attracted many scholars, e.g., [2–4,8–11,15–23], the stability of bidirectional associative memory neural networks with and without delays has been studied. In [5,10,12–15],
*
Corresponding author. E-mail addresses: [email protected] (A. Chen), [email protected] (L. Huang), [email protected], [email protected] (J. Cao). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 0 9 5 - 4
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A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
some sufficient conditions have been obtained for globally asymptotic stability of delayed bidirectional associative memory networks (BAM). Moreover, author in [4] researches the periodic oscillatory solution of BAM networks with delays in the case of constant coefficients. It is well known that studies on neural dynamical systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior, almost periodic oscillatory properties, chaos, and bifurcation [17]. In applications, almost periodic oscillatory is more accordant with fact. To the best of our knowledge, few authors have considered almost periodic oscillatory solutions for BAM networks, most of them study the stability, periodic oscillation of BAM networks in the case of constant coefficients. In this paper, we discuss almost periodic oscillatory solutions of BAM networks with variable coefficients and delays, obtain some simple sufficient conditions ensuring the existence and global exponential stability of almost periodic solution. Consider the BAM networks with variable coefficients X dxi ¼ ai ðtÞxi ðtÞ þ pji ðtÞfj ðyj ðt sji ÞÞ þ Ii ðtÞ; dt j¼1
ð1:1aÞ
n X dyj ¼ bj ðtÞyj ðtÞ þ qij ðtÞfi ðxi ðt rij ÞÞ þ Jj ðtÞ; dt i¼1
ð1:1bÞ
p
where i ¼ 1; 2; . . . ; n, j ¼ 1; 2; . . . ; p, xi ðtÞ and yj ðtÞ are the activations of the ith neurons and the jth neurons, respectively. pji ðtÞ, qij ðtÞ are the connection weights at the time t, and Ii ðtÞ and Jj ðtÞ denote the external inputs at time t. sji and rij are nonnegative constants, which correspond to the finite speed of the axonal signal transmission. An analog circuit implementing of BAM equation with axonal signal transmission delays can be seen in [4,15,18]. Throughout this paper, we always assume that ai ðtÞ, bj ðtÞ, pji ðtÞ, qij ðtÞ, Ii ðtÞ and Jj ðtÞ are continuous almost periodic functions. Moreover, ai ðtÞ, bj ðtÞ are positive, 0 < inf fai ðtÞg ¼ a i ; t2R
0 < inf fbj ðtÞg ¼ b j ;
qþ ij
¼ supfjqij ðtÞjg < þ1;
Jjþ
¼ supfjJj ðtÞjg < þ1:
t2R
t2R
Iiþ
pjiþ ¼ supfjpji ðtÞjg < þ1; t2R
¼ supfjIi ðtÞjg < þ1; t2R
t2R
Furthermore, the signal function fi possesses the following properties: (H1) fi is bounded on R, for all i ¼ 1; 2; . . . ; maxfn; pg; (H2) There exists a number li > 0 such that jfi ðxÞ fi ðyÞj 6 li jx yj, i ¼ 1; 2; . . . ; maxfn; pg.
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
179
The initial conditions associated with (1.1a), (1.1b) are associated to be of the form xi ðtÞ ¼ /i ðsÞ;
s 2 ½s; 0 ;
s ¼ max max fsji g;
yj ðtÞ ¼ wi ðsÞ;
s 2 ½r; 0 ;
r ¼ max max frij g;
16i6n 16j6p
ð1:2Þ
16j6p 16i6n
where /i ðsÞ, wi ðsÞ are continuous almost periodic functions on R. For any solution zðtÞ , ðxðtÞ; yðtÞÞT , ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; yp ðtÞÞT and almost periodic solution T
z ðtÞ , ðx ðtÞ; y ðtÞÞ , ðx 1 ðtÞ; x 2 ðtÞ; . . . ; x n ðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; yp ðtÞÞ T
T
T
of system (1.1a), (1.1b), define kð/; wÞ ðx ; y Þ k as ) ( n X T T 2
ð/; wÞ ðx ; y Þ ¼ sup ð/i ðtÞ x ðtÞÞ s 6 t 6 0
i
i¼1
( þ sup r 6 t 6 0
p X
) ðwj ðtÞ
2 yj ðtÞÞ
:
j¼1
Definition 1. The almost periodic solution ðx ðtÞ; y ðtÞÞT of system (1.1a), (1.1b) is said to be global exponential stable, if there exists constants a > 0 and M P 1 such that p n X X ðxi ðtÞ x i ðtÞÞ2 þ ðyj ðtÞ yj ðtÞÞ2 6 M ð/; wÞT ðx ; y ÞT eat i¼1
j¼1
for all t > 0. Definition 2 [1,7]. Let zðtÞ : R ! Rn be continuous for t 2 R. zðtÞ is said to be almost periodic on R if, for any e > 0, it is possible to find a real number l ¼ lðeÞ > 0 such that, for any interval with length lðeÞ, there exists a number s ¼ sðeÞ in this interval such that jzðt þ sÞ zðtÞj < e for 8t 2 R: Definition 3 [1]. Let z 2 Rn and QðtÞ be a n n continuous matrix defined on R. The linear system dz ¼ QðtÞzðtÞ dt
ð1:3Þ
is said to admit an exponential dichotomy on R if there exits constants k, k > 0, projection P and the fundamental matrix ZðtÞ of (1.3) satisfying
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A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
ZðtÞPZ 1 ðsÞ 6 k ekðtsÞ for t P s; ZðtÞðI P ÞZ 1 ðsÞ 6 k ekðstÞ for t 6 s: Lemma 1 [1,7]. If the linear system (1.3) admits an exponential dichotomy, then almost periodic system dz ¼ QðtÞz þ gðtÞ dt
ð1:4Þ
has a unique almost periodic solution zðtÞ, and Z t Z þ1 zðtÞ ¼ ZðtÞPZ 1 ðsÞgðsÞ ds ZðtÞðI P ÞZ 1 ðsÞgðsÞ ds: 1
ð1:5Þ
t
Lemma 2 [1]. Assume that ci ðtÞ is an almost periodic function and Z 1 tþT M½ci ¼ lim ci ðsÞ ds > 0; i ¼ 1; 2; . . . ; n: T !þ1 T t Then the system dz ¼ diagðc1 ðtÞ; c2 ðtÞ; . . . ; cn ðtÞÞzðtÞ: dt Then the linear system (1.3) admits an exponential dichotomy. Lemma 3. Assume that fk ðxÞ ðk ¼ 1; 2; . . . ; maxðn; pÞÞ is continuous bounded function on R. Then every solution of system (1.1a), (1.1b) is bounded. Proof. Set di ¼
p X j¼1
pjiþ sup jfj ðxÞj þ Iiþ x2R
and cj ¼
n X i¼1
þ qþ ij sup jfj ðxÞj þ Jj
for i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; p:
x2R
Let T
zðtÞ , ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; yp ðtÞÞ
be a solution of (1.1a), (1.1b). Then it follows from (1.1a) that ai ðtÞxi ðtÞ di 6 x0i ðtÞ 6 ai ðtÞxi ðtÞ þ di :
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
Multiplying the inequality above by e di e
Rt 0
ai ðuÞ du
Rt 0
ai ðuÞ du
181
, we have
Rt Rt d ai ðuÞ du a ðuÞ du xi ðtÞ e 0 6 ; 6 di e 0 i dt
i ¼ 1; 2; . . . ; n:
By integrating the inequality from 0 to s P 0, we get Z t Rv Z s Rv Rs a ðuÞ du a ðuÞ du a ðuÞ du di e 0 i dv 6 xi ðsÞ e 0 i xi ð0Þ 6 di e 0 i dv; 0
0
i.e., xi ð0Þ e
Rs 0
ai ðuÞ du
di
Z
s
e
Rs v
ai ðuÞ du
0
6 xi ðsÞ 6 xi ð0Þ e
Rs 0
ai ðuÞ du
þ di
dv
Z
s
e
Rs v
ai ðuÞ du
dv
0
for s P v P 0. Therefore, di di di di jxi ð0Þj þ eai s 6 xi ðsÞ 6 jxi ð0Þj eai s þ ai ai ai ai for all s > 0. Hence, di di di di jxi ð0Þj 6 xi ðtÞ 6 þ jxi ð0Þj ai ai ai ai for all s > 0 and i ¼ 1; 2; . . . ; n. Using a similar argument, we can obtain cj cj cj cj jyj ð0Þj 6 yj ðtÞ 6 þ jyj ð0Þj bj bj bj bj for all s > 0 and j ¼ 1; 2; . . . ; p. The proof is complete.
2. Existence of almost periodic solution For an arbitrary vector zðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; yp ðtÞÞT , we define the norm kzðtÞk ¼ max1 6 i 6 n jxi ðtÞj þ max1 6 j 6 p jyj ðtÞj. Set n o T S nþp ¼ z j z ¼ ð/1 ; /2 ; . . . ; /n ; w1 ; w2 ; . . . ; wp Þ ; where /i , wj are continuous almost periodic functions on R, i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; p. For any z 2 S nþp , we define induced module kzk ¼ supt2R kzðtÞk ¼ supt2R max1 6 i 6 n j/i ðtÞj þ supt2R max1 6 j 6 p jwj ðtÞj, then S nþp is a Banach space.
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Theorem 1. In addition to (H1) and (H2), suppose further that ( Pn )
Pp þ þ j¼1 pji lj i¼1 qij li (H3) r ¼ max þ max < 1; 16i6n 16j6p a b i j R tþT R tþT (H4) M½ai ¼ limT !þ1 t ai ðsÞ ds > 0, and M½bj ¼ limT !þ1 t bj ðsÞ ds > 0, i ¼ 1; 2; . . . ; n, j ¼ 1; 2; 3; . . . ; p. Then there is a unique periodic solution of system (1.1a), (1.1b) in the region kz z0 k 6 rI=ð1 rÞ, where ( )
þ Ii Jiþ I ¼ max þ max ; 16i6n 16j6p ai b j and z0 ¼
Z
t
Rt
a1 ðuÞ du
I1 ðsÞ ds; . . . ;
Z
t
Rt
an ðuÞ du
In ðsÞ ds; 1 Z 1 Z R R t t t t b ðuÞ du b ðuÞ du e s 1 J1 ðsÞ ds; . . . ; e s p Jp ðsÞ ds : e
s
1
e
s
1
T
T
Proof. For any ð/; wÞ ¼ ð/1 ; /2 ; . . . ; /n ; w1 ; w2 ; . . . ; wp Þ 2 S nþp , we consider the almost solution zð/;wÞT of nonlinear almost periodic differential equation X dxi ¼ ai ðtÞxi ðtÞ þ pji ðtÞfj ðwj ðt sji ÞÞ þ Ii ðtÞ; dt j¼1
ð2:1aÞ
n X dyj ¼ bj ðtÞyj ðtÞ þ qij ðtÞfi ð/i ðt rij ÞÞ þ Jj ðtÞ: dt i¼1
ð2:1bÞ
p
Because M½ai > 0, M½bj > 0, by Lemma 2, the linear system dxi ¼ ai ðtÞxi ðtÞ; dt dyj ¼ bj ðtÞyj ðtÞ dt
ð2:2Þ
admits an exponential dichotomy on R. By Lemma 1, the solution zð/;wÞT of (2.1a), (2.1b) can be expressed as following (Z " # Rt p t X a1 ðuÞ du zð/;wÞT ¼ e s pj1 ðsÞfj ðwj ðs sj1 ÞÞ þ I1 ðsÞ ds; . . . ; 1
Z
t
e 1
Rt s
" an ðuÞ du
j¼1 p X j¼1
# pjn ðsÞfj ðwj ðs: sjn ÞÞ þ In ðsÞ ds;
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
Z
t
e
Rt s
" b1 ðuÞ du
1
Z
n X
183
# qj1 ðsÞfi ð/i ðs ri1 ÞÞ þ J1 ðsÞ ds; . . . ;
i¼1
t
e
Rt s
" bp ðuÞ du
1
n X
#
)
qip ðsÞfi ð/i ðs sip ÞÞ þ Jp ðsÞ ds :
ð2:3Þ
i¼1
Now we define a mapping T : S nþp ! S nþp ;
T ð/; wÞðtÞ ¼ zð/;wÞT :
Set B ¼
z j z 2 S nþp ; kz z0 k 6
rI : 1r
Then B is a closed convex subset of S nþp . According to the definition of the norm of Banach space S nþp , we have Z t Rt ai ðuÞ du s kz0 k ¼ sup max e Ii ðsÞ ds t2R 1 6 i 6 n
1
Z t Rt b ðuÞ du þ sup max e s j Jj ðsÞ ds t2R 1 6 j 6 p 1 Z t Rt a ðuÞ du 6 sup max jIi ðsÞje s i t2R 1 6 i 6 n
1
þ sup max
t2R 1 6 j 6 p
Z
Rt Jj ðsÞ e s bj ðuÞ du ds
t
1
( ) Jjþ Iiþ 6 sup max þ sup max ai b t2R 1 6 i 6 n t2R 1 6 j 6 p j ( )
þ þ Jj Ii ¼ max þ max 16i6n 16j6p ai b j
¼ I: Therefore, kzk 6 kz z0 k þ kz0 k ¼
rI I þI ¼ : 1r 1r
ð2:4Þ
First, we prove that the mapping T is a self-mapping from B to B . In fact, for any z 2 B , we have
184
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
( Z " # ) Rt p t X ai ðuÞ du kT ðzÞ z0 k ¼ sup max e s pji ðsÞfj wj ðs sji Þ ds t2R 1 6 i 6 n 1 j¼1 ) ( Z " # Rt n t X bj ðuÞ du e s qij ðsÞfi /i ðs rij Þ ds þ sup max 1 6 j 6 p t2R 1 i¼1 (Z " # ) p t X 6 sup max eai ðtsÞ pjiþ lj wj ðs sji Þ ds t2R 1 6 i 6 n
1
þ sup max
(Z
(Z
t
e
t2R 1 6 i 6 n
¼
(Z
a i ðtsÞ
qþ ij li /i ðs
p X
e
b j ðtsÞ
1
n X i¼1
þ j¼1 pji lj a i
þ max
16j6p
¼ rkzk rI ; 6 1r
# ) rij Þ ds
)
pjiþ lj kzk ds
j¼1 t
Pp max
n X i¼1
1
t2R 1 6 j 6 p
16i6n
b j ðtsÞ
1
6 sup max
(
"
e
t2R 1 6 j 6 p
þ sup max
j¼1 t
) qþ ij li k zk ds Pn
þ i¼1 qij li b j
!) kzk
ð2:5Þ
which implies that T ðzÞðtÞ 2 B . Therefore, the mapping T is a self-mapping from B to B . Next, we prove the mapping T is a contraction mapping of B . In fact, in view of (H1) and (H2), for any z1 ; z2 2 B , where T
z1 ¼ ðn1 ; n2 ; . . . ; nn ; g1 ; g2 ; . . . ; gp Þ ; T
z2 ¼ ð/1 ; /2 ; . . . ; /n ; w1 ; w2 ; . . . ; wp Þ ; we have kT ðz1 Þ T ðz2 Þk
) ( Z Rt p t X ai ðuÞ du s e pji ðsÞ fj ðnj ðs sji ÞÞ fj ð/j ðs sji ÞÞ ¼ sup max t2R 1 6 i 6 n 1 j¼1 ) ( Z Rt t n X bj ðuÞ du s e qij ðsÞ fi ðgi ðs rij ÞÞ fi ðwi ðs rij ÞÞ þ sup max 1 t2R 1 6 j 6 p i¼1 (Z " # ) p t X eai ðtsÞ pþ lj nj ðs sji Þ /j ðs sji Þ ds 6 sup max t2R 1 6 i 6 n
1
ji
j¼1
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
þ sup max
(Z
t2R 1 6 j 6 p
(Z
¼
ebj ðtsÞ
1
e (Z
a i ðtsÞ
t
1
Pp max
16i6n
ds qþ ij li gi ðs rij Þ wi ðs rij Þ
p X
þ j¼1 pji lj a i
b j ðtsÞ
n X i¼1
!
þ max
16j6p
)
)
pjiþ lj kz1
z2 k ds
j¼1
e
t2R 1 6 j 6 p
n X
185
#
i¼1
1
þ sup max (
t
t
6 sup max
t2R 1 6 i 6 n
"
) qþ ij li kz1
z2 k ds
Pn
þ i¼1 qij li b j
!) kz 1 z 2 k
¼ rkz1 z2 k:
ð2:6Þ
Noting that r < 1, it is clear that T is a contraction mapping. Thus, T possesses a unique fixed point z 2 B , that is T ðz Þ ¼ z . By (2.3), (2.1a) and (2.1b), T T ðx ; y Þ satisfies (1.1a), (1.1b). So ðx ; y Þ is a unique periodic solution of
(1.1a) and (1.1b) in B . The proof is complete.
3. Global asymptotic stability of almost periodic solution In this section, we discuss the global asymptotic stability of BAM networks. Set ui ðtÞ ¼ xi ðtÞ x i ðtÞ;
vj ðtÞ ¼ yj ðtÞ yj ðtÞ;
gi ðui ðt rij ÞÞ ¼ fi ðui ðt rij Þ þ x i ðtÞÞ fi ðx i ðtÞÞ; gj ðvj ðt sij ÞÞ ¼ fj ðvj ðt sji Þ þ yj ðtÞÞ fj ðyj ðtÞÞ: Then jgi ðui ðt rij ÞÞj 6 li jui ðt rij Þj;
jgj ðvj ðt sij ÞÞj 6 lj jvj ðt sji Þj
for i ¼ 1; 2; . . . ; n, and j ¼ 1; 2; . . . ; p. It is easy to see that system (1.1a), (1.1b) can be reduced to the following system X dui ¼ ai ðtÞui ðtÞ þ pji ðtÞgj ðvj ðt sij ÞÞ; dt j¼1 p
n X dvj ¼ bj ðtÞvj ðtÞ þ qij ðtÞgi ðui ðt rij ÞÞ; dt i¼1
ð3:1Þ
Theorem 2. Assume that the signal function fi ði ¼ 1; 2; . . . ; maxfn; pgÞ satisfies the hypotheses (H1) and (H2), suppose furthermore that (H3) and (H4) hold. If the system parameters satisfy the following conditions
186
ðH5Þ
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193 p X ðpjiþ þ l2i qþ ij Þ < 2ai ;
n X 2 þ ðqþ ij þ lj pji Þ < 2bj ;
j¼1
i¼1
then the almost periodic solution of the system (1.1a), (1.1b) is global asymptotically stable. Proof. Consider the Lyapunov functional V ðtÞ defined by V ðtÞ ¼ V1 ðtÞ þ V2 ðtÞ; p n X X V1 ðtÞ ¼ u2i ðtÞ þ v2j ðtÞ; j¼1
i¼1
V2 ðtÞ ¼
p n X X i¼1
l2j pjiþ
Z
p X n X
t
v2j ðsÞ ds þ
tsji
j¼1
j¼1
l2i qþ ij
Z
ð3:2Þ t
u2i ðsÞ ds:
trij
i¼1
Calculating the derivative of the Vi ðtÞ, i ¼ 1; 2, along the solutions of (3.1), respectively. p n X X dV1 0 ¼ 2 u ðtÞu ðtÞ þ 2 vj ðtÞv0j ðtÞ i i dt ð3:1Þ j¼1 i¼1 ( " #) p n X X ¼2 ui ðtÞ ai ðtÞui ðtÞ þ pji ðtÞgj ðt sij Þ i¼1
þ
p X
(
j¼1
"
vj ðtÞ bj ðtÞvj ðtÞ þ
j¼1
6
n X
2ai ðtÞu2i ðtÞ þ
þ
p X
6
2bi ðtÞv2j ðtÞ
(
i¼1
þ
p X j¼1
qij ðtÞgi ðt rij Þ
2a i þ (
p X
) h i jpji ðtÞj u2i ðtÞ þ l2j v2j ðt sji Þ
j¼1
(
j¼1 n X
#)
i¼1
(
i¼1
n X
p X
)
h i jqij ðtÞj v2j ðtÞ þ l2i u2i ðt rji Þ
n X i¼1
)
i¼1
pjiþ u2i ðtÞ þ
j¼1
2b j þ
þ
n X
)
p n X X i¼1
2 qþ ij vj ðtÞ þ
pjiþ l2j v2j ðt sji Þ
j¼1
p X
n X
j¼1
i¼1
2 2 qþ ij li ui ðt rij Þ:
ð3:3Þ
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
187
p n X n o X dV2 2 þ 2 2 ¼ l p v ðtÞ v ðt s Þ ji j ji j j dt ð3:1Þ i¼1 j¼1 p X n X
þ
i¼1
j¼1
¼
n X
p X
i¼1
j¼1
þ
2 2 l2i qþ ij ui ðtÞ ui ðt rij Þ
l2j pjiþ v2j ðtÞ
p n X X i¼1
p X
n X
j¼1
i¼1
2 l2i qþ ij ui ðtÞ
l2j pjiþ v2j ðt sji Þ
j¼1
p X
n X
j¼1
i¼1
2 l2i qþ ij ui ðt rij Þ:
ð3:4Þ
Note that ri , 2a i
p X
pjiþ l2i
j¼1
sj , 2b i
p X
qþ ij < 0;
j¼1
n X
2 qþ ij lj
i¼1
n X
pjiþ < 0:
i¼1
Combine with (3.3), (3.4) and (H5), we have ( ) p p n X X X dV þ 2 þ 2ai þ 6 pji þ li qij u2i ðtÞ dt ð3:1Þ j¼1 j¼1 i¼1 ( ) p n n X X X þ 2 þ þ qij þ lj pji v2j ðtÞ 2bj þ i¼1
j¼1
¼
n X
ri u2i ðtÞ
i¼1
p X
i¼1
sj v2j
j¼1
ð3:5Þ
< 0:
By Lemma 1, every solutions of (1.1a), (1.1b) remains bounded for all t P 0, then the derivatives dui ðtÞ=dt and dvj ðtÞ=dt also remain bounded for t P 0, which implies that ui ðtÞ and vj ðtÞ are uniformly continuous on ½0; þ1Þ. It follows from (3.5) that V ðtÞ þ
Z 0
t
n X i¼1
ri u2i ðsÞ ds þ
Z 0
t
p X j¼1
sj v2j ðsÞ ds 6 V ð0Þ:
188
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
Thus, ui ðtÞ; vj ðtÞ 2 L1 ½0; þ1Þ. By BarbalattÕs Lemma (see [6]), we can get that lim ui ðtÞ ¼ 0;
lim vi ðtÞ ¼ 0:
t!þ1
t!þ1
This completes the proof.
4. Global exponential stability of almost periodic solution Theorem 3. Assume that the signal function fi ði ¼ 1; 2; . . . ; maxfn; pgÞ satisfies (H1) and (H2). Conditions (H3) and (H4) hold. If there exist constants ki and knþj (i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; p) satisfy the following conditions: (H6) ! p p X X þ ki 2ai þ pji lj þ knþj qþ ij li < 0; j¼1
knþj
2b j þ
n X
j¼1
! qþ ij li
þ
n X
i¼1
ki pjiþ lj < 0:
i¼1
Then the almost periodic solution of system (1.1a), (1.1b) is global exponential stable. Proof. In view of (H3), we can choose a suitable constant a > 0 such that ! p n X X þ ki a 2ai þ pji lj þ ear knþj qþ ij li < 0; j¼1
knþj a
2b j
þ
n X
i¼1
! qþ ij li
þ eas
i¼1
n X
ki pjiþ lj < 0:
i¼1
Consider a Lyapunov functional V ðtÞ defined by V ðtÞ ¼ V1 ðtÞ þ V2 ðtÞ; ( Z p n X X ki u2i ðtÞ eat þ pjiþ lj V1 ðtÞ ¼ i¼1
V2 ðtÞ ¼
p X j¼1
(
knþj v2j ðtÞ eat þ
i¼1
v2j ðsÞ eaðsþsji Þ ds
tsji
j¼1 n X
)
t
qþ ij li
Z
t
; )
ð4:1Þ
u2i ðsÞ eaðsþrij Þ ds :
trij
Calculating the derivative of the Vi ðtÞ ði ¼ 1; 2Þ along solutions of (3.1), we have
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
189
(
n X dV1 ¼ ki a eat u2i ðtÞ þ 2ui ðtÞu0i ðtÞ eat dt ð3:1Þ i¼1 þ
p X
pjiþ lj
h
v2j ðtÞ eaðtþsji Þ
j¼1
¼
n X
(
v2j ðt
sji Þ e
aðtþssji Þ
"
ki a eat u2i ðtÞ þ 2 eat ui ðtÞ ai ðtÞui ðtÞ þ
i¼1 p X j¼1
6e
n X
pjiþ lj v2j ðtÞ eaðtþsji Þ
(
ki
pji ðtÞgj ðvj ðt sji ÞÞ
p X
pjiþ lj v2j ðt
)
aðtþssji Þ
sji Þ e
p h i X 2 þ 2 2 a 2a ðtÞ þ p l u ðtÞ þ v ðt s Þ u ji j i i ji i j j¼1
as
þe
p X
pjiþ lj v2j ðtÞ
e
aðssji Þ
j¼1
¼e
#
j¼1
i¼1
at
p X
)
j¼1
þ
at
i
n X
) pjiþ lj v2j ðt
sji Þ
j¼1
(
ki
p X
a
2a i
i¼1
þ
p X
!
pjiþ lj
u2i ðtÞ
þe
as
j¼1
p X
) pjiþ lj v2j ðtÞ
j¼1
ð4:2Þ and ( ) ! p n n X X X dV2 at þ 2 ar þ 2 ¼e knþj a 2bj þ qij li vj ðtÞ þ e qij li ui ðtÞ : dt ð3:1Þ j¼1 i¼1 i¼1 ð4:3Þ From (4.2) and (4.3), we obtain ( " ! # p p n X X X dV at þ ar þ 6e ki a 2ai þ pji lj þ e knþj qij li u2i ðtÞ dt j¼1 j¼1 i¼1 " ) ! # p n n X X X þ knþj a 2b qþ ki pjiþ lj v2j ðtÞ þ eas j þ ij li i¼1
j¼1
i¼1
ð4:4Þ
< 0: Namely, V ðtÞ 6 V ð0Þ for t P 0. Clearly, ( V ðtÞ P
min fkk g e
1 6 k 6 nþp
at
n X i¼1
u2i ðtÞ
þ
p X j¼1
) v2j ðtÞ
:
ð4:5Þ
190
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
On the other hand, from (4.1), we have ( " Z p n X X V ð0Þ 6 max fkk g u2i ð0Þ þ pjiþ lj 1 6 k 6 nþp
þ
p X
v2j ð0Þ
n X
þ
j¼1
6
max fkk g
n X
1 6 k 6 nþp
þ
pjiþ lj eas
p X n X
qþ ij li
Z
s
Z
rij p X
v2j ð0Þ
)
0
u2i ðsÞ eas ds
p n X X
pjiþ lj
j¼1
i¼1
j¼1
u2i ðsÞ eaðsþrij Þ ds
v2j ðsÞ eas ds
r
1 6 k 6 nþp
p X n X
#)
0
0
max fkk g 1 þ s eas
þ a ear
v2j ðsÞ eaðsþsji Þ ds
j¼1
i¼1
j¼1
6
u2i ð0Þ þ
j¼1
i¼1
þe
Z
i¼1
p n X X
ar
qþ ij li
i¼1
(
sji
j¼1
i¼1
"
#
0
! qþ ij li
T
T
kð/; wÞ ðx ; y Þ k
i¼1
T
T
¼ M kð/; wÞ ðx ; y Þ k;
ð4:6Þ
where M ¼ 1 þ s eas
p n X X i¼1
pjiþ lj þ a ear
j¼1
p X n X j¼1
qþ ij li :
i¼1
Set M¼
max1 6 k 6 nþp fkk g M : min1 6 k 6 nþp fkk g
Obviously, M > 1: It follows from (4.5) and (4.6) that n X i¼1
u2i ðtÞ þ
p X
v2j ðtÞ 6 M ð/; wÞT ðx ; Y ÞT eat
j¼1
for t P 0. This completes the proof.
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
191
5. An example In this section, we give an example to illustrate that our results are feasible. Consider the following simple BAM networks with periodic coefficients and delays, 3 X dxi ¼ ai ðtÞxi ðtÞ þ pji ðtÞfj ðyj ðt 2pÞÞ þ Ii ðtÞ; dt j¼1
ð5:1aÞ
3 X dyj ¼ bj ðtÞyj ðtÞ þ qij ðtÞfi ðxi ðt 2pÞÞ þ Jj ðtÞ; dt i¼1
ð5:1bÞ
where Ii ðtÞ ¼ sin t
ði ¼ 1; 2; 3Þ;
rij ¼ sji ¼ 2p
Jj ðtÞ ¼ cos t
ðj ¼ 1; 2; 3Þ;
ði ¼ 1; 2; 3; j ¼ 1; 2; 3Þ:
To take fi ðxÞ ¼ 12ðjx þ 1j jx 1jÞ, we have li ¼ 1 ði ¼ 1; 2; 3; 4; 5; 6Þ. Again take ða1 ðtÞ; a2 ðtÞ; a3 ðtÞÞT ¼ ð2 sin t; 2 cos t; 2 sin tÞT ; T
T
ðb1 ðtÞ; b2 ðtÞ; b3 ðtÞÞ ¼ ð2 cos t; 2 sin t; 2 cos tÞ we obtain T
T
ðb 1 ; b2 ; b3 Þ ¼ ð1; 1; 1Þ ;
T
T
and M½ai > 0; M½bj > 0:
ða 1 ; a2 ; a3 Þ ¼ ð1; 1; 1Þ ; þ þ ðbþ 1 ; b2 ; b3 Þ ¼ ð3; 3; 3Þ ;
T
T
Let 0
1 0 1 q11 ðtÞ q12 ðtÞ q13 ðtÞ 0:25 sin t 0:05 cos t 0:15 sin t @ q21 ðtÞ q22 ðtÞ q23 ðtÞ A ¼ @ 0:05 cos t 0:50 sin t 0:05 cos t A; q31 ðtÞ q32 ðtÞ q33 ðtÞ 0:25 sin t 0:05 cos t 0:25 sin t 0 1 0 1 0:05 sin t 0:10 cos t 0:15 sin t p11 ðtÞ p12 ðtÞ p13 ðtÞ @ p21 ðtÞ p22 ðtÞ p23 ðtÞ A ¼ @ 0:15 cos t 0:05 sin t 0:15 cos t A: p31 ðtÞ p32 ðtÞ p33 ðtÞ 0:05 sin t 0:15 cos t 0:05 sin t Then 0
qþ 11 @ qþ 21 qþ 31
qþ 12 qþ 22 qþ 32
1 0 qþ 0:25 13 A ¼ @ 0:05 qþ 23 qþ 0:25 33
0:05 0:50 0:05
1 0:15 0:05 A 0:25
192
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
and 0
þ p11 @ pþ 21 þ p31
þ p12 þ p22 þ p32
1 0 þ p13 0:05 þ A ¼ @ 0:15 p23 þ 0:05 p33
0:10 0:05 0:15
1 0:15 0:15 A: 0:05
Moreover, ( P3
þ j¼1 pji lj a i
r ¼ max
16i63
)
( P3 þ max
16j63
þ i¼1 qij li b j
) ¼ 0:95 < 1:
Set ki ¼ 1 ði ¼ 1; 2; 3; 4; 5; 6Þ, we get ! 3 3 X X þ pj1 lj þ k3þj qþ k1 2a1 þ 1j l1 ¼ 1:3 < 0; j¼1
k2
2a 2 þ
3 X
j¼1
! þ pj2 lj
þ
j¼1
k3
2a 3 þ
3 X
k4
2b 1 þ
! þ pj3 lj
þ
k5
þ
3 X
! qþ i1 li
þ
2b 3 þ
3 X i¼1
k3þj qþ 3j l3 ¼ 1:1 < 0;
3 X
ki p1iþ l1 ¼ 1:15 < 0;
i¼1
! qþ i2 li
þ
i¼1
k6
3 X j¼1
i¼1
2b 2
k3þj qþ 2j l2 ¼ 1:1 < 0;
j¼1
j¼1 3 X
3 X
3 X
ki p2iþ l2 ¼ 1:05 < 0;
i¼1
! qþ i3 li
þ
3 X
ki p3iþ l3 ¼ 1:3 < 0:
i¼1
Thus, it follows from Theorems 1 and 2 that the unique periodic solution (5.1a), (5.1b) is globally exponential stable.
Acknowledgements Supported by the NNSF of China (10071016)and Foundation for University Key Teacher by the Ministry of Education of China, as also supported by the Foundation of Southeast University and the Natural Science Foundations of Jiangsu Province and Yunnan Province, China, and by the Foundation of professor project of Chenzhou Teachers College.
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193
References [1] A.M. Fink, Almost Periodic Differential Equation, Springer-Verlag, Berlin, Heidleberg, New York, 1974. [2] J.D. Cao, Global stability analysis in delayed cellular neural networks, Phys. Rev. E 59 (1999) 5940–5944. [3] J.D. Cao, On stability of delayed cellular neural networks, Phys. Lett. A 261 (1999) 303–308. [4] J.D. Cao, Periodic oscillatory solution of bidirectional associative memory networks with delays, Phys. Rev. E 61 (2000) 1825–1828. [5] K. Gopalsamy, X.Z. He, Delay-independent stability in bi-directional associative memory networks, IEEE Trans. Neural Networks 5 (1994) 998–1002. [6] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population, Kluwer, Dordrecht, 1992. [7] C.Y. He, Almost Periodic Differential Equation, Higher Education Publishing House, Beijing, 1992 (in Chinese). [8] C. Jin, Stability analysis of discrete-time Hopfield BAM neural networks, Acta. Auto. Sinica 25 (5) (1999) 606–612. [9] T. Kohonen, Self-Organization and Associative Memory, Springer-Verlag, New York, 1988. [10] D.G. Kelly, Stability in contractive nonlinear neural networks, IEEE Trans. Biomed. Eng. 37 (1990) 231–242. [11] B. Kosto, in: Neural Networks and Fuzzy Systems – A Dynamical System Approach to Machine Intelligence, Prentice-Hall, Englewood Cliffs, NJ, 1992, pp. 38–108. [12] B. Kosto, Adaptive bi-directional associative memories, IEEE Trans. Syst. Man Cybernet. 18 (1988) 49. [13] B. Kosto, Bi-directional associative memories, IEEE Trans. Syst. Man Cybernet. 18 (1988) 49– 60. [14] X.F. Liao, J.B. Yu, Qualitative analysis of bidrectional associative memory with time delays, Int. J. Circuit Theory Appl. 26 (1998) 219–229. [15] X.X. Liao, Stability of the Hopfield neural networks, Sci. China 23 (5) (1993) 523–532. [16] X.B. Liang, Global exponential stability and application of the Hopfield neural networks, Sci. China 25 (5) (1995) 523–532. [17] J.J. Wei, S.G. Ruan, Stability and bifurcation in a neural network model with two delays, Phys. D 130 (1999) 255–272. [18] J. Cao, L. Wang, Exponential stability and periodic oscillatory solution in BAM networks with delays, IEEE Trans. Neural Networks 13 (2) (2002). [19] J. Cao, A set of stability criteria for delayed cellular neural networks, IEEE Trans. Circuits Syst. I 48 (4) (2001) 494–498. [20] J. Cao, Global stability conditions for delayed CNNs, IEEE Trans. Circuits Syst. I 48 (11) (2001) 1330–1333. [21] J. Cao, Periodic oscillation and exponential stability of delayed CNNs, Phys. Lett. A 270 (3–4) (2000) 157–163. [22] J. Cao, Exponential stability and periodic solution of delayed cellular neural networks, Science in China (Series E) 43 (3) (2000) 328–336. [23] J. Cao, Global exponential stability of Hopfield neural networks, Int. J. Syst. Sci. 32 (2) (2001) 233–236.
Existence and stability of almost periodic solution for BAM neural networks with delays Anping Chen a, Lihong Huang b, Jinde Cao a
c,*
Department of Mathematics, Chenzhou Teacher’s College, Chenzhou, Hunan 423000, China b College of Mathematics and Econometrics, Hunan University, Changsha 410082, China c Department of Applied Mathematics, Southeast University, Nanjing 210096, China
Abstract By using the Banach fixed point theorem and constructing suitable Lyapunov function, some sufficient conditions are obtained ensuring existence, uniqueness and global stability of almost periodic solution of the BAM neural networks with variable coefficients and delays. These results are helpful to design global exponential stable BAM networks and almost periodic oscillatory BAM networks. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Almost periodic solution; Global stability; BAM neural networks
1. Introduction Recently, a class of two-layer heteroassociative networks called bidirectional associative memory (BAM) networks [5,12–15] with and without axonal signal transmission delays has been proposed and used in many fields such as pattern recognition and automatic control. At the same time, it has attracted many scholars, e.g., [2–4,8–11,15–23], the stability of bidirectional associative memory neural networks with and without delays has been studied. In [5,10,12–15],
*
Corresponding author. E-mail addresses: [email protected] (A. Chen), [email protected] (L. Huang), [email protected], [email protected] (J. Cao). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 0 9 5 - 4
178
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some sufficient conditions have been obtained for globally asymptotic stability of delayed bidirectional associative memory networks (BAM). Moreover, author in [4] researches the periodic oscillatory solution of BAM networks with delays in the case of constant coefficients. It is well known that studies on neural dynamical systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior, almost periodic oscillatory properties, chaos, and bifurcation [17]. In applications, almost periodic oscillatory is more accordant with fact. To the best of our knowledge, few authors have considered almost periodic oscillatory solutions for BAM networks, most of them study the stability, periodic oscillation of BAM networks in the case of constant coefficients. In this paper, we discuss almost periodic oscillatory solutions of BAM networks with variable coefficients and delays, obtain some simple sufficient conditions ensuring the existence and global exponential stability of almost periodic solution. Consider the BAM networks with variable coefficients X dxi ¼ ai ðtÞxi ðtÞ þ pji ðtÞfj ðyj ðt sji ÞÞ þ Ii ðtÞ; dt j¼1
ð1:1aÞ
n X dyj ¼ bj ðtÞyj ðtÞ þ qij ðtÞfi ðxi ðt rij ÞÞ þ Jj ðtÞ; dt i¼1
ð1:1bÞ
p
where i ¼ 1; 2; . . . ; n, j ¼ 1; 2; . . . ; p, xi ðtÞ and yj ðtÞ are the activations of the ith neurons and the jth neurons, respectively. pji ðtÞ, qij ðtÞ are the connection weights at the time t, and Ii ðtÞ and Jj ðtÞ denote the external inputs at time t. sji and rij are nonnegative constants, which correspond to the finite speed of the axonal signal transmission. An analog circuit implementing of BAM equation with axonal signal transmission delays can be seen in [4,15,18]. Throughout this paper, we always assume that ai ðtÞ, bj ðtÞ, pji ðtÞ, qij ðtÞ, Ii ðtÞ and Jj ðtÞ are continuous almost periodic functions. Moreover, ai ðtÞ, bj ðtÞ are positive, 0 < inf fai ðtÞg ¼ a i ; t2R
0 < inf fbj ðtÞg ¼ b j ;
qþ ij
¼ supfjqij ðtÞjg < þ1;
Jjþ
¼ supfjJj ðtÞjg < þ1:
t2R
t2R
Iiþ
pjiþ ¼ supfjpji ðtÞjg < þ1; t2R
¼ supfjIi ðtÞjg < þ1; t2R
t2R
Furthermore, the signal function fi possesses the following properties: (H1) fi is bounded on R, for all i ¼ 1; 2; . . . ; maxfn; pg; (H2) There exists a number li > 0 such that jfi ðxÞ fi ðyÞj 6 li jx yj, i ¼ 1; 2; . . . ; maxfn; pg.
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
179
The initial conditions associated with (1.1a), (1.1b) are associated to be of the form xi ðtÞ ¼ /i ðsÞ;
s 2 ½s; 0 ;
s ¼ max max fsji g;
yj ðtÞ ¼ wi ðsÞ;
s 2 ½r; 0 ;
r ¼ max max frij g;
16i6n 16j6p
ð1:2Þ
16j6p 16i6n
where /i ðsÞ, wi ðsÞ are continuous almost periodic functions on R. For any solution zðtÞ , ðxðtÞ; yðtÞÞT , ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; yp ðtÞÞT and almost periodic solution T
z ðtÞ , ðx ðtÞ; y ðtÞÞ , ðx 1 ðtÞ; x 2 ðtÞ; . . . ; x n ðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; yp ðtÞÞ T
T
T
of system (1.1a), (1.1b), define kð/; wÞ ðx ; y Þ k as ) ( n X T T 2
ð/; wÞ ðx ; y Þ ¼ sup ð/i ðtÞ x ðtÞÞ s 6 t 6 0
i
i¼1
( þ sup r 6 t 6 0
p X
) ðwj ðtÞ
2 yj ðtÞÞ
:
j¼1
Definition 1. The almost periodic solution ðx ðtÞ; y ðtÞÞT of system (1.1a), (1.1b) is said to be global exponential stable, if there exists constants a > 0 and M P 1 such that p n X X ðxi ðtÞ x i ðtÞÞ2 þ ðyj ðtÞ yj ðtÞÞ2 6 M ð/; wÞT ðx ; y ÞT eat i¼1
j¼1
for all t > 0. Definition 2 [1,7]. Let zðtÞ : R ! Rn be continuous for t 2 R. zðtÞ is said to be almost periodic on R if, for any e > 0, it is possible to find a real number l ¼ lðeÞ > 0 such that, for any interval with length lðeÞ, there exists a number s ¼ sðeÞ in this interval such that jzðt þ sÞ zðtÞj < e for 8t 2 R: Definition 3 [1]. Let z 2 Rn and QðtÞ be a n n continuous matrix defined on R. The linear system dz ¼ QðtÞzðtÞ dt
ð1:3Þ
is said to admit an exponential dichotomy on R if there exits constants k, k > 0, projection P and the fundamental matrix ZðtÞ of (1.3) satisfying
180
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
ZðtÞPZ 1 ðsÞ 6 k ekðtsÞ for t P s; ZðtÞðI P ÞZ 1 ðsÞ 6 k ekðstÞ for t 6 s: Lemma 1 [1,7]. If the linear system (1.3) admits an exponential dichotomy, then almost periodic system dz ¼ QðtÞz þ gðtÞ dt
ð1:4Þ
has a unique almost periodic solution zðtÞ, and Z t Z þ1 zðtÞ ¼ ZðtÞPZ 1 ðsÞgðsÞ ds ZðtÞðI P ÞZ 1 ðsÞgðsÞ ds: 1
ð1:5Þ
t
Lemma 2 [1]. Assume that ci ðtÞ is an almost periodic function and Z 1 tþT M½ci ¼ lim ci ðsÞ ds > 0; i ¼ 1; 2; . . . ; n: T !þ1 T t Then the system dz ¼ diagðc1 ðtÞ; c2 ðtÞ; . . . ; cn ðtÞÞzðtÞ: dt Then the linear system (1.3) admits an exponential dichotomy. Lemma 3. Assume that fk ðxÞ ðk ¼ 1; 2; . . . ; maxðn; pÞÞ is continuous bounded function on R. Then every solution of system (1.1a), (1.1b) is bounded. Proof. Set di ¼
p X j¼1
pjiþ sup jfj ðxÞj þ Iiþ x2R
and cj ¼
n X i¼1
þ qþ ij sup jfj ðxÞj þ Jj
for i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; p:
x2R
Let T
zðtÞ , ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; yp ðtÞÞ
be a solution of (1.1a), (1.1b). Then it follows from (1.1a) that ai ðtÞxi ðtÞ di 6 x0i ðtÞ 6 ai ðtÞxi ðtÞ þ di :
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
Multiplying the inequality above by e di e
Rt 0
ai ðuÞ du
Rt 0
ai ðuÞ du
181
, we have
Rt Rt d ai ðuÞ du a ðuÞ du xi ðtÞ e 0 6 ; 6 di e 0 i dt
i ¼ 1; 2; . . . ; n:
By integrating the inequality from 0 to s P 0, we get Z t Rv Z s Rv Rs a ðuÞ du a ðuÞ du a ðuÞ du di e 0 i dv 6 xi ðsÞ e 0 i xi ð0Þ 6 di e 0 i dv; 0
0
i.e., xi ð0Þ e
Rs 0
ai ðuÞ du
di
Z
s
e
Rs v
ai ðuÞ du
0
6 xi ðsÞ 6 xi ð0Þ e
Rs 0
ai ðuÞ du
þ di
dv
Z
s
e
Rs v
ai ðuÞ du
dv
0
for s P v P 0. Therefore, di di di di jxi ð0Þj þ eai s 6 xi ðsÞ 6 jxi ð0Þj eai s þ ai ai ai ai for all s > 0. Hence, di di di di jxi ð0Þj 6 xi ðtÞ 6 þ jxi ð0Þj ai ai ai ai for all s > 0 and i ¼ 1; 2; . . . ; n. Using a similar argument, we can obtain cj cj cj cj jyj ð0Þj 6 yj ðtÞ 6 þ jyj ð0Þj bj bj bj bj for all s > 0 and j ¼ 1; 2; . . . ; p. The proof is complete.
2. Existence of almost periodic solution For an arbitrary vector zðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ; y1 ðtÞ; y2 ðtÞ; . . . ; yp ðtÞÞT , we define the norm kzðtÞk ¼ max1 6 i 6 n jxi ðtÞj þ max1 6 j 6 p jyj ðtÞj. Set n o T S nþp ¼ z j z ¼ ð/1 ; /2 ; . . . ; /n ; w1 ; w2 ; . . . ; wp Þ ; where /i , wj are continuous almost periodic functions on R, i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; p. For any z 2 S nþp , we define induced module kzk ¼ supt2R kzðtÞk ¼ supt2R max1 6 i 6 n j/i ðtÞj þ supt2R max1 6 j 6 p jwj ðtÞj, then S nþp is a Banach space.
182
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
Theorem 1. In addition to (H1) and (H2), suppose further that ( Pn )
Pp þ þ j¼1 pji lj i¼1 qij li (H3) r ¼ max þ max < 1; 16i6n 16j6p a b i j R tþT R tþT (H4) M½ai ¼ limT !þ1 t ai ðsÞ ds > 0, and M½bj ¼ limT !þ1 t bj ðsÞ ds > 0, i ¼ 1; 2; . . . ; n, j ¼ 1; 2; 3; . . . ; p. Then there is a unique periodic solution of system (1.1a), (1.1b) in the region kz z0 k 6 rI=ð1 rÞ, where ( )
þ Ii Jiþ I ¼ max þ max ; 16i6n 16j6p ai b j and z0 ¼
Z
t
Rt
a1 ðuÞ du
I1 ðsÞ ds; . . . ;
Z
t
Rt
an ðuÞ du
In ðsÞ ds; 1 Z 1 Z R R t t t t b ðuÞ du b ðuÞ du e s 1 J1 ðsÞ ds; . . . ; e s p Jp ðsÞ ds : e
s
1
e
s
1
T
T
Proof. For any ð/; wÞ ¼ ð/1 ; /2 ; . . . ; /n ; w1 ; w2 ; . . . ; wp Þ 2 S nþp , we consider the almost solution zð/;wÞT of nonlinear almost periodic differential equation X dxi ¼ ai ðtÞxi ðtÞ þ pji ðtÞfj ðwj ðt sji ÞÞ þ Ii ðtÞ; dt j¼1
ð2:1aÞ
n X dyj ¼ bj ðtÞyj ðtÞ þ qij ðtÞfi ð/i ðt rij ÞÞ þ Jj ðtÞ: dt i¼1
ð2:1bÞ
p
Because M½ai > 0, M½bj > 0, by Lemma 2, the linear system dxi ¼ ai ðtÞxi ðtÞ; dt dyj ¼ bj ðtÞyj ðtÞ dt
ð2:2Þ
admits an exponential dichotomy on R. By Lemma 1, the solution zð/;wÞT of (2.1a), (2.1b) can be expressed as following (Z " # Rt p t X a1 ðuÞ du zð/;wÞT ¼ e s pj1 ðsÞfj ðwj ðs sj1 ÞÞ þ I1 ðsÞ ds; . . . ; 1
Z
t
e 1
Rt s
" an ðuÞ du
j¼1 p X j¼1
# pjn ðsÞfj ðwj ðs: sjn ÞÞ þ In ðsÞ ds;
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
Z
t
e
Rt s
" b1 ðuÞ du
1
Z
n X
183
# qj1 ðsÞfi ð/i ðs ri1 ÞÞ þ J1 ðsÞ ds; . . . ;
i¼1
t
e
Rt s
" bp ðuÞ du
1
n X
#
)
qip ðsÞfi ð/i ðs sip ÞÞ þ Jp ðsÞ ds :
ð2:3Þ
i¼1
Now we define a mapping T : S nþp ! S nþp ;
T ð/; wÞðtÞ ¼ zð/;wÞT :
Set B ¼
z j z 2 S nþp ; kz z0 k 6
rI : 1r
Then B is a closed convex subset of S nþp . According to the definition of the norm of Banach space S nþp , we have Z t Rt ai ðuÞ du s kz0 k ¼ sup max e Ii ðsÞ ds t2R 1 6 i 6 n
1
Z t Rt b ðuÞ du þ sup max e s j Jj ðsÞ ds t2R 1 6 j 6 p 1 Z t Rt a ðuÞ du 6 sup max jIi ðsÞje s i t2R 1 6 i 6 n
1
þ sup max
t2R 1 6 j 6 p
Z
Rt Jj ðsÞ e s bj ðuÞ du ds
t
1
( ) Jjþ Iiþ 6 sup max þ sup max ai b t2R 1 6 i 6 n t2R 1 6 j 6 p j ( )
þ þ Jj Ii ¼ max þ max 16i6n 16j6p ai b j
¼ I: Therefore, kzk 6 kz z0 k þ kz0 k ¼
rI I þI ¼ : 1r 1r
ð2:4Þ
First, we prove that the mapping T is a self-mapping from B to B . In fact, for any z 2 B , we have
184
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
( Z " # ) Rt p t X ai ðuÞ du kT ðzÞ z0 k ¼ sup max e s pji ðsÞfj wj ðs sji Þ ds t2R 1 6 i 6 n 1 j¼1 ) ( Z " # Rt n t X bj ðuÞ du e s qij ðsÞfi /i ðs rij Þ ds þ sup max 1 6 j 6 p t2R 1 i¼1 (Z " # ) p t X 6 sup max eai ðtsÞ pjiþ lj wj ðs sji Þ ds t2R 1 6 i 6 n
1
þ sup max
(Z
(Z
t
e
t2R 1 6 i 6 n
¼
(Z
a i ðtsÞ
qþ ij li /i ðs
p X
e
b j ðtsÞ
1
n X i¼1
þ j¼1 pji lj a i
þ max
16j6p
¼ rkzk rI ; 6 1r
# ) rij Þ ds
)
pjiþ lj kzk ds
j¼1 t
Pp max
n X i¼1
1
t2R 1 6 j 6 p
16i6n
b j ðtsÞ
1
6 sup max
(
"
e
t2R 1 6 j 6 p
þ sup max
j¼1 t
) qþ ij li k zk ds Pn
þ i¼1 qij li b j
!) kzk
ð2:5Þ
which implies that T ðzÞðtÞ 2 B . Therefore, the mapping T is a self-mapping from B to B . Next, we prove the mapping T is a contraction mapping of B . In fact, in view of (H1) and (H2), for any z1 ; z2 2 B , where T
z1 ¼ ðn1 ; n2 ; . . . ; nn ; g1 ; g2 ; . . . ; gp Þ ; T
z2 ¼ ð/1 ; /2 ; . . . ; /n ; w1 ; w2 ; . . . ; wp Þ ; we have kT ðz1 Þ T ðz2 Þk
) ( Z Rt p t X ai ðuÞ du s e pji ðsÞ fj ðnj ðs sji ÞÞ fj ð/j ðs sji ÞÞ ¼ sup max t2R 1 6 i 6 n 1 j¼1 ) ( Z Rt t n X bj ðuÞ du s e qij ðsÞ fi ðgi ðs rij ÞÞ fi ðwi ðs rij ÞÞ þ sup max 1 t2R 1 6 j 6 p i¼1 (Z " # ) p t X eai ðtsÞ pþ lj nj ðs sji Þ /j ðs sji Þ ds 6 sup max t2R 1 6 i 6 n
1
ji
j¼1
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
þ sup max
(Z
t2R 1 6 j 6 p
(Z
¼
ebj ðtsÞ
1
e (Z
a i ðtsÞ
t
1
Pp max
16i6n
ds qþ ij li gi ðs rij Þ wi ðs rij Þ
p X
þ j¼1 pji lj a i
b j ðtsÞ
n X i¼1
!
þ max
16j6p
)
)
pjiþ lj kz1
z2 k ds
j¼1
e
t2R 1 6 j 6 p
n X
185
#
i¼1
1
þ sup max (
t
t
6 sup max
t2R 1 6 i 6 n
"
) qþ ij li kz1
z2 k ds
Pn
þ i¼1 qij li b j
!) kz 1 z 2 k
¼ rkz1 z2 k:
ð2:6Þ
Noting that r < 1, it is clear that T is a contraction mapping. Thus, T possesses a unique fixed point z 2 B , that is T ðz Þ ¼ z . By (2.3), (2.1a) and (2.1b), T T ðx ; y Þ satisfies (1.1a), (1.1b). So ðx ; y Þ is a unique periodic solution of
(1.1a) and (1.1b) in B . The proof is complete.
3. Global asymptotic stability of almost periodic solution In this section, we discuss the global asymptotic stability of BAM networks. Set ui ðtÞ ¼ xi ðtÞ x i ðtÞ;
vj ðtÞ ¼ yj ðtÞ yj ðtÞ;
gi ðui ðt rij ÞÞ ¼ fi ðui ðt rij Þ þ x i ðtÞÞ fi ðx i ðtÞÞ; gj ðvj ðt sij ÞÞ ¼ fj ðvj ðt sji Þ þ yj ðtÞÞ fj ðyj ðtÞÞ: Then jgi ðui ðt rij ÞÞj 6 li jui ðt rij Þj;
jgj ðvj ðt sij ÞÞj 6 lj jvj ðt sji Þj
for i ¼ 1; 2; . . . ; n, and j ¼ 1; 2; . . . ; p. It is easy to see that system (1.1a), (1.1b) can be reduced to the following system X dui ¼ ai ðtÞui ðtÞ þ pji ðtÞgj ðvj ðt sij ÞÞ; dt j¼1 p
n X dvj ¼ bj ðtÞvj ðtÞ þ qij ðtÞgi ðui ðt rij ÞÞ; dt i¼1
ð3:1Þ
Theorem 2. Assume that the signal function fi ði ¼ 1; 2; . . . ; maxfn; pgÞ satisfies the hypotheses (H1) and (H2), suppose furthermore that (H3) and (H4) hold. If the system parameters satisfy the following conditions
186
ðH5Þ
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193 p X ðpjiþ þ l2i qþ ij Þ < 2ai ;
n X 2 þ ðqþ ij þ lj pji Þ < 2bj ;
j¼1
i¼1
then the almost periodic solution of the system (1.1a), (1.1b) is global asymptotically stable. Proof. Consider the Lyapunov functional V ðtÞ defined by V ðtÞ ¼ V1 ðtÞ þ V2 ðtÞ; p n X X V1 ðtÞ ¼ u2i ðtÞ þ v2j ðtÞ; j¼1
i¼1
V2 ðtÞ ¼
p n X X i¼1
l2j pjiþ
Z
p X n X
t
v2j ðsÞ ds þ
tsji
j¼1
j¼1
l2i qþ ij
Z
ð3:2Þ t
u2i ðsÞ ds:
trij
i¼1
Calculating the derivative of the Vi ðtÞ, i ¼ 1; 2, along the solutions of (3.1), respectively. p n X X dV1 0 ¼ 2 u ðtÞu ðtÞ þ 2 vj ðtÞv0j ðtÞ i i dt ð3:1Þ j¼1 i¼1 ( " #) p n X X ¼2 ui ðtÞ ai ðtÞui ðtÞ þ pji ðtÞgj ðt sij Þ i¼1
þ
p X
(
j¼1
"
vj ðtÞ bj ðtÞvj ðtÞ þ
j¼1
6
n X
2ai ðtÞu2i ðtÞ þ
þ
p X
6
2bi ðtÞv2j ðtÞ
(
i¼1
þ
p X j¼1
qij ðtÞgi ðt rij Þ
2a i þ (
p X
) h i jpji ðtÞj u2i ðtÞ þ l2j v2j ðt sji Þ
j¼1
(
j¼1 n X
#)
i¼1
(
i¼1
n X
p X
)
h i jqij ðtÞj v2j ðtÞ þ l2i u2i ðt rji Þ
n X i¼1
)
i¼1
pjiþ u2i ðtÞ þ
j¼1
2b j þ
þ
n X
)
p n X X i¼1
2 qþ ij vj ðtÞ þ
pjiþ l2j v2j ðt sji Þ
j¼1
p X
n X
j¼1
i¼1
2 2 qþ ij li ui ðt rij Þ:
ð3:3Þ
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
187
p n X n o X dV2 2 þ 2 2 ¼ l p v ðtÞ v ðt s Þ ji j ji j j dt ð3:1Þ i¼1 j¼1 p X n X
þ
i¼1
j¼1
¼
n X
p X
i¼1
j¼1
þ
2 2 l2i qþ ij ui ðtÞ ui ðt rij Þ
l2j pjiþ v2j ðtÞ
p n X X i¼1
p X
n X
j¼1
i¼1
2 l2i qþ ij ui ðtÞ
l2j pjiþ v2j ðt sji Þ
j¼1
p X
n X
j¼1
i¼1
2 l2i qþ ij ui ðt rij Þ:
ð3:4Þ
Note that ri , 2a i
p X
pjiþ l2i
j¼1
sj , 2b i
p X
qþ ij < 0;
j¼1
n X
2 qþ ij lj
i¼1
n X
pjiþ < 0:
i¼1
Combine with (3.3), (3.4) and (H5), we have ( ) p p n X X X dV þ 2 þ 2ai þ 6 pji þ li qij u2i ðtÞ dt ð3:1Þ j¼1 j¼1 i¼1 ( ) p n n X X X þ 2 þ þ qij þ lj pji v2j ðtÞ 2bj þ i¼1
j¼1
¼
n X
ri u2i ðtÞ
i¼1
p X
i¼1
sj v2j
j¼1
ð3:5Þ
< 0:
By Lemma 1, every solutions of (1.1a), (1.1b) remains bounded for all t P 0, then the derivatives dui ðtÞ=dt and dvj ðtÞ=dt also remain bounded for t P 0, which implies that ui ðtÞ and vj ðtÞ are uniformly continuous on ½0; þ1Þ. It follows from (3.5) that V ðtÞ þ
Z 0
t
n X i¼1
ri u2i ðsÞ ds þ
Z 0
t
p X j¼1
sj v2j ðsÞ ds 6 V ð0Þ:
188
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
Thus, ui ðtÞ; vj ðtÞ 2 L1 ½0; þ1Þ. By BarbalattÕs Lemma (see [6]), we can get that lim ui ðtÞ ¼ 0;
lim vi ðtÞ ¼ 0:
t!þ1
t!þ1
This completes the proof.
4. Global exponential stability of almost periodic solution Theorem 3. Assume that the signal function fi ði ¼ 1; 2; . . . ; maxfn; pgÞ satisfies (H1) and (H2). Conditions (H3) and (H4) hold. If there exist constants ki and knþj (i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; p) satisfy the following conditions: (H6) ! p p X X þ ki 2ai þ pji lj þ knþj qþ ij li < 0; j¼1
knþj
2b j þ
n X
j¼1
! qþ ij li
þ
n X
i¼1
ki pjiþ lj < 0:
i¼1
Then the almost periodic solution of system (1.1a), (1.1b) is global exponential stable. Proof. In view of (H3), we can choose a suitable constant a > 0 such that ! p n X X þ ki a 2ai þ pji lj þ ear knþj qþ ij li < 0; j¼1
knþj a
2b j
þ
n X
i¼1
! qþ ij li
þ eas
i¼1
n X
ki pjiþ lj < 0:
i¼1
Consider a Lyapunov functional V ðtÞ defined by V ðtÞ ¼ V1 ðtÞ þ V2 ðtÞ; ( Z p n X X ki u2i ðtÞ eat þ pjiþ lj V1 ðtÞ ¼ i¼1
V2 ðtÞ ¼
p X j¼1
(
knþj v2j ðtÞ eat þ
i¼1
v2j ðsÞ eaðsþsji Þ ds
tsji
j¼1 n X
)
t
qþ ij li
Z
t
; )
ð4:1Þ
u2i ðsÞ eaðsþrij Þ ds :
trij
Calculating the derivative of the Vi ðtÞ ði ¼ 1; 2Þ along solutions of (3.1), we have
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
189
(
n X dV1 ¼ ki a eat u2i ðtÞ þ 2ui ðtÞu0i ðtÞ eat dt ð3:1Þ i¼1 þ
p X
pjiþ lj
h
v2j ðtÞ eaðtþsji Þ
j¼1
¼
n X
(
v2j ðt
sji Þ e
aðtþssji Þ
"
ki a eat u2i ðtÞ þ 2 eat ui ðtÞ ai ðtÞui ðtÞ þ
i¼1 p X j¼1
6e
n X
pjiþ lj v2j ðtÞ eaðtþsji Þ
(
ki
pji ðtÞgj ðvj ðt sji ÞÞ
p X
pjiþ lj v2j ðt
)
aðtþssji Þ
sji Þ e
p h i X 2 þ 2 2 a 2a ðtÞ þ p l u ðtÞ þ v ðt s Þ u ji j i i ji i j j¼1
as
þe
p X
pjiþ lj v2j ðtÞ
e
aðssji Þ
j¼1
¼e
#
j¼1
i¼1
at
p X
)
j¼1
þ
at
i
n X
) pjiþ lj v2j ðt
sji Þ
j¼1
(
ki
p X
a
2a i
i¼1
þ
p X
!
pjiþ lj
u2i ðtÞ
þe
as
j¼1
p X
) pjiþ lj v2j ðtÞ
j¼1
ð4:2Þ and ( ) ! p n n X X X dV2 at þ 2 ar þ 2 ¼e knþj a 2bj þ qij li vj ðtÞ þ e qij li ui ðtÞ : dt ð3:1Þ j¼1 i¼1 i¼1 ð4:3Þ From (4.2) and (4.3), we obtain ( " ! # p p n X X X dV at þ ar þ 6e ki a 2ai þ pji lj þ e knþj qij li u2i ðtÞ dt j¼1 j¼1 i¼1 " ) ! # p n n X X X þ knþj a 2b qþ ki pjiþ lj v2j ðtÞ þ eas j þ ij li i¼1
j¼1
i¼1
ð4:4Þ
< 0: Namely, V ðtÞ 6 V ð0Þ for t P 0. Clearly, ( V ðtÞ P
min fkk g e
1 6 k 6 nþp
at
n X i¼1
u2i ðtÞ
þ
p X j¼1
) v2j ðtÞ
:
ð4:5Þ
190
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
On the other hand, from (4.1), we have ( " Z p n X X V ð0Þ 6 max fkk g u2i ð0Þ þ pjiþ lj 1 6 k 6 nþp
þ
p X
v2j ð0Þ
n X
þ
j¼1
6
max fkk g
n X
1 6 k 6 nþp
þ
pjiþ lj eas
p X n X
qþ ij li
Z
s
Z
rij p X
v2j ð0Þ
)
0
u2i ðsÞ eas ds
p n X X
pjiþ lj
j¼1
i¼1
j¼1
u2i ðsÞ eaðsþrij Þ ds
v2j ðsÞ eas ds
r
1 6 k 6 nþp
p X n X
#)
0
0
max fkk g 1 þ s eas
þ a ear
v2j ðsÞ eaðsþsji Þ ds
j¼1
i¼1
j¼1
6
u2i ð0Þ þ
j¼1
i¼1
þe
Z
i¼1
p n X X
ar
qþ ij li
i¼1
(
sji
j¼1
i¼1
"
#
0
! qþ ij li
T
T
kð/; wÞ ðx ; y Þ k
i¼1
T
T
¼ M kð/; wÞ ðx ; y Þ k;
ð4:6Þ
where M ¼ 1 þ s eas
p n X X i¼1
pjiþ lj þ a ear
j¼1
p X n X j¼1
qþ ij li :
i¼1
Set M¼
max1 6 k 6 nþp fkk g M : min1 6 k 6 nþp fkk g
Obviously, M > 1: It follows from (4.5) and (4.6) that n X i¼1
u2i ðtÞ þ
p X
v2j ðtÞ 6 M ð/; wÞT ðx ; Y ÞT eat
j¼1
for t P 0. This completes the proof.
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
191
5. An example In this section, we give an example to illustrate that our results are feasible. Consider the following simple BAM networks with periodic coefficients and delays, 3 X dxi ¼ ai ðtÞxi ðtÞ þ pji ðtÞfj ðyj ðt 2pÞÞ þ Ii ðtÞ; dt j¼1
ð5:1aÞ
3 X dyj ¼ bj ðtÞyj ðtÞ þ qij ðtÞfi ðxi ðt 2pÞÞ þ Jj ðtÞ; dt i¼1
ð5:1bÞ
where Ii ðtÞ ¼ sin t
ði ¼ 1; 2; 3Þ;
rij ¼ sji ¼ 2p
Jj ðtÞ ¼ cos t
ðj ¼ 1; 2; 3Þ;
ði ¼ 1; 2; 3; j ¼ 1; 2; 3Þ:
To take fi ðxÞ ¼ 12ðjx þ 1j jx 1jÞ, we have li ¼ 1 ði ¼ 1; 2; 3; 4; 5; 6Þ. Again take ða1 ðtÞ; a2 ðtÞ; a3 ðtÞÞT ¼ ð2 sin t; 2 cos t; 2 sin tÞT ; T
T
ðb1 ðtÞ; b2 ðtÞ; b3 ðtÞÞ ¼ ð2 cos t; 2 sin t; 2 cos tÞ we obtain T
T
ðb 1 ; b2 ; b3 Þ ¼ ð1; 1; 1Þ ;
T
T
and M½ai > 0; M½bj > 0:
ða 1 ; a2 ; a3 Þ ¼ ð1; 1; 1Þ ; þ þ ðbþ 1 ; b2 ; b3 Þ ¼ ð3; 3; 3Þ ;
T
T
Let 0
1 0 1 q11 ðtÞ q12 ðtÞ q13 ðtÞ 0:25 sin t 0:05 cos t 0:15 sin t @ q21 ðtÞ q22 ðtÞ q23 ðtÞ A ¼ @ 0:05 cos t 0:50 sin t 0:05 cos t A; q31 ðtÞ q32 ðtÞ q33 ðtÞ 0:25 sin t 0:05 cos t 0:25 sin t 0 1 0 1 0:05 sin t 0:10 cos t 0:15 sin t p11 ðtÞ p12 ðtÞ p13 ðtÞ @ p21 ðtÞ p22 ðtÞ p23 ðtÞ A ¼ @ 0:15 cos t 0:05 sin t 0:15 cos t A: p31 ðtÞ p32 ðtÞ p33 ðtÞ 0:05 sin t 0:15 cos t 0:05 sin t Then 0
qþ 11 @ qþ 21 qþ 31
qþ 12 qþ 22 qþ 32
1 0 qþ 0:25 13 A ¼ @ 0:05 qþ 23 qþ 0:25 33
0:05 0:50 0:05
1 0:15 0:05 A 0:25
192
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
and 0
þ p11 @ pþ 21 þ p31
þ p12 þ p22 þ p32
1 0 þ p13 0:05 þ A ¼ @ 0:15 p23 þ 0:05 p33
0:10 0:05 0:15
1 0:15 0:15 A: 0:05
Moreover, ( P3
þ j¼1 pji lj a i
r ¼ max
16i63
)
( P3 þ max
16j63
þ i¼1 qij li b j
) ¼ 0:95 < 1:
Set ki ¼ 1 ði ¼ 1; 2; 3; 4; 5; 6Þ, we get ! 3 3 X X þ pj1 lj þ k3þj qþ k1 2a1 þ 1j l1 ¼ 1:3 < 0; j¼1
k2
2a 2 þ
3 X
j¼1
! þ pj2 lj
þ
j¼1
k3
2a 3 þ
3 X
k4
2b 1 þ
! þ pj3 lj
þ
k5
þ
3 X
! qþ i1 li
þ
2b 3 þ
3 X i¼1
k3þj qþ 3j l3 ¼ 1:1 < 0;
3 X
ki p1iþ l1 ¼ 1:15 < 0;
i¼1
! qþ i2 li
þ
i¼1
k6
3 X j¼1
i¼1
2b 2
k3þj qþ 2j l2 ¼ 1:1 < 0;
j¼1
j¼1 3 X
3 X
3 X
ki p2iþ l2 ¼ 1:05 < 0;
i¼1
! qþ i3 li
þ
3 X
ki p3iþ l3 ¼ 1:3 < 0:
i¼1
Thus, it follows from Theorems 1 and 2 that the unique periodic solution (5.1a), (5.1b) is globally exponential stable.
Acknowledgements Supported by the NNSF of China (10071016)and Foundation for University Key Teacher by the Ministry of Education of China, as also supported by the Foundation of Southeast University and the Natural Science Foundations of Jiangsu Province and Yunnan Province, China, and by the Foundation of professor project of Chenzhou Teachers College.
A. Chen et al. / Appl. Math. Comput. 137 (2003) 177–193
193
References [1] A.M. Fink, Almost Periodic Differential Equation, Springer-Verlag, Berlin, Heidleberg, New York, 1974. [2] J.D. Cao, Global stability analysis in delayed cellular neural networks, Phys. Rev. E 59 (1999) 5940–5944. [3] J.D. Cao, On stability of delayed cellular neural networks, Phys. Lett. A 261 (1999) 303–308. [4] J.D. Cao, Periodic oscillatory solution of bidirectional associative memory networks with delays, Phys. Rev. E 61 (2000) 1825–1828. [5] K. Gopalsamy, X.Z. He, Delay-independent stability in bi-directional associative memory networks, IEEE Trans. Neural Networks 5 (1994) 998–1002. [6] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population, Kluwer, Dordrecht, 1992. [7] C.Y. He, Almost Periodic Differential Equation, Higher Education Publishing House, Beijing, 1992 (in Chinese). [8] C. Jin, Stability analysis of discrete-time Hopfield BAM neural networks, Acta. Auto. Sinica 25 (5) (1999) 606–612. [9] T. Kohonen, Self-Organization and Associative Memory, Springer-Verlag, New York, 1988. [10] D.G. Kelly, Stability in contractive nonlinear neural networks, IEEE Trans. Biomed. Eng. 37 (1990) 231–242. [11] B. Kosto, in: Neural Networks and Fuzzy Systems – A Dynamical System Approach to Machine Intelligence, Prentice-Hall, Englewood Cliffs, NJ, 1992, pp. 38–108. [12] B. Kosto, Adaptive bi-directional associative memories, IEEE Trans. Syst. Man Cybernet. 18 (1988) 49. [13] B. Kosto, Bi-directional associative memories, IEEE Trans. Syst. Man Cybernet. 18 (1988) 49– 60. [14] X.F. Liao, J.B. Yu, Qualitative analysis of bidrectional associative memory with time delays, Int. J. Circuit Theory Appl. 26 (1998) 219–229. [15] X.X. Liao, Stability of the Hopfield neural networks, Sci. China 23 (5) (1993) 523–532. [16] X.B. Liang, Global exponential stability and application of the Hopfield neural networks, Sci. China 25 (5) (1995) 523–532. [17] J.J. Wei, S.G. Ruan, Stability and bifurcation in a neural network model with two delays, Phys. D 130 (1999) 255–272. [18] J. Cao, L. Wang, Exponential stability and periodic oscillatory solution in BAM networks with delays, IEEE Trans. Neural Networks 13 (2) (2002). [19] J. Cao, A set of stability criteria for delayed cellular neural networks, IEEE Trans. Circuits Syst. I 48 (4) (2001) 494–498. [20] J. Cao, Global stability conditions for delayed CNNs, IEEE Trans. Circuits Syst. I 48 (11) (2001) 1330–1333. [21] J. Cao, Periodic oscillation and exponential stability of delayed CNNs, Phys. Lett. A 270 (3–4) (2000) 157–163. [22] J. Cao, Exponential stability and periodic solution of delayed cellular neural networks, Science in China (Series E) 43 (3) (2000) 328–336. [23] J. Cao, Global exponential stability of Hopfield neural networks, Int. J. Syst. Sci. 32 (2) (2001) 233–236.