exponential stability of stochastic reaction–diffusion cohen–grossberg neural networks with...
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Applied Mathematics and Computation 206 (2008) 818–824
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Applied Mathematics and Computation
journal homepage: www.elsevier .com/ locate /amc
Exponential stability of stochastic reaction–diffusion Cohen–Grossbergneural networks with delays
Li Wan a,*, Qinghua Zhou b
a Department of Mathematics and Physics, Wuhan University of Science and Engineering, Wuhan Fangzhi Road No. 1, Wuhan 430073, Chinab Department of Mathematics, Zhaoqing University, Zhaoqing 526061, China
a r t i c l e i n f o a b s t r a c t
Keywords:Cohen–Grossberg neural networksReaction–diffusionMean value exponential stabilityTime-varying delays
0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.10.002
* Corresponding author.E-mail addresses: [email protected] (L. Wa
In this paper, stochastic Cohen–Grossberg neural network with time-varying delays andreaction–diffusion terms is considered. The sufficient conditions are obtained to guaranteethe mean value exponential stability of an equilibrium solution.
� 2008 Elsevier Inc. All rights reserved.
1. Introduction
The stability analysis problem for Cohen–Grossberg neural network (CGNN) has gained much research attention, and alarge amount of results related to this problem have been published, see e.g. [2,4,5,7,8,17,21,23,28,29,33,38]. However,strictly speaking, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric elec-tromagnetic fields. Therefore we must consider that the activations vary in space as well as in time. In [9–11,15,18,19,26,27,31,35,36,39], the authors have considered the stability of neural networks with reaction–diffusion terms.
On the other hand, a real system is usually affected by external perturbations which in many cases are of great uncer-tainty and hence may be treated as random, as pointed out by Haykin [3] that in real nervous systems, the synaptic trans-mission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilisticcauses. It has also been known that a neural network could be stabilized or destabilized by certain stochastic inputs [1].Hence, the stability analysis problem for stochastic neural network becomes increasingly significant, and some results onstability have been derived, see e.g. [13,14,16,20,22,24,25,30,32,34,37].
To the best of our knowledge, however, few authors have considered the exponential stability for stochastic CGNN withdelays and diffusion terms. Motivated by the above discussions, our objective in this paper is to present the sufficient con-ditions ensuring the mean value exponential stability of stochastic CGNN.
2. Model description and assumptions
Consider the following stochastic CGNN with delays and diffusion terms
dyiðt; xÞ ¼Xm
k¼1
o
oxkDik
oyiðt; xÞoxk
� �dt � diðyiðt; xÞÞ ciðyiðt; xÞÞ �
Xn
j¼1
aijfjðyjðt; xÞÞ �Xn
j¼1
bijgjðyjðt � sijðtÞ; xÞÞ þ Ji
!dt
þXn
j¼1
rijðyiðt; xÞÞdwjðtÞ; x 2 X;oyi
on:¼ oyi
ox1; . . . ;
oyi
oxm
� �T
¼ 0; x 2 oX; ð2:1Þ
yiðs; xÞ ¼ niðs; xÞ;�s 6 s 6 0; x 2 X;
. All rights reserved.
n), [email protected] (Q. Zhou).
L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824 819
for 1 6 i 6 n and t P 0. In the above model, n P 2 is the number of neurons in the network; xi is space variable; yiðt; xÞ is thestate variable of the ith neuron at time t and in space x; fjðyjðt; xÞÞ and gjðyjðt; xÞÞ denote the activation functions of the jth unitat time t and in space x; smooth function Dik ¼ Dikðt; x; yÞP 0 is diffusion operator; diðyiðt; xÞÞ presents an amplification func-tion; ciðyiðt; xÞÞ is an appropriately behavior function; aij and bij denote the connection strengths of the jth unit on the ith unit,respectively; sijðtÞ corresponds to the transmission delay and satisfies 0 6 sijðtÞ 6 sij 6 sðsij and s are constants); Ji denotesthe external bias on the ith unit; X is a compact set with smooth boundary oX and measure mes X > 0 in Rm; niðs; xÞ is theinitial boundary value; wðtÞ ¼ ðw1ðtÞ; . . . ;wnðtÞÞT is n-dimensional Brownian motion defined on a complete probability spaceðX;F;PÞ with a natural filtration fFtgtP0 generated by fwðsÞ : 0 6 s 6 tg, where we associate X with the canonical spacegenerated by all fwiðtÞg, and denote by F the associated r-algebra generated by fwðtÞg with the probability measure P.
Let L2ðXÞ be the space of real Lebesgue measurable functions on X and be a Banach space for the L2-norm
kuðtÞk22 ¼
ZX
u2ðt; xÞdx:
Note that, n ¼ fðn1ðs; xÞ; . . . ; nnðs; xÞÞT : �s 6 s 6 0g is Cð½�s;0� � Rm; RnÞ-valued function and F0-measurable Rn-valued ran-dom variable, where F0 ¼Fs on ½�s;0�, Cð½�s;0� � Rm; RnÞ is the space of all continuous Rn-valued functions defined on½�s;0� � Rm with a norm kniðtÞk2
2 ¼R
X n2i ðt; xÞdx.
Throughout this paper, for system (2.1), we have the following assumptions:
(A1) fj; gj and rij are Lipschitz continuous with Lipschitz constant aj > 0; bj > 0 and Lij > 0, respectively, andjfjj 6 Mf
j ; jgjj 6 Mgj ; for i; j ¼ 1;2; . . . ;n.
(A2) ci 2 C1ðR;RÞ and ci ¼ infu2Rc0iðuÞ > 0; for i ¼ 1;2; . . . ;n.(A3) di is bounded, positive and continuous, i.e. there exist constants di; di such that 0 < di 6 diðuÞ 6 di; for
u 2 R; i ¼ 1;2; . . . ;n.
When rij � 0; i; j ¼ 1;2; . . . ;n; system (2.1) becomes the deterministic system
dyiðtÞ ¼Xm
k¼1
o
oxkDik
oyi
oxk
� �dt � diðyiðtÞÞ � fciðyiðtÞÞ �
Xn
j¼1
aijfjðyjðt; xÞÞ �Xn
j¼1
bijgjðyjðt � sijðtÞ; xÞÞ þ Jigdt; x 2 X: ð2:2Þ
Furthermore when Dik � 0; i;¼ 1;2; . . . ;n, k ¼ 1;2; . . . ;m, system (2.2) becomes
dyiðtÞ ¼ �diðyiðtÞÞ � ciðyiðtÞÞ �Xn
j¼1
aijfjðyjðt; xÞÞ �Xn
j¼1
bijgjðyjðt � sijðtÞ; xÞÞ þ Ji
( )dt; x 2 X: ð2:3Þ
From [28], we know that under the assumptions (A1)–(A3), system (2.3) has an equilibrium solution y� ¼ ðy�1; . . . ; y�nÞT: It is
easy to see that y� ¼ ðy�1; . . . ; y�nÞT is also the equilibrium solution of system (2.2).
Now we assume that
(A4) rijðy�i Þ ¼ 0; for i; j ¼ 1;2; . . . ;n.
Then system (2.1) admits an equilibrium solution y� ¼ ðy�1; . . . ; y�nÞT and can be described by
dðyiðtÞ � y�i Þ ¼Xm
k¼1
o
oxkDik
oðyiðtÞ � y�i Þoxk
� �dt þ diðyiðtÞÞf�½ciðyiðtÞ � ciðy�i Þ� þ
Xn
j¼1
aij½fjðyjðt; xÞÞ � fjðy�j Þ�
þXn
j¼1
bij½gjðyjðt � sijðtÞ; xÞÞ � gjðy�j Þ�gdt þXn
j¼1
rijðyiðt; xÞÞdwjðtÞ; x 2 X;
since �Ji ¼ ciðy�i Þ �Pn
j¼1aijfjðy�j Þ �Pn
j¼1bijgjðy�j Þ.It follows from [12] that under the conditions (A1)–(A4), system (2.1) has the global solution on t P 0, which is denoted
by yðt; nÞ; or, yðtÞ, if no confusion occurs.
Definition 2.1. The equilibrium solution y� of system (2.1) is said to be pth moment exponentially stable if there exists a pairof positive constants k and C such that for any n
Ejyðt; nÞ � y�jp 6 CEjn� y�jpe�kt; t P 0;
where jyðtÞj2 ¼Pn
i¼1kyiðtÞk22: When p ¼ 1 it is usually called the mean value exponential stability.
For the convenience, we introduce some notations. For n� n matrix A, A�1 denotes the inverse of A, A P 0 means A is anonnegative matrix.
Lemma 2.2 [6]. If M P 0 and qðMÞ < 1, then ðI �MÞ�1 P 0; where I denotes the identity matrix and qðMÞ denotes the spectralradius of a square matrix M.
820 L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824
3. Mean value exponential stability
For system (2.1), we have the following result.
Theorem 3.1. Suppose that system (2.1) satisfies the assumptions ðA1Þ—ðA4Þ and
(A5) C > 0; qðC�1ðD1Aþaþ D1BþbÞ < 1;
whereC ¼ diag ðd1; d2; . . . ; dnÞ; di ¼ dici � 1
2
Pnj¼1L2
ij; i ¼ 1;2; . . . ;n,D1 ¼ diag ðd1; d2; . . . ; dnÞ;Aþ ¼ ðjaijjÞn�n;Bþ ¼ ðjbijjÞn�n,a ¼ diag ða1;a2;
. . . ;anÞ; b ¼ diag ðb1; b2; . . . ; bnÞ.
Then the equilibrium solution y� of system (2.1) is mean value exponentially stable.
Proof. Let ziðt; xÞ ¼ yiðt; xÞ � y�i ; i ¼ 1;2; . . . ;n: Applying Ito formula to z2i ðt; xÞ and integrating with respect to x, we have
dkziðtÞk22 ¼ 2
ZX
ziðtÞXm
k¼1
o
oxkDik
ozi
oxk
� �dxdt þ 2
ZX
ziðtÞdiðyiðtÞÞf�½ciðyiðtÞ � ciðy�i Þ� þXn
j¼1
aij½fjðyjðt; xÞÞ � fjðy�j Þ�
þXn
j¼1
bij½gjðyjðt � sijðtÞ; xÞÞ � gjðy�j Þ�gdxdt þZ
X
Xn
j¼1
r2ijðyiðt; xÞÞdxdt þ 2
ZX
Xn
j¼1
ziðtÞrijðyiðt; xÞÞdxdwjðtÞ:
Applying Ito formula to ðkziðtÞk22Þ
1=2, we have
dkziðtÞk2 ¼ �dikziðtÞk2 þ dikziðtÞk2 þ1
kziðtÞk2
ZX
zi
Xm
k¼1
o
oxkDik
oziðtÞoxk
� �dxdt þ 1
kziðtÞk2
ZX
ziðtÞdiðyiðtÞÞf�½ciðyiðtÞ � ciðy�i Þ�
þXn
j¼1
aij½fjðyjðt; xÞÞ � fjðy�j Þ� þXn
j¼1
bij½gjðyjðt � sijðtÞ; xÞÞ � gjðy�j Þ�gdxdt þ 12kziðtÞk2
ZX
Xn
j¼1
r2ijðyiðt; xÞÞdxdt
þ 1kziðtÞk2
ZX
Xn
j¼1
ziðtÞrijðyiðt; xÞÞdxdwjðtÞ �1
2kziðtÞk32
Xn
j¼1
ZX
ziðtÞrijðyiðt; xÞÞdx� �2
dt:
Notice that, it follows from the boundary condition that
Xm
k¼1
ZX
zio
oxkDik
ozi
oxk
� �dx ¼
ZX
zir � Dikozi
oxk
� �m
k¼1dx ¼
ZXr � ziDik
ozi
oxk
� �m
k¼1dx�
ZX
Dikozi
oxk
� �m
k¼1� rzidx
¼Xm
k¼1
ZoX
ziDikozi
oxk
� �m
k¼1� ds�
Xm
k¼1
ZX
Dikozi
oxk
� �2
dx ¼ �Xm
k¼1
ZX
Dikozi
oxk
� �2
dx;
where ‘‘�” is inner product, r ¼ ð oox1; . . . ; o
oxmÞ is the gradient operator, and
Dikozi
oxk
� �m
k¼1¼ Di1
ozi
ox1
� �; . . . ; Dim
ozi
oxm
� �� �T
:
Then, by using the method of variation parameter, Hölder inequality and (A1)–(A4), we obtain
kziðtÞk2 ¼ e�di tkzið0Þk2 þZ t
0e�diðt�sÞ dikziðsÞk2 �
1kziðsÞk2
Xm
k¼1
ZX
DikoziðsÞoxk
� �2
dx
(
þ 1kziðsÞk2
ZX
ziðtÞdiðyiðsÞÞf�½ciðyiðsÞ � ciðy�i Þ� þXn
j¼1
aij½fjðyjðs; xÞÞ � fjðy�j Þ�
þXn
j¼1
bij½gjðyjðs� sijðsÞ; xÞÞ � gjðy�j Þ�gdxþ 12kziðsÞk2
ZX
Xn
j¼1
r2ijðyiðs; xÞÞdx
)ds
þ NiðtÞ �Z t
0e�diðt�sÞ 1
2kziðsÞk32
Xn
j¼1
ZX
ziðsÞrijðyiðs; xÞÞdx� �2
ds 6 e�di tkzið0Þk2
þZ t
0e�diðt�sÞ dikziðsÞk2 þ
1kziðsÞk2
f�dicikziðsÞk22 þ di
Xn
j¼1
jaijjajkziðsÞk2kzjðsÞk2
(
L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824 821
þ di
Xn
j¼1
jbijjbjkziðsÞk2kzjðs� sijðsÞÞk2 þ12
Xn
j¼1
L2ijkziðsÞk2
2g)
dsþ NiðtÞ ¼ e�di tkzið0Þk2
þ di
Z t
0e�diðt�sÞ
Xn
j¼1
jaijjajkzjðsÞk2 þXn
j¼1
jbijjbjkzjðs� sijðsÞÞk2
( )dsþ NiðtÞ;
where
NiðtÞ ¼Z t
0e�diðt�sÞ 1
kziðsÞk2
ZX
Xn
j¼1
ziðsÞrijðyjðs; xÞÞdxdwjðsÞ:
Therefore, for a sufficiently small positive constant k < minfdi : 1 6 i 6 ng; we have
EkziðtÞk2ekt6 eðk�diÞtEkzið0Þk2 þ
Z t
0eðk�diÞðt�sÞdi
Xn
j¼1
jaijjaj½EkzjðsÞk2eks� þXn
j¼1
jbijjeksijðsÞbj½Ekzjðs� sijðsÞÞk2ekðs�sijðsÞÞ�( )
ds
6 Ekzið0Þk2 þZ t
0eðk�diÞðt�sÞdsdi
Xn
j¼1
jaijjaj þXn
j¼1
jbijjbjeks
( )GjðtÞ
6 Ekzið0Þk2 þ1
di � kdi
Xn
j¼1
jaijjaj þXn
j¼1
jbijjbjeks
( )GjðtÞ;
which yields
GiðtÞ 6 Ekzið0Þk2 þ1
di � kdi
Xn
j¼1
jaijjaj þXn
j¼1
jbijjbjeks
( )GjðtÞ;
that is,
GðtÞ 6 Ekzð0Þk þ ðC � kIÞ�1ðD1Aþaþ D1BþbeksÞGðtÞ;
where
GjðtÞ ¼ sup�s6h6t
EkzjðhÞk2ekh; j ¼ 1;2; . . . ; n;
GðtÞ ¼ðG1ðtÞ;G2ðtÞ; . . . ;GnðtÞÞT ;Ekzð0Þk ¼ðEkz1ð0Þk2;Ekz2ð0Þk2; . . . ;Ekznð0Þk2Þ
T:
Since
qðC�1ðD1Aþaþ D1BþbÞÞ < 1; C�1ðD1Aþaþ D1BþbÞP 0;
from Lemma 2.2, it deduces
ðI � C�1ðD1Aþaþ D1BþbÞÞ�1 P 0:
Then there exists a sufficiently small positive constant c < k such that
MðcÞ :¼ ðI � ðC � cIÞ�1ðD1Aþaþ D1BþbecsÞÞ�1 P 0:
Therefore, one can derive that
EkzðtÞk 6 MðcÞEkzð0Þke�kt6 MðcÞEkzð0Þke�ct ;
in which
EkzðtÞk ¼ ðEkz1ðtÞk2;Ekz2ðtÞk2; . . . ;EkznðtÞk2ÞT:
From
jzðtÞj2 ¼Xn
i¼1
kziðtÞk22 6
Xn
i¼1
kziðtÞk2
!2
;
it follows
EjzðtÞj 6Xn
i¼1
EkziðtÞk2 6 e�ctXn
i¼1
Xn
j¼1
MijðcÞEkzjð0Þk2:
The proof is completed. h
822 L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824
Notice that qðAÞ 6 kAk for any A 2 Rn�n, in which k�k is an arbitrary matrix norm. Moreover, for any matrix norm and anynonsingular matrix S, a matrix norm kAkS can be given by kAkS ¼ kS
�1ASk. For the convenience of calculation, in general, tak-ing S ¼ diag fs1; . . . ; sng > 0: Therefore, corresponding to the matrix norm widely applied— the row norm, column norm andFrobenius norm, we can obtain the following sufficient conditions to guarantee kAkS < 1, respectively.
(1)Pn
j¼1ðsisjjaijjÞ < 1, for 1 6 i 6 n;
(2)Pn
i¼1ðsisjjaijjÞ < 1, for 1 6 j 6 n;
(3)Pn
i¼1
Pnj¼1ð
sisjjaijjÞ2 < 1.
From the above statement, we have
Corollary 3.2. The equilibrium solution y�of system (2.1) is mean value exponentially stable provided that one of the followingconditions holds: there exist positive real numbers s1; . . . ; sn such that
(1)
Xn
j¼1
si
sjdi
Pnj¼1jaijjaj þ
Pnj¼1jbijjbj
di
0BBB@
1CCCA
26664
37775 < 1; for 1 6 i 6 n;
(2)
Xn
i¼1
si
sjdi
Pnj¼1jaijjaj þ
Pnj¼1jbijjbj
di
0BBB@
1CCCA
26664
37775 < 1; for 1 6 j 6 n;
(3)
Xn
i¼1
Xn
j¼1
si
sjdi
Pnj¼1jaijjaj þ
Pnj¼1jbijjbj
di
0BBB@
1CCCA
26664
37775
2
< 1:
From Theorem 3.1, we obtain the following results discussed in [28].
Corollary 3.3. Suppose that system (2.2) (or (2.3)) satisfies the assumptions (A1)–(A3). Then the equilibrium solution y� of system(2.2) (or (2.3)) is unique and globally exponentially stable provided that the following condition holds:
(A6) qðC�11 ðD1Aþaþ D1BþbÞ < 1, where C1 ¼ diag ðd1c1; d2c2; . . . ; dncnÞ.
4. One example
The following illustrative example will demonstrate the effectiveness of our results.
Example 1. Consider the following stochastic Cohen–Grossberg neural networks
dy1ðtÞy2ðtÞ
� �¼
2þ sin y1 00 2þ cos y2
� �� �
3 00 3
� �y1ðtÞy2ðtÞ
� �dt
�
þ0:2 0:10:3 0:1
� �sinðy1ðtÞÞcosðy2ðtÞÞ
� �dt þ
�0:3 0:20:1 0:3
� �
�tanhðy1ðt � s1ðtÞÞÞtanhðy2ðt � s2ðtÞÞÞ
� �dt�þ
L11y1ðtÞ L12y1ðtÞL21y2ðtÞ L22y2ðtÞ
� �dwðtÞ
þD11
oy1ox1
D12oy1ox2
D21oy2ox1
D22oy2ox2
!o
ox1
oox2
!dt
oyi
on:¼ oyi
ox1; � � � ; oyi
oxm
� �T
¼ 0; t P 0; x 2 oX;
yiðs; xÞ ¼ niðs; xÞ;�sij 6 s 6 0; x 2 X;
L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824 823
where tanhðxÞ ¼ ex�e�x
exþe�x ;Dik ¼ xk or constant, i; k ¼ 1;2. It is clear that dj ¼ 3; dj ¼ 1; cj ¼ 3;aj ¼ bj ¼ 1; j ¼ 1;2, and
C ¼3�
P2j¼1
L21j 0
0 3�P2j¼1
L22j
0BBBB@
1CCCCA; D1 ¼
3 00 3
� �;
Aþ ¼0:2 0:10:3 0:1
� �; Bþ ¼
0:3 0:20:1 0:3
� �;
Therefore,
C�1ðD1Aþaþ D1BþbÞ ¼
3
3�P2
j¼1
L21j
0
0 3
3�P2
j¼1
L22j
0BBBB@
1CCCCA
0:5 0:30:4 0:4
� �:
Taking
L ¼0 0ffiffiffiffiffiffiffi0:4p ffiffiffiffiffiffiffi
0:2p
� �;
we have
C�1ðD1Aþaþ D1BþbÞ ¼0:5 0:30:5 0:5
� �;
qðC�1ðD1Aþaþ D1BþbÞ ¼ 0:5þffiffiffiffiffiffiffiffiffiffi0:15p
< 1:
It follows from Theorem 3.1 that the equilibrium solution of such system is mean value exponentially stable.
5. Conclusions
The results in this paper show that, the sufficient criteria on the mean value exponential stability of stochastic CGNN withdelays and diffusion terms are independent of the magnitude of the delays and diffusion effect, but are dependent of themagnitude of noise, and therefore, in the above content, diffusion and delays are harmless, but noisy fluctuations areimportant.
Acknowledgements
The authors would like to thank the referee and the editor for their helpful comments and suggestions. This research waspartly supported by the National Natural Science Foundation of China (10801109) and the Natural Science Foundation ofWuhan University of Science and Engineering (2008Z25).
References
[1] S. Blythe, X. Mao, X. Liao, Stability of stochastic delay neural networks, J. Franklin Inst. 338 (2001) 481–495.[2] Z. Chen, J. Ruan, Global dynamic analysis of general Cohen- Grossberg neural networks with impulse, Chaos Soliton. Fract. 32 (2007) 1830–1837.[3] S. Haykin, Neural Networks, Prentice-Hall, NJ, 1994.[4] C. Hwang, C. Cheng, T. Liao, Globally exponential stability of generalized Cohen–Grossberg neural networks with delays, Phys. Lett. A 319 (2003) 157–
166.[5] T. Huang, A. Chan, Y. Huang, J. Cao, Stability of Cohen–Grossberg neural networks with time-varying delays, Neural Net. 20 (2007) 868–873.[6] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, London, 1985.[7] T. Huang, C. Li, G. Chen, Stability of Cohen–Grossberg neural networks with unbounded distributed delays, Chaos Soliton. Fract. 34 (2007) 992–996.[8] M. Jiang, Y. Shen, X. Liao, Boundedness and global exponential stability for generalized Cohen–Grossberg neural networks with variable delay, Appl.
Math. Comput. 172 (2006) 379–393.[9] L. Liang, J. Cao, Global exponential stability of reaction–diffusion recurrent neural networks with time-varying delays, Phys. Lett. A 314 (2003) 434–
442.[10] X. Lou, B. Cui, Boundedness and exponential stability for nonautonomous cellular neural networks with reaction–diffusion terms, Chaos Soliton. Fract.
33 (2007) 653–662.[11] K. Li, Z. Li, X. Zhang, Exponential stability of reaction–diffusion generalized Cohen–Grossberg neural networks with both variable and distributed
delays, Int. Math. Forum 2 (2007) 1397–1414.[12] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, 1997.[13] R. Rakkiyappan, P. Balasubramaniam, Delay-dependent asymptotic stability for stochastic delayed recurrent neural networks with time varying delays,
Appl. Math. Comput. 198 (2008) 526–533.[14] Q. Song, Z. Wang, Stability analysis of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays, Phys. A 387 (13) (2008) 3314–
3326.
824 L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824
[15] Q. Song, J. Zhang, Global exponential stability of impulsive Cohen-Grossberg neural network with time-varying delays, Nonlinear Anal-Real. 9 (2)(2008) 500–510.
[16] Y. Sun, J. Cao, pth moment exponential stability of stochastic recurrent neural networks with time-varying delays, Nonlinear Anal-Real. 8 (2007) 1171–1185.
[17] Q. Song, J. Cao, Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays, J. Comput. Appl.Math. 197 (2006) 188–203.
[18] Q. Song, J. Cao, Global exponential stability and existence of periodic solutions in BAM networks with delays and reaction–diffusion terms, ChaosSoliton. Fract. 23 (2005) 421–430.
[19] Q. Song, J. Cao, Z. Zhao, Periodic solutions and its exponential stability of reaction–diffusion recurrent neural networks with continuously distributeddelays, Nonlinear Anal-Real. 7 (2006) 65–80.
[20] J. Sun, L. Wan, Convergence dynamics of stochastic reaction–diffusion recurrent neural networks with delays, Inter. J. Bifurcat. Chaos 15 (7) (2005)2131–2144.
[21] J. Sun, L. Wan, Global exponential stability and periodic solutions of Cohen-Grossberg neural networks with continuously distributed delays, Phys. D208 (2005) 1–20.
[22] Z. Wang, J. Fang, X. Liu, Global stability of stochastic high-order neural networks with discrete and distributed delays, Chaos Soliton. Fract. 36 (2008)388–396.
[23] A. Wan, H. Qiao, J. Peng, M. Wang, Delay-independent criteria for exponential stability of generalized Cohen–Grossberg neural networks with discretedelays, Phys. Lett. A 353 (2006) 151–157.
[24] L. Wan, J. Sun, Mean square exponential stability of stochastic delayed Hopfield neural networks, Phys. Lett. A 343 (4) (2005) 306–318.[25] L. Wan, J. Sun, Global asymptotic stability of Cohen–Grossberg neural network with continuously distributed delays, Phys. Lett. A 343 (4) (2005) 331–
340.[26] L. Wang, D. Xu, Global exponential stability of Hopfield reaction–diffusion neural networks with time-varying delays, Science in China (Series F) 46
(2003) 466–474.[27] L. Wang, D. Xu, Asymptotic behavior of a class of reaction–diffusion equations with delays, J. Math. Anal. Appl. 28 (2003) 439–453.[28] L. Wang, X. Zou, Exponential stability of Cohen–Grossberg neural networks, Neural Net. 15 (2002) 415–422.[29] L. Wang, X. Zou, Harmless delays in Cohen–Grossberg neural networks, Phys. D 170 (2002) 162–173.[30] L. Wan, Q. Zhou, Convergence analysis of stochastic hybrid bidirectional associative memory neural networks with delays, Phys. Lett. A 370 (2007)
423–432.[31] L. Wan, Q. Zhou, Global exponential stability of BAM neural networks with time-varying delays and diffusion terms, Phys. Lett. A 371 (2007) 83–89.[32] L. Wan, Q. Zhou, J. Sun, Mean value exponential stability of stochastic reaction–diffusion generalized Cohen–Grossberg neural networks with time-
varying delay, Inter. J. Bifurcat. Chaos 17 (9) (2007) 3219–3227.[33] H. Ye, A. Michel, K. Wang, Qualitative analysis of Cohen–Grossberg neural networks with multiple delays, Phys. Rev. E 51 (1995) 2611–2618.[34] H. Zhao, N. Ding, Dynamic analysis of stochastic bidirectional associative memory neural networks with delays, Chaos Soliton. Fract. 32 (2007) 1692–
1702.[35] Z. Zhao, Q. Song, J. Zhang, Exponential periodicity and stability of neural networks with reaction–diffusion terms and both variable and unbounded
delays, Comput. Math. Appl. 51 (2006) 475–486.[36] H. Zhao, K. Wang, Dynamical behaviors of Cohen–Grossberg neural networks with delays and reaction–diffusion terms, Neurocom. 70 (2006) 536–543.[37] Q. Zhou, L. Wan, Exponential stability of stochastic delayed Hopfield neural networks, Appl. Math. Comput. 199 (2008) 84–89.[38] Q. Zhou, L. Wan, J. Sun, Exponential stability of reaction–diffusion generalized Cohen–Grossberg neural networks with time-varying delays, Chaos
Soliton. Fract. 32 (2007) 1713–1719.[39] Q. Zhou, L. Wan, J. Sun, Exponential stability of reaction–diffusion fuzzy recurrent neural networks with time-varying delays, Inter. J. Bifurcat. Chaos 17
(9) (2007) 3099–3108.