# Periodic Oscillation of Fuzzy  Cohen-Grossberg Neural  Networks with Distributed Delay and Variable Coefficients

Authors: Hongjun Xiang; Jinde Cao
Journal: Journal of Applied Mathematics
 (2008)

---

## Abstract

A class of fuzzy Cohen-Grossberg neural networks with distributed delay and variable coefficients is discussed.
It is neither employing coincidence degree theory nor constructing Lyapunov functionals, instead, by applying matrix theory and inequality analysis, some sufficient conditions are obtained to ensure the existence, uniqueness, global attractivity and global exponential stability of the periodic solution for the fuzzy Cohen-Grossberg neural networks. The method is very concise and practical. Moreover, two examples are posed to illustrate the effectiveness of our results.


---

## Body

## 1. Introduction

Recently,
Cohen-Grossberg neural networks (CGNNs) and fuzzy cellular neural networks
(FCNNs) with their various models have attracted many scholars' attention due
to their potential applications in classification, associative memory, parallel
computation, image processing, and pattern recognition, especially in white
blood cell detection and in the solution of some optimization problems, presented
by [1–22]. It is well known that FCCNs [16–18] have been proposed by  T. Yang
and  L. Yang in 1996. Unlike the traditional CNNs structures, FCNNs have fuzzy
logic between their template input and output besides the sum of product
operation. It is worth noting that studies have  indicated FCNNs' potential
applications in many fields such as image and signal processing, pattern
recognition, white blood cell detection, and so on. Some results on stability
have been derived from the FCNNs models without or with time delays; see [16–21].
In [22], the authors obtained a new detection algorithm based on FCNNs for
white blood cell detection, with which one can detect almost all white blood
cells and the contour of each detected cell is nearly completed.It is also noted that CGNNs are presented by Cohen and
Grossberg in 1983 [1]. This model is a very common class of neural networks,
which includes lots of famous neural networks such as Lotka-Volterra ecological
system, Hopfield neural networks, and cellular neural networks, and so on. Up to
now, some useful results were given to ensure the existence and stability of
the equilibrium point of CGNNs without or with time delays, presented by
[1–11, 13, 14, 15]. In [4], the boundedness and stability were analyzed for a class of
CGNNs with time-varying delays using the inequalities technique and Lyapunov
method. In [2, 8], by constructing suitable Lyaponuv functionals and using the
linear matrix inequality (LMI) technique, the authors investigated a class of
delayed CGNNs and obtained several criteria to guarantee the global asymptotic
stability of the equilibrium point for this system. In [3, 6, 7, 9, 13, 15], the
global exponential stability is discussed for a class of delayed CGNNs via
nonsmooth analysis. In [14], the authors integrated fuzzy logic into the
structure of CGNNs, maintained local connectedness among cells which are called
fuzzy Cohen-Grossberg neural networks (FCGNNs), and studied impulsive effects
on stability of FCGNNs with time-varying delays.Due to the presence of an amount of parallel pathways
of a variety of axon sizes and lengths, neural networks usually have spatial
nature so it is desired to model them by introducing continuously distributed
delays over a certain duration of time so that the distant past has less
influence compared with the recent behavior of the state [28]. Hence, some
authors have researched neural networks with distributed delays; see, for
example, [5, 20]. In [20], stability was considered for a class of FCNNs with
distributed delay. The condition for feedback kernel is ∫
0
τ
k
(
s
)
d
s
=
1. In [5], following the idea of vector Lyapunov
function, M-matrix theory, and inequality technique, authors studied
Cohen-Grossberg neural network model with both time-varying and continuously
distributed delays and obtained several sufficient conditions to ensure the
existence, uniqueness, and global exponential stability of equilibrium point
for this system. The delay kernel K
i
j
:
[
0
,
+
∞
)
→
[
0
,
+
∞
) is real-valued
nonnegative continuous function and satisfies ∫
0
+
∞
e
β
s
K
i
j
(
s
)
d
s
=
p
i
j
(
β
), where p
i
j
(
β
) is continuous
function in [
0
,
δ
)
,
δ
>
0 and p
i
j
(
0
)
=
1
,
i
,
j
=
1
,
2
,
…
,
n.Moreover, studies on neural dynamical systems do not
only involve a discussion of stability properties, but also many other dynamic
behaviors such as periodic oscillatory; see [21, 23, 24] and references therein.
Although many results were derived from testing the stability of CGNNs [1–14], to
the best of our knowledge, FCGNNs with distributed delay and variable
coefficients are seldom considered. Motivated by the above discussions, the
objective of this paper is to study the periodic oscillatory solutions of fuzzy
Cohen-Grossberg neural networks with distributed delay and variable coefficients
and to obtain several novel sufficient conditions to ensure the existence,
uniqueness, global attractivity, and global exponential stability of periodic
solutions for the model with periodic external inputs. In this paper, we do not
use coincidence degree theory nor apply Lyapunov method but use inequality
analysis and matrix theory to discuss them.The rest of the paper is organized as follows. In
Section2, we will describe the model and introduce some necessary notations,
definitions, and preliminaries which will be used later. In Section 3, several
sufficient conditions are derived from the existence, uniqueness, global
attractivity, and global exponential stability of periodic solutions. In
Section
4, two examples are given to show the effectiveness of the obtained
results. Finally, we make the conclusion in Section 5.


## 2. Model Description and Preliminaries

In this paper,
we will consider the following fuzzy Cohen-Grossberg neural networks with
periodic coefficients and distributed delays:(2.1)
x
˙
i
(
t
)
=
−
a
i
(
x
i
(
t
)
)
[
b
i
(
x
i
(
t
)
)
−
∑
j
=
1
n
c
i
j
(
t
)
f
j
(
x
j
(
t
)
)
−
∑
j
=
1
n
d
i
j
(
t
)
u
j
(
t
)
−
⋀
j
=
1
n
α
i
j
(
t
)






×
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s
−
⋁
j
=
1
n
β
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s






−
⋀
j
=
1
n
G
i
j
(
t
)
u
j
(
t
)
−
⋁
j
=
1
n
H
i
j
(
t
)
u
j
(
t
)
−
I
i
(
t
)
]
, where i
=
1
,
2
,
…
,
n,  n
≥
2 is the number
of neurons in the networks, x
i
(
t
) denotes the
neuron state vector; a
i
(
⋅
) is an
amplification function, b
i
(
⋅
) denotes an
appropriately behaved function, f
i is the activation
function of the neurons, and α
i
j
(
t
)
,
β
i
j
(
t
)
,
G
i
j
(
t
)
,
H
i
j
(
t
) are elements of
fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed-forward
MIN template, and fuzzy feed-forward MAX template at the time t, respectively. C
=
(
c
i
j
(
t
)
)
n
×
n and D
=
(
d
i
j
(
t
)
)
n
×
n, c
i
j
(
t
)
,
d
i
j
(
t
) are elements of
feedback template and feed-forward template at the time t, respectively; α
=
(
α
i
j
(
t
)
)
n
×
n and β
=
(
β
i
j
(
t
)
)
n
×
n, α
i
j
(
t
)
,
β
i
j
(
t
) are elements of
fuzzy feedback MIN template and fuzzy feedback MAX template, respectively; G
=
(
G
i
j
(
t
)
)
n
×
n and H
=
(
H
i
j
(
t
)
)
n
×
n, G
i
j
(
t
)
,
H
i
j
(
t
) are elements of
fuzzy feed-forward MIN template and fuzzy feed-forward MAX template,
respectively; ⋀ and ⋁  denote the
fuzzy AND and fuzzy OR operations, respectively; u
i
(
t
)
,
I
i
(
t
) denote input
and bias of the ith neurons at
the time t, respectively; k
i
(
s
)
≥
0 is the feedback
kernel, defined on the interval [
0
,
τ
] when τ is a positive
finite number or [
0
,
+
∞
) while τ is infinite.Letℂ
=
C
(
[
−
τ
,
0
]
;
R
n
) denote the
Banach space of continuous mapping from [
−
τ
,
0
] to R
n with the
topology of uniform convergence. For a given s
∈
[
0
,
∞
) and a
continuous function x
:
[
−
τ
,
∞
)
→
R
n, we define x
s
:
[
−
τ
,0
]
→
R
n by x
s
(
θ
)
=
x
(
s
+
θ
) for θ
∈
[
−
τ
,0
]. The initial conditions of (2.1) are of the form x
i
(
t
)
=
ϕ
i
(
t
)
,
−
τ
≤
t
≤
0, where ϕ
i is bounded and
continuous on [
−
τ
,
0
]. For matrix P
=
(
p
i
j
(
t
)
)
n
×
n, let ρ
(
P
) denote the
spectral radius of P. A matrix or a vector P
≥
0 means that all
the elements of P are greater
than or equal to zero, similarly define P
>
0. For x
(
t
)
=
(
x
1
(
t
)
,
x
2
(
t
)
,
…
,
x
n
(
t
)
)
T define |
x
(
t
)
|
=
∑
i
=
1
n
|
x
i
(
t
)
|, ∥
x
(
t
)
∥
=
max
1
≤
i
≤
n
{
∥
x
i
(
t
)
∥
}, [
x
(
t
)
]
+
=
(
∥
x
1
(
t
)
∥
,
∥
x
2
(
t
)
∥
,
…
,
∥
x
n
(
t
)
∥
)
T
, where ∥
x
i
(
t
)
∥
=
max
−
τ
≤
s
≤
0
{
|
x
i
(
t
+
s
)
|
}
,
i
=
1
,
2
,
…
,
n
.Remark 2.1.
ℂ
=
C
(
[
−
τ
,
0
]
;
R
n
) when
τ is a positive
finite number or ℂ
=
C
(
(
−
∞
,
0
]
;
R
n
) while τ is infinite.
To obtain our results, we give the following
assumptions.(A1)
The functionsa
i
(
⋅
)
(
i
=
1
,
2
,
…
,
n
) are
continuously bounded, that is, there exist positive constants a
i
− and a
i
+ so that(2.2)
0
<
a
i
−
≤
a
i
(
x
)
≤
a
i
+
,

∀
x
∈
R
,
i
=
1
,
2
,
…
,
n
.
(A2)
Eachb
i
(
⋅
) with b
i
(
0
)
=
0 is continuous
and there exists a constant δ
i so that(2.3)
b
i
(
x
)
−
b
i
(
y
)
x
−
y
≥
δ
i
>
0
,

∀
x
,
y
∈
R
,
x
≠
y
,
i
=
1
,
2
,
…
,
n
.
(A3)
For activation
functionf
i
(
x
) with f
i
(
0
)
=
0, there exist l
i
>
0 so that(2.4)
l
i
=
sup
⁡
x
≠
y
|
f
i
(
x
)
−
f
i
(
y
)
x
−
y
|
,

∀
x
,
y
∈
R
,
x
≠
y
,
i
=
1
,
2
,
…
,
n
.
(A4)
The feedback
kernels satisfyk
j
(
s
)
≥
0 and ∫
0
τ
k
j
(
s
)
d
s
=
1
,
j
=
1
,
2
,
…
,
n
. Where τ is a positive
finite number or infinite.In addition, we
will use the following notations throughout this paper:(2.5)
c
i
j
=
max
⁡
t
≥
t
0
|
c
i
j
(
t
)
|
,


α
i
j
=
max
⁡
t
≥
t
0
|
α
i
j
(
t
)
|
,


β
i
j
=
max
⁡
t
≥
t
0
|
β
i
j
(
t
)
|
,
I
i
(
t
)
=
∑
j
=
1
n
d
i
j
(
t
)
u
j
(
t
)
+
⋀
j
=
1
n
G
i
j
(
t
)
u
j
(
t
)
+
⋁
j
=
1
n
H
i
j
(
t
)
u
j
(
t
)
+
I
i
(
t
)
,


I
i
=
max
⁡
t
≥
t
0
{
|
I
i
(
t
)
|
}
.Definition 2.2 (see [25]).
LetR
+
=
[
0
,
∞
). Suppose that ℂ is a Banach
space and that ϱ
:
R
×
ℂ
×
R
+
→
ℂ is a given
mapping. Define Θ
(
u
,
t
)
:
ℂ
→
ℂ
,
∀
u
∈
R
,
t
∈
R
+ by Θ
(
u
,
t
)
x
=
ϱ
(
u
,
x
,
t
). A process on ℂ is a mapping ϱ
:
R
×
ℂ
×
R
+
→
ℂ satisfying the
following properties: (i)   ϱ is continuous; (ii)            Θ
(
u
,0
)
=
E is the
identity; (iii)        Θ
(
u
+
s
,
t
)
Θ
(
u
,
s
)
=
Θ
(
u
,
s
+
t
)
.A processϱ is said to be
an ω-periodic
process if there is an ω
>
0 such that Θ
(
u
+
ω
,
t
)
=
Θ
(
u
,
t
)
,
for
all
u
∈
R and t
∈
R
+.Remark 2.3.
Suppose thatF
:
R
×
ℂ
→
R
n is completely
continuous and let x
(
t
0
,
φ
) denote a
solution of the system
(*)
x
˙
(
t
)
=
F
(
t
,
x
t
) through (
t
0
,
φ
) and assume x is uniquely
defined for t
≥
t
0
−
τ. If ϱ
(
t
0
,
φ
,
t
)
=
x
t
0
+
t
(
t
0
,
φ
) for (
t
0
,
φ
,
t
)
∈
R
×
ℂ
×
R
+, then ϱ is a process on ℂ. If there is an ω
>
0 such that F
(
t
0
+
ω
,
φ
)
=
F
(
t
0
,
φ
) for all (
t
0
,
φ
)
∈
R
×
ℂ, then the process generated by F
(
t
,
x
t
) is an ω-periodic
process. Furthermore, if Θ is defined as
before, then Θ
(
t
0
,
s
)
=
T
(
t
0
+
s
,
t
0
), where T
(
t
,
t
0
) is the solution
operator of x
˙
(
t
)
=
F
(
t
,
x
t
) satisfying T
(
t
,
t
0
)
φ
=
x
t
(
t
0
,
φ
).Definition 2.4 (see [2.44]).
A continuous mapϒ
:
ℂ
→
ℂ is said to be
point dissipative if there exists a bounded set ℂ
0
⊂
ℂ such that ℂ
0 attracts each
point of ℂ.Lemma 2.5 (see [25]).
If anω-periodic F
(
t
,
x
t
) generates an ω-periodic
process ϱ on ℂ, Θ
(
u
,
t
) is a bounded
map for each u
,
t and is point
dissipative, then there exists a compact, connected, global attractor. Also,
there is an ω-periodic
solution of F
(
t
,
x
t
).ForF
(
t
,
x
t
), Lemma 2.5 provides the existence of a periodic
solution under the weak assumption of point dissipative. Let system
(2.1) be
rewritten as the system * and one can obtain that system (2.1) can generate an ω-periodic
process ϱ on ℂ (see [24, 25]).
Under some assumptions, one can view system (2.1) as a dissipative system and
apply Lemma 2.5 to it.Lemma 2.6 (see [2.66]).
Ifρ
(
A
)
<
1 for A
≥
0, then (
E
−
A
)
−
1
≥
0, where E denotes the
identity matrix of size n, A denotes a
square matrix of size n.Definition 2.7.
For system (2.1), an ω-periodic
solution x
*
(
t
)
=
(
x
1
*
(
t
)
,
x
2
*
(
t
)
,
…
,
x
n
*
(
t
)
)
T is said to be
globally exponentially stable if there exist constants λ
>
0 and M
>
0 so that(2.6)
|
x
(
t
)
−
x
*
(
t
)
|
≤
M
∥
ϕ
−
x
*
∥
e
−
λ
(
t
−
t
0
)
,

t
≥
t
0
, for the solution x
(
t
) of system (2.1)
with any initial value ϕ
∈
ℂ. Moreover, λ is called
globally exponentially convergent rate.Lemma 2.8 (see [18]).
Suppose thatx and x
' are two states
of system (2.1), then one has(2.7)
|
⋀
j
=
1
n
α
i
j
f
j
(
x
j
)
−
⋀
j
=
1
n
α
i
j
f
j
(
x
j
'
)
|
≤
∑
j
=
1
n
|
α
i
j
|
|
f
j
(
x
j
)
−
f
j
(
x
j
'
)
|
,
|
⋁
j
=
1
n
β
i
j
f
j
(
x
j
)
−
⋁
j
=
1
n
β
i
j
f
j
(
x
j
'
)
|
≤
∑
j
=
1
n
|
β
i
j
|
|
f
j
(
x
j
)
−
f
j
(
x
j
'
)
|
.


## 3. Main Results and Proofs

In this
section, by making use of some inequal analysis and matrix knowledge, we will
derive some sufficient conditions ensuring the existence, uniqueness, global
attractivity and global exponential stability of periodic solution of system
(2.1). First and foremost, in order to ensure the existence of periodic
attractor, we should find out an invariant set where the expected periodic
attractor * is located.Theorem 3.1.
Assume that assumptions (A1)–(A4)  hold and(A5)
ρ
(
S
)
<
1, where S
=
(
s
i
j
)
n
×
n and s
i
j
=
(
l
j
a
i
+
/
δ
i
a
i
−
)
(
c
i
j
+
α
i
j
+
β
i
j
);then the set Ω
=
{
ϕ
∈
ℂ
;
[
ϕ
]
+
≤
P
=
(
E
−
S
)
−
1
Q
} is a positively
invariant set of system (2.1), where vector Q
=
(
Q
1
,
Q
2
,
…
,
Q
n
)
T, and Q
i
=
(
a
i
+
I
i
/
δ
i
a
i
−
), E denotes the
identity matrix of size n.Proof.
Associated with system (2.1), (A1)–(A4), and
Lemma 2.8, we have(3.1)
d
|
x
i
(
t
)
|
d
t
≤
−
δ
i
a
i
−
|
x
i
(
t
)
|
+
a
i
+
{
∑
j
=
1
n
c
i
j
l
j
|
x
j
(
t
)
|
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
∫
t
−
τ
t
k
j
(
t
−
s
)
|
x
j
(
s
)
|
d
s









+
|
∑
j
=
1
n
d
i
j
(
t
)
u
j
(
t
)
+
⋀
j
=
1
n
G
i
j
(
t
)
u
j
(
t
)
+
⋁
j
=
1
n
H
i
j
(
t
)
u
j
(
t
)
+
I
i
(
t
)
|
}
≤
−
δ
i
a
i
−
|
x
i
(
t
)
|
+
a
i
+
{
∑
j
=
1
n
c
i
j
l
j
|
x
j
(
t
)
|
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
∫
t
−
τ
t
k
j
(
t
−
s
)
|
x
j
(
s
)
|
d
s
+
I
i
}
.
 Applying the variation of constant formula for the
above, we can derive that(3.2)
|
x
i
(
t
)
|
≤
exp
⁡
[
−
δ
i
a
i
−
(
t
−
t
0
)
]
|
x
i
(
t
0
)
|

+
a
i
+
∫
t
0
t
exp
⁡
[
−
δ
i
a
i
−
(
t
−
s
)
]
{
∑
j
=
1
n
c
i
j
l
j
|
x
j
(
s
)
|
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)













×
∫
s
−
τ
s
k
j
(
s
−
θ
)
|
x
j
(
θ
)
|
d
θ
+
I
i
}
d
s
,

t
≥
t
0
. Together with ρ
(
S
)
<
1 and Lemma 2.6,
one has (
E
−
S
)
−
1
≥
0. In view of Q
>
0, then P
=
(
E
−
S
)
−
1
Q
>
0. We will prove that if [
ϕ
]
+
≤
P, then(3.3)
[
x
(
t
)
]
+
≤
P
,

t
≥
t
0
. It suffices to show that for all λ
>
1, if [
ϕ
]
+
<
λ
P, then(3.4)
[
x
(
t
)
]
+
<
λ
P
,

t
≥
t
0
. Using the method of contrary. If not, there exist some i
∈
{
1
,
2
,
…
,
n
} and t
1
≥
t
0 such that(3.5)
|
x
i
(
t
1
)
|
=
λ
P
i
,

|
x
i
(
t
)
|
≤
λ
P
i
,

t
∈
[
t
0
−
τ
,
t
1
)
,
(3.6)
[
x
(
t
)
]
+
≤
λ
P
,

t
∈
[
t
0
,
t
1
]
, where P
i is the ith component of
vector P. Noticing that S
P
+
Q
=
P or ∑
j
=
1
n
s
i
j
P
j
+
Q
i
=
P
i, then according to (3.5) and (3.2), we have(3.7)
λ
P
i
=
|
x
i
(
t
1
)
|
≤
exp
⁡
[
−
δ
i
a
i
−
(
t
1
−
t
0
)
]
∥
ϕ
i
∥
a
i
+
∫
t
0
t
1
exp
⁡
[
−
δ
i
a
i
−
(
t
1
−
s
)
]

×
{
∑
j
=
1
n
c
i
j
l
j
∥
x
j
(
s
)
∥
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
∫
s
−
τ
s
k
j
(
s
−
θ
)
d
θ
∥
x
j
(
s
)
∥
+
I
i
}
d
s
≤
exp
⁡
[
−
δ
i
a
i
−
(
t
1
−
t
0
)
]
∥
ϕ
i
∥

+
a
i
+
∫
t
0
t
1
exp
⁡
[
−
δ
i
a
i
−
(
t
1
−
s
)
]
δ
i
a
i
−
{
∑
j
=
1
n
(
c
i
j
+
α
i
j
+
β
i
j
)
l
j
δ
i
a
i
−
∥
x
j
(
s
)
∥
+
I
i
δ
i
a
i
−
}
d
s
≤
exp
⁡
[
−
δ
i
a
i
−
(
t
1
−
t
0
)
]
λ
P
i
+
(
∑
j
=
1
n
s
i
j
λ
P
j
+
Q
i
)
∫
t
0
t
1
exp
⁡
[
−
δ
i
a
i
−
(
t
1
−
s
)
]
δ
i
a
i
−
d
s
=
exp
⁡
[
−
δ
i
a
i
−
(
t
1
−
t
0
)
]
[
λ
P
i
−
(
∑
j
=
1
n
s
i
j
λ
P
j
+
Q
i
)
]
+
(
∑
j
=
1
n
s
i
j
λ
P
j
+
Q
i
)
. In view of exp
[
−
δ
i
a
i
−
(
t
1
−
t
0
)
]
<
1, we can easily conclude that λ
P
i
<
λ
P
i from (3.7), which
is a contradiction. Therefore, (3.4) holds. Let λ
→
1, then we imply that (3.3) holds.
The proof of Theorem 3.1
is completed.In order to investigate the global attractivity of the
periodic solutions, it is necessary to require global attractivity of the
invariant setΩ.Theorem 3.2.
Under the assumptions of Theorem3.1,
the set Ω is a global
attractivity set of system (2.1).Proof.
Using similar discussion to Theorem3.1, for any given
initial value ϕ
∈
ℂ, there exists some constant λ
>
1 such that if [
ϕ
]
+
<
λ
P, then(3.8)
[
x
(
t
)
]
+
<
λ
P
,

t
≥
t
0
. If this is not true, there exists a nonnegative vector σ
=
(
σ
1
,
σ
2
,
…
,
σ
n
)
T such that(3.9)
lim
⁡
t
→
∞
sup
⁡
[
x
(
t
)
]
+
=
σ
+
P
. By the definition of upper limit of x
(
t
) and (3.8), for
any sufficiently small value ε
(
0
<
ε
≪
1
), there exists some t
2
≥
t
0 such that(3.10)
|
x
i
(
t
−
τ
)
|
<
P
i
+
(
1
+
ε
)
σ
i
,

t
≥
t
2
,
i
=
1
,
2
,
…
,
n
. By the continuity of function exp
(
⋅
), set T
>
−
ln
ε
/
δ
i
a
i
−
>
0, then we have(3.11)
exp
⁡
[
−
∫
t
−
T
t
δ
i
a
i
−
d
s
]
<
ε
,


or


exp
⁡
(
−
T
δ
i
a
i
−
)
<
ε
, for all t
>
t
2
+
T.
According to (2.1) and (3.9)–(3.11), we have (3.12)
|
x
i
(
t
)
|
≤
exp
⁡
[
−
δ
i
a
i
−
(
t
−
t
0
)
]
|
x
i
(
t
0
)
|
+
a
i
+
∫
t
0
t
exp
⁡
[
−
δ
i
a
i
−
(
t
−
s
)
]

×
{
∑
j
=
1
n
c
i
j
l
j
|
x
j
(
s
)
|
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
∫
0
τ
k
j
(
θ
)
|
x
j
(
s
−
θ
)
|
d
θ
+
I
i
}
d
s
≤
exp
⁡
[
−
δ
i
a
i
−
(
t
−
t
0
)
]
|
x
i
(
t
0
)
|
+
a
i
+
(
∫
t
0
t
−
T
+
∫
t
−
T
t
)
exp
⁡
[
−
δ
i
a
i
−
(
t
−
s
)
]

×
{
∑
j
=
1
n
c
i
j
l
j
|
x
j
(
s
)
|
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
∥
x
j
(
s
−
τ
)
∥
+
I
i
}
d
s
≤
exp
⁡
[
−
δ
i
a
i
−
(
t
−
t
0
)
]
|
x
i
(
t
0
)
|
+
(
exp
⁡
(
−
δ
i
a
i
−
T
)
−
exp
⁡
[
−
δ
i
a
i
−
(
t
−
t
0
)
]
)

×
a
i
+
δ
i
a
i
−
{
∑
j
=
1
n
c
i
j
l
j
(
P
j
+
σ
j
)
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
(
P
j
+
(
1
+
ε
)
σ
j
)
+
I
i
}

+
(
1
−
exp
⁡
(
−
δ
i
a
i
−
T
)
)
a
i
+
δ
i
a
i
−
{
∑
j
=
1
n
c
i
j
l
j
(
P
j
+
σ
j
)
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
(
P
j
+
(
1
+
ε
)
σ
j
)
+
I
i
}
≤
exp
⁡
[
−
δ
i
a
i
−
(
t
−
t
0
)
]
|
x
i
(
t
0
)
|
+
a
i
+
δ
i
a
i
−

×
{
∑
j
=
1
n
c
i
j
l
j
(
P
j
+
σ
j
)
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
(
P
j
+
(
1
+
ε
)
σ
j
)
+
I
i
}
=
exp
⁡
[
−
δ
i
a
i
−
(
t
−
t
0
)
]
|
x
i
(
t
0
)
|
+
∑
j
=
1
n
s
i
j
P
j
+
∑
j
=
1
n
s
i
j
σ
j
+
Q
i

+
ε
{
a
i
+
δ
i
a
i
−
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
σ
j
}
≤
exp
⁡
[
−
δ
i
a
i
−
(
t
−
t
0
)
]
∥
ϕ
i
∥
+
∑
j
=
1
n
s
i
j
P
j
+
(
1
+
ε
)
∑
j
=
1
n
s
i
j
σ
j
+
Q
i
.   It follows from (3.9) and the properties of the upper
limit of x
(
t
) that there
exists a sequence t
k so that t
k
≥
t
2
+
T and(3.13)
lim
⁡
t
k
→
∞
|
x
i
(
t
k
)
|
=
σ
i
+
P
i
. Let t
k
→
∞
,
ε
→
0, then we have(3.14)
σ
i
+
P
i
≤
∑
j
=
1
n
s
i
j
P
j
+
∑
j
=
1
n
s
i
j
σ
j
+
Q
i
. Noting that P
i
=
∑
j
=
1
n
s
i
j
P
j
+
Q
i, we can easily obtain that σ
i
≤
∑
j
=
1
n
s
i
j
σ
j
,
i
=
1
,
2
,
…
,
n
, that is, σ
≤
S
σ. By  [27, Theorem 8.3.2], we have ρ
(
S
)
≥
1, which is a contradiction. Thus, σ
≡
0, this completes the proof of the theorem.Theorem3.2 implies that under the assumptions of
Theorem 3.1, system (2.1) is a dissipative system. Associated with
Lemma 2.5 and
Theorem 3.2, we can easily deduce the sufficient conditions for the existence of
global periodic attractor of system (2.1) as follows.Theorem 3.3.
Under the assumptions of Theorem3.1, there exists a
global attractor of system (2.1). Moreover, it is ω-periodic and
belongs to the positively invariant set Ω.Proof.
Obviously, system (2.1) can generate an ω-periodic
process by Lemma 2.5 and Theorem 3.2. Moreover, together with the assumptions of
Theorem 3.1, we can conclude that there exists ω-periodic
solution denoted by x
*
(
t
).
Letx
(
t
) be an arbitrary
solution of model (2.1) and use the substitution y
(
t
)
=
x
(
t
)
−
x
*
(
t
), the neural network model (2.1) can be rewritten as(3.15)
y
˙
i
(
t
)
=
−
a
i
(
y
i
(
t
)
)
[
b
i
(
y
i
(
t
)
)
−
∑
j
=
1
n
c
i
j
(
t
)
F
j
(
y
j
(
t
)
)





−
(
⋀
j
=
1
n
α
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s
−
⋀
j
=
1
n
α
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
*
(
s
)
)
d
s
)





−
(
⋁
j
=
1
n
β
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s
−
⋁
j
=
1
n
β
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
*
(
s
)
)
d
s
)
]
, where a
i
(
y
i
(
t
)
)
=
a
i
(
y
i
(
t
)
+
x
i
*
(
t
)
), b
i
(
y
i
(
t
)
)
=
b
i
(
y
i
(
t
)
+
x
i
*
(
t
)
)
−
b
i
(
x
i
*
(
t
)
), F
j
(
y
j
(
t
)
)
=
f
j
(
y
j
(
t
)
+
x
j
*
(
t
)
)
−
f
j
(
x
j
*
(
t
)
), i
=
1
,
2
,
…
,
n
.
Associated with
system (3.15), (A1–A4), and Lemma 2.8, we
have(3.16)
d
|
y
i
(
t
)
|
d
t
≤
−
δ
i
a
i
−
|
y
i
(
t
)
|
+
a
i
+
{
∑
j
=
1
n
c
i
j
l
j
|
y
j
(
t
)
|
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
∫
t
−
τ
t
k
j
(
t
−
s
)
|
y
j
(
s
)
|
d
s
}
, where i
=
1
,
2
,
…
,
n
. Repeating the
above arguments, it is easy to see that system (3.15) satisfies all conditions of
Theorem 3.2 with Q
=
{
0
}. Thus, lim
t
→
∞
y
i
=
0
,
i
=
1
,
2
,
…
,
n, which implies
that the periodic solution x
*
(
t
) is globally
attracting. Thus, we complete the proof of this theorem.Remark 3.4.
By Theorem3.3, all other solutions of system (2.1)
converge to the
ω-periodic
solution as t
→
∞. Theorem 3.3 provides a guideline on the choice of
interconnection matrices for a neural network which is desired to have a unique
global attractor. Moreover, if the parameters c
i
j
(
t
)
,
α
i
j
(
t
)
,
β
i
j
(
t
)
, and I
i
(
t
) are not
periodic in time t, under the assumptions of (A1)–(A4)  and ∫
t
0
+
∞
δ
i
a
i
−
d
t
=
+
∞ for all i
=
1
,
2
,
…
,
n, Theorems 3.1 and 3.2 are also achievable.Theorem 3.5.
Under the assumptions of Theorem3.1, the ω-periodic
solution x
*
(
t
) of system (2.1)
is globally exponentially stable if the following condition holds:(3.17)
δ
i
a
i
−
>
a
i
+
∑
j
=
1
n
(
c
i
j
+
α
i
j
+
β
i
j
)
l
j
,

i
=
1
,
2
,
…
,
n
.Proof.
According to Theorem3.3, we only prove that (
0
,
0
,
…
,0
)
T of the system
(3.15) is globally exponentially stable. Consider the function g
i
(
ξ
) given by(3.18)
g
i
(
ξ
)
=
δ
i
a
i
−
−
ξ
−
a
i
+
[
∑
j
=
1
n
c
i
j
l
j
+
∑
j
=
1
n
(
α
i
j
+
β
i
j
)
l
j
e
ξ
τ
]
,

i
=
1
,
2
,
…
,
n
. By (3.17), we have(3.19)
g
i
(
0
)
=
δ
i
a
i
−
−
a
i
+
[
∑
j
=
1
n
c
i
j
l
j
+
∑
j
=
1
n
(
α
i
j
+
β
i
j
)
l
j
]
>
0 and g
i
(
ξ
) is continuous, g
i
(
ξ
)
→
−
∞ as ξ
→
+
∞. Thus, there exists a ξ
i
>
0 so that g
i
(
ξ
i
)
=
0. Without loss of generality, set ξ
i
=
min
{
ξ
>
0
∣
g
i
(
ξ
)
=
0
}, so g
i
(
ξ
)
>
0 when ξ
∈
(
0
,
ξ
i
). Now let ξ
=
min
{
ξ
i
,
i
=
1
,
2
,
…
,
n
}, when ξ
0
∈
(
0
,
ξ
), we have g
i
(
ξ
0
)
>
0
,
i
=
1
,
2
,
…
,
n, that is,(3.20)
δ
i
a
i
−
−
ξ
0
−
a
i
+
[
∑
j
=
1
n
c
i
j
l
j
+
∑
j
=
1
n
(
α
i
j
+
β
i
j
)
l
j
e
ξ
0
τ
]
>
0
,

i
=
1
,
2
,
…
,
n
. Let y
(
t
)
=
(
y
1
(
t
)
,
y
2
(
t
)
,
…
,
y
n
(
t
)
)
T be a solution
of system (3.15) with any initial value ϕ
∈
ℂ, then we have(3.21)
|
y
i
(
t
)
|
≤
exp
⁡
[
−
δ
i
a
i
−
(
t
−
t
0
)
]
|
y
i
(
t
0
)
|

+
a
i
+
∫
t
0
t
exp
⁡
[
−
δ
i
a
i
−
(
t
−
s
)
]
{
∑
j
=
1
n
c
i
j
l
j
|
y
j
(
s
)
|
+
∑
j
=
1
n
(
α
i
j
l
j
+
β
i
j
l
j
)
∫
0
τ
k
j
(
θ
)
|
y
j
(
s
−
θ
)
|
d
θ
}
d
s
, for t
≥
t
0 and i
=
1
,
2
,
…
,
n. Set(3.22)
w
i
(
t
)
=
{
e
ξ
0
(
t
−
t
0
)
|
y
i
(
t
)
|
,
t
>
t
0
,
|
y
i
(
t
)
|
,
t
0
−
τ
≤
t
≤
t
0
. Then, it follows from
(3.21) and (3.22) that for t
>
t
0 and i
=
1
,
2
,
…
,
n, (3.23)
w
i
(
t
)
≤
exp
⁡
[
−
(
δ
i
a
i
−
−
ξ
0
)
(
t
−
t
0
)
]
w
i
(
t
0
)

+
a
i
+
∫
t
0
t
exp
⁡
[
−
(
δ
i
a
i
−
−
ξ
0
)
(
t
−
s
)
]
{
∑
j
=
1
n
[
c
i
j
l
j
+
(
α
i
j
l
j
+
β
i
j
l
j
)
e
τ
ξ
0
]
∥
w
j
(
s
)
∥
}
d
s
, for t
≥
t
0
,
i
=
1
,
2
,
…
,
n
.
For the initial valueϕ
∈
ℂ, there must exist η
>
0 and r
∈
{
1
,
2
,
…
,
n
} so that ∥
ϕ
r
(
0
)
∥
=
η and ∥
ϕ
i
(
0
)
∥
≤
η for i
=
1
,
2
,
…
,
n
.
We will show that for any sufficiently small constantε
>
0,(3.24)
w
i
(
t
)
<
η
+
ε
,

t
≥
t
0
,
i
=
1
,
2
,
…
,
n
. If it is not true, there must exist some t
1
≥
t
0 and k
∈
{
1
,
2
,
…
,
n
} so that(3.25)
w
k
(
t
1
)
=
η
+
ε
,

w
i
(
t
)
≤
η
+
ε
,

t
∈
[
t
0
,
t
1
)
,
i
=
1
,
2
,
…
,
n
. By (3.23) and (3.20), we have(3.26)
η
+
ε
=
w
k
(
t
1
)
≤
exp
⁡
[
−
(
δ
i
a
i
−
−
ξ
0
)
(
t
1
−
t
0
)
]
w
k
(
t
0
)

+
a
i
+
∫
t
0
t
1
exp
⁡
[
−
(
δ
i
a
i
−
−
ξ
0
)
(
t
1
−
s
)
]
{
∑
j
=
1
n
[
c
i
j
l
j
+
(
α
i
j
l
j
+
β
i
j
l
j
)
e
τ
ξ
0
]
∥
w
j
(
s
)
∥
}
d
s
≤
(
η
+
ε
)
exp
⁡
[
−
(
δ
i
a
i
−
−
ξ
0
)
(
t
1
−
t
0
)
]
+
(
η
+
ε
)
a
i
+
δ
i
a
i
−
−
ξ
0
∑
j
=
1
n
[
c
i
j
l
j
+
(
α
i
j
l
j
+
β
i
j
l
j
)
e
τ
ξ
0
]


×
{
1
−
exp
⁡
[
−
(
δ
i
a
i
−
−
ξ
0
)
(
t
1
−
t
0
)
]
}
<
η
+
ε
, which is a contradiction. Thus (3.24) holds. Let ε
→
0, we have(3.27)
w
i
(
t
)
≤
η
,

t
≥
t
0
,
i
=
1
,
2
,
…
,
n
, that is,(3.28)
w
i
(
t
)
≤
∥
ϕ
r
(
0
)
∥
,

t
≥
t
0
,
i
=
1
,
2
,
…
,
n
. This implies that there exists Γ
>
1 so that(3.29)
|
w
(
t
)
|
≤
Γ
∥
ϕ
(
0
)
∥
,

t
≥
t
0
. From (3.22), there exists some constant ζ
≥
1 so that(3.30)
|
y
(
t
)
|
≤
ζ
∥
ϕ
(
0
)
∥
e
ξ
0
(
t
0
−
t
)
,

t
≥
t
0
, which implies that the solution (
0
,
0
,
…
,0
)
T of the system
(3.15) is globally exponentially stable. The proof of Theorem 3.5 is completed.Corollary 3.6.
Leta
0
=
min
1
≤
i
≤
n
{
a
i
−
}
,
δ
0
=
min
1
≤
i
≤
n
{
δ
i
}. Under the assumptions  of Theorem 3.1, the ω-periodic
solution x
*
(
t
) of system (2.1)
is globally exponentially stable if the following condition holds:(3.31)
a
0
δ
0
>
a
i
+
∑
j
=
1
n
(
c
i
j
+
α
i
j
+
β
i
j
)
l
j
,

i
=
1
,
2
,
…
,
n
.Proof.
Letg
i
(
ξ
)
=
δ
0
a
0
−
ξ
−
a
i
+
[
∑
j
=
1
n
c
i
j
l
j
+
∑
j
=
1
n
(
α
i
j
+
β
i
j
)
l
j
e
ξ
τ
]
,
i
=
1
,
2
,
…
,
n. It can be directly obtained by repeating the
argument of Theorem 3.5. Here we omit it.Remark 3.7.
ξ
0 is called
exponentially convergent rate which can be estimated by (3.20). Global
exponential stability means that system (2.1) can converge to the periodic
solution associated with the inputs when the external inputs are provided to
the system, irrespective of the initial values. The conditions of the above
theorems and Corollary 3.6 are dependant on the spectral radius which needs δ
i
a
i
−
>
0 for i
=
1
,
2
,
…
,
n and t
≥
t
0. They may be invalid when δ
i
a
i
−
≥
0 and is not
identically zero. Therefore, we present other criteria to overcome the
limitation.
Now we consider the following system:(3.32)
x
˙
i
(
t
)
=
−
a
−
[
b
i
(
x
i
(
t
)
)
−
∑
j
=
1
n
c
i
j
(
t
)
f
j
(
x
j
(
t
)
)
−
∑
j
=
1
n
d
i
j
(
t
)
u
j
(
t
)
−
⋀
j
=
1
n
α
i
j
(
t
)




×
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s
−
⋁
j
=
1
n
β
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s




−
⋀
j
=
1
n
G
i
j
(
t
)
u
j
(
t
)
−
⋁
j
=
1
n
H
i
j
(
t
)
u
j
(
t
)
−
I
i
(
t
)
]
.
(A6)
Functionb
i
(
x
i
(
t
)
) is continuous
and there exist continuous positive ω-periodic
functions b
i
(
t
) so that(3.33)
b
i
(
u
)
−
b
i
(
u
)
u
−
u
≥
b
i
(
t
)
≥
0,

∀
u
,
u
∈
R
,
u
≠
u
,
i
=
1
,
2
,
…
,
n
.
Set(3.34)
z
i
(
t
,
s
)
=
a
i
+
exp
⁡
(
∫
0
ω
a
i
−
b
i
(
θ
)
d
θ
)
exp
⁡
(
∫
0
ω
a
i
−
b
i
(
θ
)
d
θ
)
−
1
exp
⁡
(
∫
t
s
a
i
−
b
i
(
θ
)
d
θ
)
,
v
i
j
=
sup
⁡
t
≥
t
0
∫
t
t
+
ω
z
i
(
t
,
s
)
[
|
c
i
j
(
s
)
|
+
|
α
i
j
(
s
)
|
+
|
β
i
j
(
s
)
|
]
l
j
d
s
,

i
,
j
=
1
,
2
,
…
,
n
.Theorem 3.8.
Assume that∫
0
ω
a
i
−
b
i
(
θ
)
d
θ
>
0 for all i
=
1
,
2
,
…
,
n and  (A1), (A3), (A4), (A6) hold and
(A7)
ρ
(
V
)
<
1, where V
=
(
v
i
j
)
n
×
n,
then system (3.32) has exactly one ω-periodic
solution.Proof.
By (A7),E
−
V is an M-matrix [29].
Therefore, there exists a diagonal matrix Q
=
diag
(
q
1
,
q
2
,
…
,
q
n
)
>
0 so that (
E
−
V
)
Q is strictly
diagonally dominant with positive diagonal entries, that is,(3.35)
∑
j
=
1
n
v
i
j
q
i
−
1
q
j
<
1
,

i
=
1
,
2
,
…
,
n
. Associated with systems (A1), (A3), (A4), (A6), and Lemma 2.8, we
have(3.36)
|
x
i
(
t
+
ω
)
|
≤
exp
⁡
(
−
∫
t
t
+
ω
a
i
−
b
i
(
s
)
d
s
)
|
x
i
(
t
)
|

+
a
i
+
∫
t
t
+
ω
exp
⁡
(
∫
t
s
a
i
−
b
i
(
θ
)
d
θ
)
{
∑
j
=
1
n
|
c
i
j
(
s
)
|
|
f
j
(
x
j
(
s
)
)
|
+
∑
j
=
1
n
(
|
α
i
j
(
s
)
|
+
|
β
i
j
(
s
)
|
)















×
∫
s
−
τ
s
k
j
(
s
−
θ
)
|
f
j
(
x
j
(
θ
)
)
|
d
θ
+
|
I
i
(
s
)
|
}
d
s
. If x
i
(
t
+
ω
)
=
x
i
(
t
), we have(3.37)
|
x
i
(
t
)
|
≤
exp
⁡
(
∫
0
ω
a
i
−
b
i
(
s
)
d
s
)
a
i
+
exp
⁡
(
∫
0
ω
a
i
−
b
i
(
s
)
d
s
)
−
1

×
∫
t
t
+
ω
⁢
exp
⁡
(
∫
t
s
a
i
−
b
i
(
θ
)
d
θ
)
{
∑
j
=
1
n
|
c
i
j
(
s
)
|
|
f
j
(
x
j
(
s
)
)
|
+
∑
j
=
1
n
(
|
α
i
j
(
s
)
|
+
|
β
i
j
(
s
)
|
)
















×
∫
s
−
τ
s
k
j
(
s
−
θ
)
|
f
j
(
x
j
(
θ
)
)
|
d
θ
+
|
I
i
(
s
)
|
}
d
s
.
LetΛ
ω
=
{
φ
;
φ
=
(
φ
1
,
φ
2
,
…
,
φ
n
)
T
}, φ
i is a continuous ω-periodic
function on R. With the norm ∥
φ
∥
=
max
1
≤
i
≤
n
{
q
i
−
1
sup
t
∈
R
|
φ
i
(
t
)
|
}, then Λ
ω is a Banach
space. Define operator T
:
Λ
ω
→
Λ
ω as follows:(3.38)
T
(
φ
1
,
φ
2
,
…
,
φ
n
)
=
(
(
T
φ
)
1
,
(
T
φ
)
2
,
…
,
(
T
φ
)
n
)
T
, where(3.39)
(
T
φ
)
i
(
t
)
=
∫
t
t
+
ω
z
i
(
t
,
s
)
[
∑
j
=
1
n
c
i
j
(
s
)
f
j
(
φ
j
(
s
)
)
+
⋀
j
=
1
n
α
i
j
(
s
)
∫
s
−
τ
s
k
j
(
s
−
θ
)
f
j
(
φ
j
(
θ
)
)
d
θ






+
⋁
j
=
1
n
β
i
j
(
s
)
∫
s
−
τ
s
k
j
(
s
−
θ
)
f
j
(
φ
j
(
θ
)
)
d
θ
+
I
i
(
s
)
]
d
s
, for t
≥
t
0 and i
=
1
,
2
,
…
,
n
. Clearly,   φ is an ω-periodic
solution of system (3.32) if and only if φ is the fixed
point of operator T. For all φ
,
ψ
∈
Λ
ω, we have(3.40)
∥
T
φ
−
T
ψ
∥

=
max
⁡
1
≤
i
≤
n
{
q
i
−
1
sup
⁡
t
∈
R
|
(
T
φ
)
i
(
t
)
−
(
T
ψ
)
i
(
t
)
|
}

≤
max
⁡
1
≤
i
≤
n
{
q
i
−
1
sup
⁡
t
∈
R
∫
t
t
+
ω
z
i
(
t
,
s
)
[
∑
j
=
1
n
|
c
i
j
(
s
)
|
|
f
j
(
φ
j
)
(
s
)
−
f
j
(
ψ
j
)
(
s
)
|













+
|
⋀
j
=
1
n
α
i
j
(
s
)
∫
s
−
τ
s
k
j
(
s
−
θ
)
f
j
(
φ
j
(
θ
)
)
d
θ














−
⋀
j
=
1
n
α
i
j
(
s
)
∫
s
−
τ
s
k
j
(
s
−
θ
)
f
j
(
ψ
j
(
θ
)
)
d
θ
|













+
|
⋁
j
=
1
n
β
i
j
(
s
)
∫
s
−
τ
s
k
j
(
s
−
θ
)
f
j
(
φ
j
(
θ
)
)
d
θ














−
⋁
j
=
1
n
β
i
j
(
s
)
∫
s
−
τ
s
k
j
(
s
−
θ
)
f
j
(
ψ
j
(
θ
)
)
d
θ
|
]
d
s
}

≤
max
⁡
1
≤
i
≤
n
{
q
i
−
1
sup
⁡
t
∈
R
∫
t
t
+
ω
z
i
(
t
,
s
)
∑
j
=
1
n
(
|
c
i
j
(
s
)
|
+
|
α
i
j
(
s
)
|
+
|
β
i
j
(
s
)
|
)
l
j
d
s
sup
⁡
t
∈
R
|
φ
j
(
s
)
−
ψ
j
(
s
)
|
}



=
max
⁡
1
≤
i
≤
n
{
q
i
−
1
∑
j
=
1
n
v
i
j
sup
⁡
t
∈
R
|
φ
j
(
s
)
−
ψ
j
(
s
)
|
}

≤
max
⁡
1
≤
i
≤
n
{
∑
j
=
1
n
q
i
−
1
q
j
v
i
j
}
max
⁡
1
≤
j
≤
n
{
q
j
−
1
sup
⁡
t
∈
R
|
φ
j
(
s
)
−
ψ
j
(
s
)
|
}

=
max
⁡
1
≤
i
≤
n
{
∑
j
=
1
n
q
i
−
1
q
j
v
i
j
}
∥
φ
−
ψ
∥
.   In view of (3.35), it is easily concluded that max
1
≤
i
≤
n
{
∑
j
=
1
n
q
i
−
1
q
j
v
i
j
}
<
1. This implies that T is a
contraction operator. By the well-known contraction mapping principle, there
exists a unique fixed point φ
* for T which is the
unique ω-periodic
solution of system (3.32). The proof of Theorem 3.8 is completed.Obviously, Cohen-Grossberg-type neural networks,
cellular neural networks, and so on are special forms of system (2.1). It is not
difficult to see that the conditions in our results are a little generation and improvement of their theorems; see [21, 24].Leta
i
(
x
i
(
t
)
)
≡
1,
b
i
(
x
i
(
t
)
)
=
b
i
(
t
)
x
i
(
t
)
,
i
=
1
,
2
,
…
,
n
, system (2.1)
reduces to the following fuzzy cellular neural networks:(3.41)
x
˙
i
(
t
)
=
−
b
i
(
t
)
x
i
(
t
)
+
∑
j
=
1
n
c
i
j
(
t
)
f
j
(
x
j
(
t
)
)
+
∑
j
=
1
n
d
i
j
(
t
)
u
j
(
t
)
+
⋀
j
=
1
n
α
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s


+
⋁
j
=
1
n
β
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s
+
⋀
j
=
1
n
G
i
j
(
t
)
u
j
(
t
)
+
⋁
j
=
1
n
H
i
j
(
t
)
u
j
(
t
)
+
I
i
(
t
)
, where b
i
(
t
)
>
0
,
for
all
t
≥
t
0
,
i
=
1
,
2
,
…
,
n
. From (3.41), we
can derive the following results.Corollary 3.9.
Assume that (A3) and (A4) hold. Moreover,(A8)
b
i
(
t
)
≥
b
i
>
0
,
for
all
t
∈
R
,
i
=
1
,
2
,
…
,
n
;
(A9)
ρ
(
S
)
<
1, where S
=
(
s
i
j
)
n
×
n and
s
i
j
=
(
c
i
j
l
j
+
α
i
j
l
j
+
β
i
j
l
j
)
/
b
i; set Q
=
(
Q
1
,
Q
2
,
…
,
Q
n
)
T, and Q
i
=
I
i
/
b
i,then (i) the set Ω
=
{
ϕ
∈
ℂ
;
[
ϕ
]
+
≤
P
=
(
E
−
S
)
−
1
Q
} is a positively
invariant set of system (3.41); (ii) the set Ω is a globally
attracting set of system (3.41); (iii) there exists a
global attractor of system (3.41), moreover, it is ω-periodic and
belongs to the positively invariant set Ω.The proof is completely similar to
Theorems3.1–3.3, here
we omit it.Corollary 3.10.
Under the assumptions of Corollary3.9, the ω-periodic
solution x
*
(
t
) of system (3.41)
is globally exponentially stable if the following condition holds:(3.42)
b
i
>
∑
j
=
1
n
(
c
i
j
+
α
i
j
+
β
i
j
)
l
j
,

i
=
1
,
2
,
…
,
n
. It easily
follows from Theorem 3.5, so the proof is omitted.


## 4. Two Examples

In this
section, we give two examples to demonstrate the efficiencies of our criteria.Example 4.1.
In the following, we consider the periodic solutions
of the two neurons fuzzy networks with delay:(4.1)
x
˙
i
(
t
)
=
−
d
i
(
t
)
x
i
(
t
)
+
∑
j
=
1
2
a
i
j
(
t
)
f
j
(
x
j
(
t
)
)
+
∑
j
=
1
2
b
i
j
(
t
)
u
j
(
t
)
+
⋀
j
=
1
2
α
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s

+
⋁
j
=
1
2
β
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s
+
⋀
j
=
1
2
G
i
j
(
t
)
u
j
(
t
)
+
⋁
j
=
1
2
H
i
j
(
t
)
u
j
(
t
)
+
I
i
(
t
)
,   where d
i
(
t
)
,
a
i
j
(
t
)
,
b
i
j
(
t
)
,
α
i
j
(
t
)
,
β
i
j
(
t
)
,
G
i
j
(
t
)
,
H
i
j
(
t
), and u
i
(
t
)
(
i
,
j
=
1
,
2
) are
continuously periodic solutions with period 2
π. Let(4.2)
D
(
t
)
=
(
1
0
0
1
)
,


A
(
t
)
=
(
a
i
j
(
t
)
)
=
(
0.21
sin
t
−
0.2
cos
t
−
0.15
cos
t
0.19
sin
t
)
,
α
(
t
)
=
(
α
i
j
(
t
)
)
=
(
0.1
sin
t
−
0.11
⁢
cos
t
0.12
⁢
cos
t
0.11
⁢
sin
t
)
,
β
(
t
)
=
(
β
i
j
(
t
)
)
=
(
0.15
⁢
sin
t
0.1
⁢
cos
t
−
0.1
⁢
cos
t
0.12
⁢
sin
t
)
,
(
b
i
j
(
t
)
)
=
(
G
i
j
(
t
)
)
=
(
H
i
j
(
t
)
)
=
(
0
0
0
0
)
,


u
(
t
)
=
2
I
(
t
)
=
(
sin
t
cos
t
)
,
f
i
(
x
i
)
=
f
(
x
)
=
1
2
(
|
x
+
1
|
−
|
x
−
1
|
)
,

∫
0
τ
k
i
(
s
)
d
s
=
1
,

i
=
1,2.   By simple computation, one can easily obtain (4.3)
S
=
(
0.46
0.41
0.37
0.42
)
,


Q
=
(
0.5
0.5
)
,


P
=
(
E
−
S
)
−
1
Q
=
(
3.0650
2.8173
)
,
ξ
(
S
)
1
=
0.83
,


ξ
(
S
)
2
=
0.05
,       where ξ
(
S
)
i
(
i
=
1
,
2
) denotes the
eigenvalues of matrix S. Therefore, ρ
(
S
)
<
1. By Corollary 3.9, we know that the set Ω
=
{
ϕ
∈
ℂ
;
[
ϕ
]
+
≤
P
=
(
E
−
S
)
−
1
Q
} is a positively
invariant set of system (3.41), which is a globally attracting set of system
(3.41), that is, there exists a global attractor of system (3.41). Moreover, it is 2
π-periodic and
belongs to the positively invariant set Ω. In addition, one can easily obtain(4.4)
∑
j
=
1
2
(
a
1
j
+
α
1
j
+
β
1
j
)
l
j
=
0.88
<
1
=
d
1
,


∑
j
=
1
2
(
a
2
j
+
α
2
j
+
β
2
j
)
l
j
=
0.99
<
1
=
d
2
. According to
Corollary 3.10, the 2
π-periodic
solution x
*
(
t
) of system (3.41)
is globally exponentially stable.Example 4.2.
We consider the periodic solutions of the following
fuzzy Cohen-Grossberg neural networks with delay:(4.5)
x
˙
i
(
t
)
=
−
a
i
(
x
i
(
t
)
)
[
b
i
(
x
i
(
t
)
)
−
∑
j
=
1
2
c
i
j
(
t
)
f
j
(
x
j
(
t
)
)
−
∑
j
=
1
2
d
i
j
(
t
)
u
j
(
t
)
−
⋀
j
=
1
2
α
i
j
(
t
)






×
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s
−
⋁
j
=
1
2
β
i
j
(
t
)
∫
t
−
τ
t
k
j
(
t
−
s
)
f
j
(
x
j
(
s
)
)
d
s






−
⋀
j
=
1
2
G
i
j
(
t
)
u
j
(
t
)
−
⋁
j
=
1
2
H
i
j
(
t
)
u
j
(
t
)
−
I
i
(
t
)
]
,   where a
i
j
(
t
)
,
b
i
j
(
t
)
,
α
i
j
(
t
)
,
β
i
j
(
t
)
,
G
i
j
(
t
)
,
H
i
j
(
t
), and u
i
(
t
)
(
i
,
j
=
1
,
2
) are
continuously periodic solutions with period 2
π. Let(4.6)
A
(
x
(
t
)
)
=
(
5
+
sin
⁡
(
x
1
(
t
)
)
0
0
4
−
cos
⁡
(
x
2
(
t
)
)
)
,


B
(
x
(
t
)
)
=
(
4
x
1
(
t
)
4
x
2
(
t
)
)
,
I
(
t
)
=
(
0.66
⁢
sin
t
0.75
⁢
cos
⁡
t
)
,
C
(
t
)
=
(
c
i
j
(
t
)
)
=
(
0.35
⁢
sin
⁡
t
−
0.25
⁢
cos
⁡
t
0.33
⁢
sin
⁡
t
0.34
⁢
cos
⁡
t
)
,
(
d
i
j
(
t
)
)
=
(
G
i
j
(
t
)
)
=
(
H
i
j
(
t
)
)
=
(
0
0
0
0
)
,
α
(
t
)
=
(
α
i
j
(
t
)
)
=
(
0.44
⁢
sin
⁡
t
−
0.38
⁢
cos
⁡
t
0.32
⁢
cos
⁡
t
0.28
⁢
sin
⁡
t
)
,
β
(
t
)
=
(
β
i
j
(
t
)
)
=
(
0.29
⁢
sin
⁡
t
0.43
⁢
cos
⁡
t
−
0.27
⁢
cos
⁡
t
0.35
⁢
sin
⁡
t
)
,
f
i
(
x
i
)
=
f
(
x
)
=
tanh
⁡
(
x
)
,

∫
0
τ
k
i
(
s
)
d
s
=
1
,

i
=
1,2.
By simple computation, we can obtain thata
1
−
=
4
,
a
2
−
=
3
,
a
1
+
=
6
,
a
2
+
=
5
,
l
1
=
l
2
=
1
,
δ
1
=
δ
2
=
4
,
(4.7)
S
=
(
0.4050
0.3975
0.3833
0.4042
)
,


Q
=
(
0.2475
0.3125
)
,


P
=
(
E
−
S
)
−
1
Q
=
(
1.3440
1.3892
)
,

ξ
(
P
)
1
=
0.7949
,


ξ
(
P
)
2
=
0.0142
,  where ξ
(
P
)
i
(
i
=
1
,
2
) denotes the
eigenvalues of matrix P. Therefore, ρ
(
P
)
<
1. By  Theorem 3.1, we know that the set Ω
=
{
ϕ
∈
ℂ
;
[
ϕ
]
+
≤
Q
=
(
E
−
P
)
−
1
M
} is a positively
invariant set of system (4.5). According to Theorems 3.2
and 3.3, one can
derive that the set Ω is a globally
attracting set of system (4.5), that is, there exists a global attractor of
system (4.5). Moreover, it is 2
π-periodic and
belongs to the positively invariant set Ω. In addition, one can easily obtain(4.8)
a
1
+
∑
j
=
1
2
(
c
1
j
+
α
1
j
+
β
1
j
)
l
j
=
12.84
<
δ
1
a
1
−
=
16
,
a
2
+
∑
j
=
1
2
(
c
2
j
+
α
2
j
+
β
2
j
)
l
j
=
9.45
<
δ
2
a
2
−
=
12. According to
Theorem 3.5, the 2
π-periodic
solution x
*
(
t
) of system (4.5)
is globally exponentially stable.


## 5. Conclusion

Periodic
oscillation is an important dynamical behavior in the applications and theories
of neural networks. We derived the novel criteria of fuzzy Cohen-Grossberg
neural networks with variable coefficients and distributed delays under easily
checkout conditions and the minimal assumption for the feedback kernel. Our
results extend some previous works and the way is common and very concise. In
the mean time, we believe that the method can be applied to research other
problems under some appropriate assumptions such as almost periodic solution,
synchronization of neural networks, and so on.



---

*Source: 453627-2008-02-28.xml*