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PureMathematics
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,2021,11(8),1559-1569
PublishedOnlineAugust2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.118174
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SynchronizationofFuzzyCellular
NeuralNetworkswithVariable
CoefficientsandTimeDelays
underImpulseControl
SonghuanZhang
∗
,YangLiu
DepartmentofMathematics,ShanghaiNormalUniversity,Shanghai
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1559-1569.DOI:10.12677/pm.2021.118174
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Received:Jul.18
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,2021;accepted:Aug.19
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,2021
Abstract
Thispaperinvestigates thesynchronization offuzzycellular neuralnetworkswithvari-
ablecoefficientsandtime-varyingdelaybydesigningaimpulsivecontrol.Bytaking
Lyapunovfunctionalmethodandthematrixinequalitymethod,thelinearmatrixin-
equality conditions are given to ensure the synchronization of the system.Meanwhile,
exponentialsynchronizationconditionsandasymptoticbehaviorofunknownparam-
etersarederived.Finally,asimulationexampleisgiventoverifytheeffectivenessof
theproposedmethod.
Keywords
FuzzyCellularNetworks,ImpulsiveControl, Synchronization, Time-Varying, Variable
Coefficients
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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n
V
j
=1
η
ij
˜
f
j
(˜
u
j
)
|≤
n
P
j
=1
|
η
ij
||
˜
f
j
(
u
j
)
−
˜
f
j
(˜
u
j
)
|
,
|
n
W
j
=1
β
ij
˜
f
j
(
u
j
)
−
n
W
j
=1
β
ij
˜
f
j
(˜
u
j
)
|≤
n
P
j
=1
|
β
ij
||
˜
f
j
(
u
j
)
−
˜
f
j
(˜
u
j
)
|
.
Ú
n
2.3.
é
u
∀
x,y
∈
R
n
Ú
½
Ý
Q
∈
R
n
×
n
,
k
2
x
T
y
≤
x
T
Qx
+
y
T
Q
−
1
y.
Ú
n
2.4.
(
Schur
Ö
Ú
n
)
é
u
©
¬
Ý
X
=
X
11
X
12
X
T
12
X
22
>
0
,
Ù
¥
X
11
,X
22
´
•
,
e
^
‡
´
d
µ
1)
X
11
>
0
,X
22
−
X
T
12
X
−
1
11
X
12
>
0
,
2)
X
22
>
0
,X
11
−
X
12
X
−
1
22
X
T
12
>
0
.
3.
Ì
‡
(
J
•
(
X
Ú
(2
.
1)
Ú
(2
.
2)
Ó
Ú
,
b
±
e
^
‡
¤
á
µ
(
H
1
)
.
˜
f
i
(
x
)
´
Lipschitz
ë
Y
,
•
3
h
i
>
0
¦
|
˜
f
i
(
x
)
−
˜
f
i
(
y
)
|≤
h
i
|
x
−
y
|
,
∀
x,y
∈
R
,
x
6
=
y
,
i
= 1
,
2
,
···
,n.
…
h
=max
1
≤
i
≤
n
(
h
2
i
)
.
(
H
2
)
.τ
(
t
)
>
0
,
0
≤
˙
τ
(
t
)
≤
µ
≤
1,
é
?
¿
t
,
ù
p
µ
´
~
ê
.
(
H
3
)
.θ
ik
∈
R
Ú
θ
ik
∈
[0
,
2]
,i
= 1
,
2
,...,n.k
∈
Z
+
.
(
H
4
)
.
e
‚
5
Ý
Ø
ª
Γ
Q
(
|
η
|
+
|
β
|
)
(
|
η
|
+
|
β
|
)
T
QE
>
0
,
(3
.
1)
DOI:10.12677/pm.2021.1181741563
n
Ø
ê
Æ
Ü
t
‚
§
4
Ù
¥
Q
=
diag
(
q
i
)
,W
=
diag
(
ω
i
)
,B
=(
b
ij
)
n
×
n
,
|
η
|
=(
η
ij
)
n
×
n
,
|
β
|
=(
|
β
ij
|
)
n
×
n
,H
=
diag
(
h
i
)
,
Γ =
QD
+
QW
−
λQ
−
1
2
(
Q
|
B
|
H
+
H
|
B
T
|
Q
)
−
1
1
−
µ
H
T
H
,
λ>
0,
E
´
ü
Ý
.
½
n
3.1.
Ä
u
(
H
1
)
−
(
H
4
).
b
¯
ω
i
(
t
)
,
¯
b
ij
(
t
)
,
¯
η
ij
(
t
)
,
¯
β
ij
(
t
)
,ε
i
(
t
)(
i,j
= 1
,
2
,
···
,n
)
÷
v
:
˙
¯
ω
i
(
t
)=
1
γ
i
e
λt
e
i
(
t,x
)˜
u
i
(
t,x
)
,
˙
¯
b
ij
(
t
)=
−
1
α
ij
e
λt
e
i
(
t,x
)˜
g
j
(˜
u
j
(
t,x
))
,
˙
¯
η
ij
(
t
)=
−
1
%
ij
e
λt
|
e
i
(
t,x
)˜
g
j
[˜
u
j
(
t
−
τ
(
t
)
,x
)]
|
sgn
(¯
η
ij
(
t
)
−
η
ij
)
,
˙
¯
β
ij
(
t
)=
−
1
σ
ij
e
λt
|
e
i
(
t,x
)˜
g
j
[˜
u
j
(
t
−
τ
(
t
)
,x
)]
|
sgn
(
¯
β
ij
(
t
)
−
β
ij
)
,
˙
ε
i
(
t
)=
−
1
δ
i
e
λt
e
2
i
(
t,x
)
.
(3
.
2)
¿
…
ε
i
(
t
k
)=
ε
i
(
t
+
k
)
,
¯
ω
i
(
t
k
)=¯
ω
i
(
t
+
k
)
,
¯
b
ij
(
t
k
)=
¯
b
ij
(
t
+
k
)
,
¯
η
ij
(
t
k
)=¯
η
ij
(
t
+
k
)
,
¯
β
ij
(
t
k
)=
¯
β
ij
(
t
+
k
).
q
i
,
ω
i
,
δ
i
,γ
i
,α
ij
,%
ij
,σ
ij
´
?
¿
~
ê
,
K
X
Ú
(2.1)
Ú
(2.2)
Ó
Ú
,
¿
…
lim
t
→∞
(¯
ω
i
(
t
)) =
ω
i
,
lim
t
→∞
(
¯
b
ij
(
t
)) =
b
ij
,
lim
t
→∞
(¯
η
ij
(
t
)) =
η
ij
,
lim
t
→∞
(
¯
β
ij
(
t
)) =
β
ij
.
(3
.
3)
y
²
.
E
Lyapunov-krasovskii
¼
ê
V
(
t
)=
Z
Ω
{
1
2
n
X
i
=1
q
i
{
e
λt
e
2
i
(
t,x
)+
γ
i
(¯
ω
i
(
t
)
−
ω
i
)
2
+
n
X
j
=1
α
ij
(
¯
b
ij
(
t
)
−
b
ij
)
2
+
n
P
j
=1
%
ij
(¯
η
ij
(
t
)
−
η
ij
)
2
+
n
P
j
=1
σ
ij
(
¯
β
ij
(
t
)
−
β
ij
)
2
+
δ
i
ε
2
i
(
t
)
}
+
Z
t
t
−
τ
(
t
)
1
1
−
µ
e
λ
(
s
+
τ
)
|
f
T
[
e
(
s,x
)]
||
f
[
e
(
s,x
)]
|
ds
}
dx.
(3
.
4)
é
u
t
≥
0
,t
6
=
t
k
,
k
∈
Z
+
,
O
Ž
(3.4)
ê
,
˙
V
(
t
)=
Z
Ω
{
n
X
i
=1
q
i
{
e
λt
e
i
(
t,x
)
∂e
i
(
t,x
)
∂t
+
1
2
λe
λt
e
2
i
(
t,x
)+
γ
i
(¯
ω
i
(
t
)
−
ω
i
)
˙
¯
ω
i
(
t
)
+
n
P
j
=1
α
ij
(
¯
b
ij
(
t
)
−
b
ij
)
˙
¯
b
ij
(
t
)+
n
P
j
=1
%
ij
(¯
η
ij
(
t
)
−
η
ij
)
˙
¯
η
ij
(
t
)+
n
P
j
=1
σ
ij
(
¯
β
ij
(
t
)
−
β
ij
)
˙
¯
β
ij
(
t
)
+
δ
i
ε
i
(
t
)˙
ε
i
(
t
)
}
+
1
1
−
µ
e
λ
(
t
+
τ
)
|
f
T
[
e
(
t,x
)]
||
f
[
e
(
t,x
)]
|
−
1
−
˙
τ
(
t
)
1
−
µ
e
λ
(
t
−
τ
(
t
)+
τ
)
|
f
T
[
e
(
t
−
τ
(
t
)
,x
)]
||
f
[
e
(
t
−
τ
(
t
)
,x
)]
|}
dx
≤
Z
Ω
{
n
X
i
=1
q
i
{
e
λt
e
i
(
t,x
)
m
X
k
=1
∂
∂x
k
(
D
ik
∂e
i
(
t,x
)
∂x
k
)
−
ω
i
e
λt
e
2
i
(
t,x
)
+
n
P
j
=1
b
ij
e
λt
e
i
(
t,x
)
f
j
(
e
j
(
t,x
))+
λe
λt
e
2
i
(
t,x
)+
n
P
j
=1
|
η
ij
|
e
λt
e
i
(
t,x
)
|
f
j
[
e
j
(
t
−
τ
(
t
)
,x
)]
|
+
n
P
j
=1
|
β
ij
|
e
λt
e
i
(
t,x
)
|
f
j
[
e
j
(
t
−
τ
(
t
)
,x
)]
|}
+
1
1
−
µ
e
λ
(
t
+
τ
)
|
f
T
[
e
(
t,x
)]
||
f
[
e
(
t,x
)]
|
−
1
−
˙
τ
(
t
)
1
−
µ
e
λ
(
t
−
τ
(
t
)+
τ
)
|
f
T
[
e
(
t
−
τ
(
t
)
,x
)]
||
f
[
e
(
t
−
τ
(
t
)
,x
)]
|}
dx,
DOI:10.12677/pm.2021.1181741564
n
Ø
ê
Æ
Ü
t
‚
§
4
d
Ú
n
2
.
1
9
>
Š
^
‡
Z
Ω
e
i
(
t,x
)
m
X
k
=1
∂
∂x
k
D
ik
∂e
i
(
t,x
)
∂x
k
dx
=
Z
Ω
e
i
(
t,x
)
m
X
k
=1
∂
∂x
k
D
ik
∂e
i
(
t,x
)
∂x
k
dx
=
Z
Ω
e
i
(
t,x
)
∇
D
ik
∂e
i
(
t,x
)
∂x
k
m
k
=1
dx
=
Z
∂
Ω
e
i
(
t,x
)
D
ik
∂e
i
(
t,x
)
∂x
k
m
k
=1
dx
−
Z
Ω
D
ik
∂e
i
(
t,x
)
∂x
k
m
k
=1
∇
e
i
(
t,x
)
dx
=
−
Z
Ω
D
ik
∂e
i
(
t,x
)
∂x
k
m
k
=1
∇
e
i
(
t,x
)
dx
=
−
m
P
k
=1
Z
Ω
D
ik
∂e
i
(
t,x
)
∂x
k
2
dx
≤−
m
P
k
=1
Z
Ω
D
ik
w
2
k
e
2
i
(
t,x
)
dx,
Ù
¥
∇
= (
∂
∂x
1
,
∂
∂x
2
,...,
∂
∂x
m
)
T
´
F
Ý
Ž
f
,
¿
…
(
D
ik
∂e
i
(
t,x
)
∂x
k
)
m
k
=1
= (
D
i
1
∂e
i
(
t,x
)
∂x
1
,D
i
2
∂e
i
(
t,x
)
∂x
2
,...D
im
∂e
i
(
t,x
)
∂x
m
)
.
Š
â
(
H
1
)
Ú
Ú
n
2
.
2,
·
‚
k
˙
V
(
t
)
≤
Z
Ω
{−
n
X
i
=1
q
i
(
m
X
k
=1
D
ik
w
2
k
+
ω
i
)
e
λt
e
2
i
(
t,x
)+
n
X
i
=1
n
X
j
=1
e
λt
q
i
|
e
i
(
t,x
)
||
b
ij
||
f
j
(
e
j
(
t,x
))
|
+
n
P
i
=1
q
i
λe
λt
e
2
i
(
t,x
)+
n
P
i
=1
n
P
j
=1
e
λt
q
i
|
e
i
(
t,x
)
||
η
ij
||
f
j
[
e
j
(
t
−
τ
(
t
)
,x
)]
|
+
n
P
i
=1
n
P
j
=1
e
λt
q
i
|
e
i
(
t,x
)
||
β
ij
||
f
j
[
e
j
(
t
−
τ
(
t
)
,x
)]
|
+
1
1
−
µ
e
λ
(
t
+
τ
)
|
f
T
[
e
(
t,x
)]
||
f
[
e
(
t,x
)]
|
−
1
−
˙
τ
(
t
)
1
−
µ
e
λ
(
t
−
τ
(
t
)+
τ
)
|
f
T
[
e
(
t
−
τ
(
t
)
,x
)]
||
f
[
e
(
t
−
τ
(
t
)
,x
)]
|}
dx
≤
Z
Ω
{−
e
λt
e
T
(
t,x
)(
QD
+
QW
)
e
(
t,x
)+
e
λt
n
X
i
=1
n
X
j
=1
q
i
|
e
i
(
t,x
)
||
b
ij
||
h
i
||
e
j
(
t,x
)
|
+
λe
λt
e
T
(
t,x
)
Qe
(
t,x
)+
e
λt
|
e
T
(
t,x
)
|
Q
(
|
η
|
+
|
β
|
)
|
f
[
e
(
t
−
τ
(
t
)
,x
)]
|
−
1
−
˙
τ
(
t
)
1
−
µ
e
λ
(
t
−
τ
(
t
)+
τ
)
|
f
T
[
e
(
t
−
τ
(
t
)
,x
)]
||
f
[
e
(
t
−
τ
(
t
)
,x
)]
|
+
1
1
−
µ
e
λ
(
t
+
τ
)
|
f
T
[
e
(
t,x
)]
||
f
[
e
(
t,x
)]
|}
dx,
DOI:10.12677/pm.2021.1181741565
n
Ø
ê
Æ
Ü
t
‚
§
4
d
(
H
2
)
Ú
Ú
n
2
.
3,
Œ
˙
V
(
t
)
≤
Z
Ω
{−
e
λt
e
T
(
t,x
)(
QD
+
QW
)
e
(
t,x
)+
e
λt
|
e
T
(
t,x
)
|
1
2
(
Q
|
B
|
H
+
H
|
B
T
|
Q
)
|
e
(
t,x
)
|
+
λe
λt
e
T
(
t,x
)
Qe
(
t,x
)+
1
2
e
λt
|
e
T
(
t,x
)
|
Q
(
|
η
|
+
|
β
|
)(
|
η
|
+
|
β
|
)
T
Q
T
|
e
(
t,x
)
|
+
1
2
e
λt
f
T
[
e
(
t
−
τ
(
t
)
,x
)]
||
f
[
e
(
t
−
τ
(
t
)
,x
)]
|
−
1
−
˙
τ
(
t
)
1
−
µ
e
λ
(
t
−
τ
(
t
)+
τ
)
|
f
T
[
e
(
t
−
τ
(
t
)
,x
)]
||
f
[
e
(
t
−
τ
(
t
)
,x
)]
|
+
1
1
−
µ
e
λ
(
t
+
τ
)
|
f
T
[
e
(
t,x
)]
||
f
[
e
(
t,x
)]
|}
dx
≤
Z
Ω
e
λt
{−|
e
T
(
t,x
)
|{
QD
+
QW
−
λQ
−
1
2
(
Q
|
B
|
H
+
H
|
B
T
|
Q
)
−
1
2
Q
(
|
η
|
+
|
β
|
)(
|
η
|
+
|
β
|
)
T
Q
T
−
1
1
−
µ
H
T
H
}|
e
(
t,x
)
|}
dx
=
−
Ke
λt
k
e
i
(
t,x
)
k
2
2
≤
0
,
Ù
¥
K
=
QD
+
QW
−
λQ
−
1
2
(
Q
|
B
|
H
+
H
|
B
T
|
Q
)
−
1
2
Q
(
|
η
|
+
|
β
|
)(
|
η
|
+
|
β
|
)
T
Q
T
−
1
1
−
µ
H
T
H.
Ï
L
Ú
n
2
.
4
,
K>
0
.
,
k
V
(
t
)
≤−
Ke
λt
k
e
i
(
t,x
)
k
2
2
≤
0
.
Ï
d
V
(
t
)
≤
V
(
t
+
k
−
1
)
é
?
¿
t
∈
(
t
k
−
1
,t
k
],
k
∈
Z
+
,
ù
p
V
(0
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) =
V
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d
(2
.
3)(3
.
2)(3
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(
H
3
),
Œ
•
V
(
t
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k
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1
2
n
X
i
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q
i
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e
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k
e
2
i
(
t
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k
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)+
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i
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t
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X
j
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ij
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¯
b
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t
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k
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b
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P
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t
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k
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σ
ij
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β
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ij
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ε
2
i
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k
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}
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t
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k
t
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t
k
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1
1
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µ
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λ
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s
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τ
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|
f
T
[
e
(
s,x
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||
f
[
e
(
s,x
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|
ds
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dx
=
Z
Ω
{
1
2
n
X
i
=1
q
i
{
e
λt
k
(1
−
θ
ik
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2
e
2
i
(
t
k
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)+
γ
i
(¯
ω
i
(
t
k
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−
ω
i
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2
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n
X
j
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α
ij
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b
ij
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t
k
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−
b
ij
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n
P
j
=1
%
ij
(¯
η
ij
(
t
k
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−
η
ij
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2
+
n
P
j
=1
σ
ij
(
¯
β
ij
(
t
k
)
−
β
ij
)
2
+
δ
i
ε
2
i
(
t
k
)
}
+
Z
t
k
t
k
−
τ
(
t
k
)
1
1
−
µ
e
λ
(
s
+
τ
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|
f
T
[
e
(
s,x
)]
||
f
[
e
(
s,x
)]
|
ds
−
Z
t
+
k
−
τ
(
t
k
)
t
k
−
τ
(
t
k
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1
1
−
µ
e
λ
(
s
+
τ
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|
f
T
[
e
(
s,x
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||
f
[
e
(
s,x
)]
|
ds
+
Z
t
+
k
t
k
1
1
−
µ
e
λ
(
s
+
τ
)
|
f
T
[
e
(
s,x
)]
||
f
[
e
(
s,x
)]
|
ds
}
dx
≤
V
(
t
k
)
.
Ï
d
,
X
Ú
(2
.
1)
Ú
(2
.
2)
´
Ó
Ú
.
½
n
3
.
1
y
.
.
í
Ø
3.1.
Ä
u
½
n
2
.
1
^
‡
,
X
J
ε
i
(0)=0
,
¯
ω
i
(0)=
ω
i
,
¯
b
ij
(0)=
b
ij
,
¯
η
ij
(0)=
η
ij
,
¯
β
ij
(0)=
β
ij
.
K
X
Ú
(2
.
1)
Ú
(2
.
2)
Û
•
ê
Ó
Ú
.
5
3.1.
θ
ik
≡
0
,i
= 1
,
2
,...,n.k
∈
Z
+
,
X
Ú
(2
.
1)
Ú
(2
.
2)
ò
z
•©
z
[3,4]
¥
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ó
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ò
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ä
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.
DOI:10.12677/pm.2021.1181741566
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4
4.
ê
Š
[
~
4.1.
3
(2.1)
ª
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D
ik
= 0
§
i
= 1
,
2.
(
ω
ij
)
2
×
2
=
10
01
!
,
(
b
ij
)
2
×
2
=
2
.
0
−
0
.
1
−
5
.
02
.
8
!
,
(
β
ij
)
2
×
2
= (
η
ij
)
2
×
2
=
−
1
.
5
−
0
.
1
−
0
.
21
−
1
.
8
!
,τ
(
t
) =
e
t
1+
e
t
,
˜
g
j
(
t
) =
tanh
(
t
)
.
Ð
©
^
‡
X
e
µ
u
1
(
t
) =
−
0
.
01
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2
(
t
) = 0
.
05
,
˜
u
1
(
t
) =
−
0
.
5
,
˜
u
2
(
t
) = 0
.
5
,
ε
1
(0) =
−
0
.
1
, ε
2
(0) = 0
.
02;
b
1
(0) =
−
0
.
02
, b
2
(0) = 4
.
46
,
η
1
(0) =
−
0
.
02
, η
2
(0) = 0
.
01;
β
1
(0) = 0
.
2
, β
2
(0) =
−
0
.
2
.θ
ik
= 0
.
1
.
X
Ú
ë
ê
ì
C
5
Æ
˙
¯
b
1
(
t
)=
−
9
.
43
×
(˜
u
1
−
u
1
)
tanh
(˜
u
1
)
,
˙
¯
b
2
(
t
) =
−
2
.
173
×
(˜
u
2
−
u
2
)
tanh
(˜
u
2
)
.
˙
¯
η
1
(
t
)=
−
6
.
65
×
sgn
(¯
η
1
+1
.
5)
|
(˜
u
1
−
u
1
)
tanh
(˜
u
1
(
t
−
τ
))
|
,
˙
¯
η
2
(
t
)=
−
1
.
45
×
sgn
(¯
η
2
+1
.
8)
|
(˜
u
2
−
u
2
)
tanh
(˜
u
2
(
t
−
τ
))
|
.
˙
¯
β
1
(
t
)=
−
9
.
22
×
sgn
(
¯
β
1
+1
.
5)
|
(˜
u
1
−
u
1
)
tanh
(˜
u
1
(
t
−
τ
))
|
,
˙
¯
β
2
(
t
)=
−
2
.
5
×
sgn
(
¯
β
2
+1
.
8)
|
(˜
u
2
−
u
2
)
tanh
(˜
u
2
(
t
−
τ
))
|
.
˙
ε
1
(
t
)=
−
0
.
5
×
(˜
u
1
−
u
1
)
2
,
˙
ε
2
(
t
) =
−
0
.
35
×
(˜
u
2
−
u
2
)
2
.
Ï
L
ã
1
Ú
ã
2
Œ
•
,
X
Ú
(2
.
1)
Ú
(2
.
2)
Ó
Ú
,
¿
…
X
Ú
(2.2)
X
ê
,
¯
b
1
,
¯
b
2
,¯
η
1
,
¯
η
2
,
¯
β
1
,
¯
β
2
,
©
O
ì
C
ª
•
u
2.0,2.8,-1.5,-1.8,-1.5,-1.8.
Figure1.
Synchronizationerrors
e
1
(
t
)and
e
2
(
t
)
ã
1.
Ø
Ó
Ú
e
1
(
t
)
Ú
e
2
(
t
)
DOI:10.12677/pm.2021.1181741567
n
Ø
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Æ
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t
‚
§
4
Figure2.
Asymptoticbehaviorofsystem(2
.
2)parameters
ã
2.
X
Ú
(2
.
2)
X
ê
ì
C
ã
”
Ä
7
‘
8
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ó
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d
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g
,
‰
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(No.12071302)
|
±
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©
z
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