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PureMathematicsnØêÆ,2022,12(9),1493-1500
PublishedOnlineSeptember2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.129163
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T‘XÚØÄ:•35"Ùg§|^‚5Ü©AŠ†-½5'XØÄ:a.!
V-ØÄ:-½5±9ƒAëê^‡"•§(Ü¥%6/½nÚ©n؃'•£?Ø
šV-ØÄ:©y–§l)flip©^‡"
'…c
ÄåÆ1•§ØÄ:§¥%6/½n§©nاFlip©
AnalysisofaClassofDifferenceEquation
ModelswithExponentialTerm
RongWang
SchoolofMathematicalSciences,ChongqingNormalUniversity,Chongqing
Received:Aug.17
th
,2022;accepted:Sep.15
th
,2022;published:Sep.22
nd
,2022
Abstract
Inthispaper,wemainlystudythedynamicbehaviorofaclassoftwo-dimensional
©ÙÚ^:N.˜a¹•ê‘©•§.©Û[J].nØêÆ,2022,12(9):1493-1500.
DOI:10.12677/pm.2022.129163
N
differenceequationmodelswithexponentialterms.Throughcalculation,wefirstly
givetheexistenceofthefixedpointofthistwo-dimensionalsystem.Secondly,the
typeoffixedpoint,thestabilityofthehyperbolicfixedpointandthecorresponding
parameterconditionsareobtainedbyusingtherelationshipbetweentheeigenvalues
ofthelinearpartandthestability.Finally,thebifurcationphenomenonofnon-
hyperbolicfixedpointisdiscussedbycombiningtheknowledgeofcentermanifold
theoremandbifurcationtheory,andtheconditionsforgeneratingflipbifurcationare
obtained.
Keywords
DynamicBehavior,FixedPoint,CenterManifoldTheorem,BifurcationTheory,Flip
Bifurcation
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
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ê‘©•§)k.5!ØC5!Û1•?1ïħ©z[6–8]鹕ê‘©•§©
y–?1ïÄ"´§3ùƒ'©Ù¥§du©•§|¥¹k•ê‘éŒU—•§|
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|kš"²ï)§ÙÄåÆ1•´Ä¬•\´Lºu´§©ò•Ä˜a¹•ê‘©•§|



x
n+1
= ax
n−1
+bx
n
e
−y
n
,
y
n+1
= cy
n−1
+dy
n
e
−x
n
,
(1.1)
ÄåÆ1•§ ùp§n•šKê§ëêa,b,c,d´~ê§Ð©Šx
0
,x
−1
,y
0
,y
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´šK¢ê"·
‚•§XÚ(1.1)½Â˜‡²¡NF: R
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+
F(x,y) −→(ax+bxe
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)
T
,(1.2)
ùp§R
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:= [0,+∞)¿…(x,y)
T
P••þ(x,y)=˜"
DOI:10.12677/pm.2022.1291631494nØêÆ
N
2.̇(J
Äk§·‚‰ÑXÚ(1.2)ØÄ:•35Ú¤kØÄ:a.!V-ØÄ:-½5±9ƒAë
ê^‡"
½n1.XÚ(1.2)–õkü‡ØÄ:E
0
(0,0), E
1
(ln
d
1−c
,ln
b
1−a
)§Ù¥E
1
…=0 <a<1,0 <
c<1¤áž•3"L1‰ÑØÄ:E
0
½55Ÿ"éuØÄ:E
1
ó§÷v^‡a+b=1½
öc+d= 1½ö(1−a)(1−c)ln
b
1−a
ln
d
1−c
= 4ž§§´šV-¶0 <a+b<1,0 <c+d<1(or
a+b>1,c+d>1),…(1−a)(1−c)ln
b
1−a
ln
d
1−c
6= 4ž§§´˜‡Q:¶0 <a+b<1,c+d>1(or
a+b>1,0 <c+d<1)ž§§´˜‡Ø-½:"
Table1.QualitativepropertiesofthefixedpointE
0
L1.ØÄ:E
0
½55Ÿ
^‡a.
0 <c+d<1-½(:
0 <a+b<1c+d= 1šV-
c+d>1Q:
a+b= 1c+d>0šV-
0 <c+d<1Q:
a+b>1c+d= 1šV-
c+d>1Ø-½(:
y²µŠâN(1.2)§ØÄ:÷v±e'X
x= ax+bxe
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,y= cy+dye
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.
OŽŒ•XÚ(1.2)–õkü‡ØÄ:E
0
(0,0),E
1
(ln
d
1−c
,ln
b
1−a
)§Ù¥E
1
…=0<a<1,
0 <c<1¤áž•3"·‚^J(x,y)L«XÚ(1.2)3(x,y)?äŒ'Ý§K
J(x,y) =



a+be
−y
−bxe
−y
−dye
−x
c+de
−x



.
3ØÄ:E
0
(0,0)?§XÚ(1.2)äŒ'Ý•
J(0,0) =



a+b0
0c+d



,
éAü‡AŠ©O•a+b,c+d§dAŠ†-½5'XŒL1¥(Ø"
DOI:10.12677/pm.2022.1291631495nØêÆ
N
3ØÄ:E
1
(ln
d
1−c
,ln
b
1−a
)?§XÚ(1.2)äŒ'Ý
J(ln
d
1−c
,ln
b
1−a
) =




1(a−1)ln
d
1−c
(c−1)ln
b
1−a
1




,
éAü‡AŠ©O•
λ
1
= 1−
r
(1−a)(1−c)ln
b
1−a
ln
d
1−c
,λ
2
= 1+
r
(1−a)(1−c)ln
b
1−a
ln
d
1−c
.
dAŠ†-½5'XŒ•§a+b=1½öc+d=1ž§λ
1,2
=1,KØÄ:E
1
´šV-¶
(1−a)(1−c)ln
b
1−a
ln
d
1−c
=4ž§λ
1
=−1,λ
2
=3§KØÄ:E
1
´šV-",˜•¡§éu
V-œ/µ
(1)XJ0<a+b<1,0<c+ d<1,…(1 −a)(1−c)ln
b
1−a
ln
d
1−c
6=4,Kln
b
1−a
ln
d
1−c
>0,
=(1−a)(1−c)ln
b
1−a
ln
d
1−c
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λ
1
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1−a
ln
d
1−c
,
u´§λ
1
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1
6= −1),λ
2
>1KE
1
´Q:"
(2)XJ0 <a+b<1,c+d>1,Kln
b
1−a
ln
d
1−c
<0,=(1−a)(1−c)ln
b
1−a
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1−c
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λ
1
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b
1−a
ln
d
1−c
,λ
2
= 1+i
r
−(1−a)(1−c)ln
b
1−a
ln
d
1−c
,
ddŒ„,λ
1,2
•þŒu1§Ïd§E
1
´˜‡Ø-½:"
(3)XJa+b>1,0 <c+d<1,Ó(2)§E
1
´˜‡Ø-½:"
(4)XJa+b>1,c+d>1,…(1−a)(1−c)ln
b
1−a
ln
d
1−c
6= 4,Ó(1)§KE
1
´Q:"y."
3þ¡L§¥§·‚©ÛØÄ:•3œ¹±9V-ØÄ:k'5Ÿ§e¡·
‚m©?ØšV-ØÄ:©y–§±9)©ëê^‡"•Äd=d
0
+ε
0
,…d
0
÷
v(1−a)(1−c)ln
b
1−a
ln
d
0
1−c
= 4,Ù¥§ε
0
Š•©ëê"P
ρ
1
=
1
2
(a+(1−a)(
−1
2
+c)ln
d
0
1−c
)
2
+
(a−1)
2
4
·(1+(−
1
2
+c)ln
d
0
1−c
)ln
d
0
1−c
,
ρ
2
=
1
d
0
ln
d
0
1−c
·
−1−a+c(1−a)ln
d
0
1−c
2
.
½n2.XJd=d
0
+ε
0
Ú(1−a)(1 −c)ln
b
1−a
ln
d
0
1−c
=4,ε
0
¿©§…ρ
1
6=0,ρ
2
6=0§KX
Ú(1.2)3ØÄ:E
1
(ln
d
1−c
,ln
b
1−a
)NC¬u)flip©"
DOI:10.12677/pm.2022.1291631496nØêÆ
N
y²µ•òXÚ(1.2)ØÄ:E
1
£:§·‚ŠC†



x= ln
d
1−c
+ξ,
y= ln
b
1−a
+η,
XÚ(1.2)C•µ

ξ
η

−→

aξ+(1−a)(ξ+ln
d
1−c
)e
−η
+(a−1)ln
d
1−c
cη+(1−c)(η+ln
b
1−a
)e
−ξ
+(c−1)ln
b
1−a

.(2.1)
••B§·‚ò(2.1)m>UVúªÐm
aξ+(1−a)(ξ+ln
d
1−c
)e
−η
+(a−1)ln
d
1−c
=aξ+(1−a)ξ+(a−1)ξη+(a−1)ln
d
1−c
η+
1−a
2
ln
d
1−c
η
2
+O(3).
ùp§O(3)L«êŒu2‘"Ón§
cη+(1−c)(η+ln
b
1−a
)e
−ξ
+(c−1)ln
b
1−a
=cη+(1−c)η+(c−1)ξη+(c−1)ln
b
1−a
ξ+
1−c
2
ln
b
1−a
ξ
2
+O(3).
=XÚ(2.1)du

ξ
η

−→

ξ+(a−1)ξη+(a−1)ln
d
1−c
η+
1−a
2
ln
d
1−c
η
2
η+(c−1)ξη+(c−1)ln
b
1−a
ξ+
1−c
2
ln
b
1−a
ξ
2

+h.o.t.,(2.2)
y3•Äd= d
0
+ε
0
,…(1−a)(1−c)ln
b
1−a
ln
d
0
1−c
=4,Ù¥§ε
0
Š•©ëê§ε
0
=0•©Š"
dž§N(2.2)¤•

ξ
η

−→



1(a−1)ln
d
0
1−c
(c−1)ln
b
1−a
1




ξ
η

+

f(ξ,η,ε
0
)
g(ξ,η,ε
0
)

,(2.3)
ùp§
f(ξ,η,ε
0
) = (a−1)ξη+(a−1)ln
d
0
+ε
0
d
0
η+
1−a
2
ln
d
0
+ε
0
1−c
η
2
+O(3),
g(ξ,η,ε
0
) = (c−1)ξη+
1−c
2
ln
b
1−a
ξ
2
+O(3).
AŠλ
1
= −1,λ
2
= 3,éAA•þ•(
1−a
2
ln
d
0
1−c
,1)
T
,(
a−1
2
ln
d
0
1−c
,1)
T
,Š‚5C†
(ξ,η)
T
= H(u,v)
T
,
DOI:10.12677/pm.2022.1291631497nØêÆ
N
KXÚ(2.3)C•µ

u
v

−→



−10
03




u
v

+H
−1

e
f(u,v,ε
0
)
eg(u,v,ε
0
)

,(2.4)
ùp§
H=



1−a
2
ln
d
0
1−c
a−1
2
ln
d
0
1−c
11



,
H
−1
=



1
(1−a)ln
d
0
1−c
1
2
−1
(1−a)ln
d
0
1−c
1
2



,
e
f(u,v,ε
0
) = (a−1)ln
d
0
+ε
0
d
0
(u+v)+(
1−a
2
ln
d
0
+ε
0
1−c
−
(1−a)
2
2
ln
d
0
1−c
)u
2
+(
1−a
2
ln
d
0
+ε
0
1−c
+
(1−a)
2
2
ln
d
0
1−c
)v
2
+(1−a)ln
d
0
+ε
0
1−c
uv,
eg(u,v,ε
0
) = (1−a)(−
1
2
+c)ln
d
0
1−c
u
2
+(1−a)(
3
2
−c)ln
d
0
1−c
v
2
+(a−1)ln
d
0
1−c
uv,
éA]!XÚ






u
ε
0
v






−→






−u
ε
0
3v






+






b
f(u,v,ε
0
)
0
bg(u,v,ε
0
)






(2.5)
ùp§
b
f(u,v,ε
0
) =
1
(1−a)ln
d
0
1−c
e
f(u,v,ε
0
)+
1
2
eg(u,v,ε
0
),
bg(u,v,ε
0
) =
−1
(1−a)ln
d
0
1−c
e
f(u,v,ε
0
)+
1
2
eg(u,v,ε
0
).
N(2.5)3:NCk˜‡‘¥%6/
v= h(u,ε
0
) = h
1
u
2
+h
2
uε
0
+h
3
ε
2
0
+O(3),
Šâ©z[9]Œ•§ùXêh
1
,h
2
,h
3
ûu•§
ℵ(h(u,ε
0
)) = h(−u+
b
f(u,h(u,ε
0
),ε
0
),ε
0
)−3h(u,ε
0
)−bg(u,h(u,ε
0
),ε
0
) = 0,
DOI:10.12677/pm.2022.1291631498nØêÆ
N
'u
2
,uε
0
,ε
2
0
X꧷‚Œ±
h
1
=
1
4
(a−1)(1+(−
1
2
+c)ln
d
0
1−c
),h
2
=
−1
4d
0
ln
d
0
1−c
,h
3
= 0.
u´§
v= h(u,ε
0
) =
1
4
(a−1)(1+(−
1
2
+c)ln
d
0
1−c
)·u
2
−
1
4d
0
ln
d
0
1−c
·uε
0
+O(3),(2.6)
Pp=
ln(d
0
+ε
0
)−lnd
0
lnd
0
−ln(1−c)
,òªf(2.6)“\XÚ(2.5)1˜‡•§§

u
ε
0

−→

−u
ε
0

+

f
∗
(u,ε
0
)
0

+h.o.t.,(2.7)
ùp§
f
∗
(u,ε
0
)=−pu+(−ph
1
+
1
2
p+
a−1
2
+
1−a
8d
0
ε
0
+
(1−a)(
−1
2
+c)ln
d
0
1−c
2
)u
2
+
p
4d
0
ln
d
0
1−c
uε
0
+
a−1
2
ln
d
0
1−c
h
1
u
3
,
XÚ(2.7)½Â˜‡˜‘Nu7→φ
ε
0
(u),ùp
φ
ε
0
(u) = −u+f
∗
(u,ε
0
)+h.o.t..(2.8)
OŽŒ
ρ
1
=[
1
2
(
∂
2
φ
ε
0
∂u
2
)
2
+
1
3
(
∂
3
φ
ε
0
∂u
3
)]|
(0,0)
=
1
2
(a+(1−a)(
−1
2
+c)ln
d
0
1−c
)
2
+
(a−1)
2
4
·(1+(−
1
2
+c)ln
d
0
1−c
)ln
d
0
1−c
,
ρ
2
=[
∂φ
ε
0
∂ε
0
∂
2
φ
ε
0
∂u
2
+2
∂
2
φ
ε
0
∂u∂ε
0
]|
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