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PureMathematicsnØêÆ,2023,13(6),1888-1896
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.136193
˜a6ÄgŒ_XÚAbelÈ©þ.
O
ÓÓÓŠŠŠ
1∗
§§§|||uuu
1
§§§ÁÁÁwww
2
1
µ>bŒÆ&Eó§Æ§ôܵ
2
2ù{HS¥Æ§ôܵ
ÂvFϵ2023c522F¶¹^Fϵ2023c623F¶uÙFϵ2023c630F
Á‡
|^Riccati•§{§ïÄ˜aº‚1/ªgŒ_XÚ(r9)3?¿3,2,1gõ‘ª6Äe
AbelÈ©á":‡êþ."( J•:33,2,1gõ‘ ª6Äeþ.´13"ù(J´
éƒc(JU?"
'…c
gŒ_XÚ§AbelÈ©§4•‚§Riccati•§
UpperBoundEstimationofAbelian
IntegralforaClassofPerturbed
QuadraticReversibleSystems
YuangenZhan
1∗
,LihuaYang
1
,LiyunZhu
2
1
Scho olofInformationandEngineering,JingdezhenCeramicUniversity,JingdezhenJiangxi
2
FuliangCountyNan’anMiddleSchool,JingdezhenJiangxi
Received:May22
nd
,2023;accepted:Jun.23
rd
,2023;published:Jun.30
th
,2023
∗ÏÕŠö"
©ÙÚ^:ÓŠ,|u,Áw.˜a6ÄgŒ_XÚAbelÈ©þ.O[J].nØêÆ,2023,13(6):
1888-1896.DOI:10.12677/pm.2023.136193
ÓŠ
Abstract
ByusingRiccatiequationmethod,theupperboundestimationofthenumberof
zerosofAbelianintegralforaclassofquadraticreversiblesystem(r9)ofgenusone
underanypolynomialperturbationofdegree3,2,1isstudied.Theresultisthatthe
upperb oundis13underpolynomialperturbationofdegree3,2,1.Theseresultsarean
improvementofthepreviousresults.
Keywords
QuadraticReversibleSystems,AbelianIntegral,LimitCycles,RiccatiEquation
Copyright
c
2023byauthor(s)andHansPublishersInc.
ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.ÚóÚ̇(Ø
•Ä6ĺ‚1/ªgŒ_XÚ
˙x=
H
y
(x,y)
M(x,y)
+εp(x,y)˙y=−
H
x
(x,y)
M(x,y)
+εq(x,y),(1)
Ù¥ε(0<ε1)´˜‡¢ê,
H
y
(x,y)
M(x,y)
,
H
x
(x,y)
M(x,y)
,p(x,y),q(x,y)Ñ´'ux,yõ‘ª,¿…
max

deg

H
y
(x,y)
M(x,y)

,deg

H
x
(x,y)
M(x,y)

=2,max{deg(p(x,y)),deg(q(x,y))}=n(n=1,2,3).
ε=0,XÚ(1)´˜‡gŒ_XÚ,…´˜‡ŒÈXÚ,§k˜‡¥%.¼êH(x,y)´Ù‘
kÈ©ÏfM(x,y)˜‡ÄgÈ©,•Ò´`,Œ±½Â˜‡ëY±Ï‚•
{Γ
h
}⊂

(x,y)∈R
2
:H(x,y)=h,h∈4

,
§‚´½Â3•Œm«m4=(h
1
,h
2
)þ.©Ì‡)û¯K´,éu?¿êε,XÚ(1)Œ
±l±Ï‚•{Γ
h
}¥©|Ñõ‡4•‚?3±Ï;?Û;—«•¥,XÚ(1)4•‚‡ê
DOI:10.12677/pm.2023.1361931889nØêÆ
ÓŠ
؇L±eAbelÈ©I(h)á":‡ê(„©z[1–4]).
I(h)=
I
Γ
h
M(x,y)[q(x,y)dx−p(x,y)dy],h∈4.(2)
©[5]•k¦^Riccati•§{ïÄXÚ(1)AbelÈ©":‡êþ..©[6]òº‚1/ª
gŒ_XÚ©¤22a,äN•(r1)-(r22).¦^Riccati•§•{:©[7]ïÄn
žM—îXÚ(r1),(r2)":‡êþ.¯K;©[8]ïÄXÚ(r3)-(r6);©[9–13]ïÄXÚ
(r9)-(r13)9(r16)-(r22).
©2gïÄXÚr(9),˜#(J(„L1).
Table1.Comparisonbetweenthenewresultsandtheoriginalresults
L1.#(J†(Jé'
XÚn#(J(J
r931339
r921339
r911339
•ÄXÚ(r9)
˙x=−xy,˙y=−
2
3
y
2
+
1
3
2
·2
4
x
2
−
1
3
2
·2
4
x.(3)
§ÄgÈ©•
H(x,y)=x
−
4
3
(
1
2
y
2
+
1
3·2
5
x
2
+
1
3·2
4
)=h,h∈(
1
2
5
,+∞),(4)
‘k˜‡È©ÏfM(x,y)=x
−
7
3
.
XÚ(r9)´˜‡ŒÈšM—îgXÚ,ÙA¤k;Ñ´8g-‚,§k˜‡¥%
(1,0),˜^È©-‚x=0,˜x±Ï;{Γ
h
}(
1
2
5
<h<+∞).
©z[10]‰ÑXÚ(r9)AbelÈ©þ.Xe½n.
½n1.1[10]éu?¿ngõ‘ªp(x,y)Úq(x,y),XÚ(r9)AbelÈ©I(h)á":
‡êþ.‚5•6un.äNœ ¹•:n>6ž,þ.•15

n
2

+6

n−1
2

−3;n=1,2,3,4,5
ž,þ.•39.
©Ì‡(J•Xe½n.
½n1.2éu?¿ngõ‘ªp(x,y)Úq(x,y)(n=1,2,3),XÚ(r9)AbelÈ©I(h)
á":‡êþ.•:n=3,2,1ž,þ.´13.
#(Jé(J´˜U?(„L1).
©¡Ü©(•:1Ü©,¼AbelÈ©I(h){üL«•{.1nÜ©,ïÄXÚ
(r9)AbelÈ©I(h)†J
2
3
(h)ƒm'X,¼ƒ'Riccati•§.1oÜ©,¦^Riccati•§•
DOI:10.12677/pm.2023.1361931890nØêÆ
ÓŠ
{y²½n1.2.1ÊÜ©,‰Ñ˜‡{á(Ø.
2.AbelÈ©I(h){üL«
bp(x,y)=
P
0≤i+j≤3
a
i,j
x
i
y
j
Úq(x,y)=
P
0≤i+j≤3
b
i,j
x
i
y
j
Ñ´?¿õ‘ª,d(2)ª•,½n
1.2¥AbelÈ©I(h)kXe/ª.
I(h)=
I
Γ
h
x
−
7
3
X
0≤i+j≤3
b
i,j
x
i
y
j
dx−
X
0≤i+j≤3
a
i,j
x
i
y
j
dy
!
,(5)
•{',P
I
i,j
(h)=
I
Γ
h
x
i−
7
3
y
j
dx,
Ù¥i=−1,0,1,2,3;j=0,1,2,3,4.j=1,PI
i,j
(h)=J
i
(h)
du
I
Γ
h
x
i−
7
3
y
j
dy=
H
Γ
h
x
i−
7
3
dy
j+1
j+1
=−
i−
7
3
j+1
I
Γ
h
x
i−
7
3
−1
y
j+1
dx=−
i−
7
3
j+1
I
i−1,j+1
(h).
Ïd,d(5)ª•,Œ±òI(h)U•
I(h)=
X
0≤i≤3,
0≤j≤3,
0≤i+j≤3
b
i,j
I
i,j
(h)+
i−
7
3
j+1
X
0≤i≤3,
0≤j≤3,
0≤i+j≤3
a
i,j
I
i−1,j+1
(h)=
X
−1≤i≤3,
0≤j≤4,
0≤i+j≤3
e
i,j
I
i,j
(h),(6)
Ù¥e
i,j
=b
i,j
+
3i−4
3j
a
i+1,j−1
,a
i,−1
=0(i=1−4),b
−1,j
=0(j=0−4).
du±Ï;Γ
h
'ux¶é¡,Ïdj•óêž,I
i,j
(h)=0,•I•Äj•Ûêœ¹.
=
I(h)=e
−1,1
J
−1
(h)+e
0,1
J
0
(h)+e
1,1
J
1
(h)+
e
2,1
J
2
(h)+e
−1,3
I
−1,3
(h)+e
0,3
I
0,3
(h),
(7)
Ù¥e
−1,1
=−
7a
0,0
3
,e
0,1
=b
0,1
−
4a
1,0
3
,e
1,1
=b
1,1
−
a
2,0
3
,e
2,1
=b
2,1
+
2a
3,0
3
,e
−1,3
=−
7a
0,2
9
,
e
0,3
=b
0,3
−
4a
1,2
9
.
©z[10]¥(18)Ú(20)ªXeµ
I
m,j
(h)=
2jA
3m+2j−4
[I
m+2,j−2
(h)−I
m+1,j−2
(h)],(8)
DOI:10.12677/pm.2023.1361931891nØêÆ
ÓŠ
A(3m+5)J
m+2
(h)=(3m+2)hJ
m−
1
2
(h)−A(6m+1)J
m+1
(h),(9)
Ù¥A=
1
3·2
5
.
3(8)¥,-(m,j)=(−1,3),(0,3),Œ
I
−1,3
(h)=6AJ
0
(h)−6AJ
1
(h),(10)
I
0,3
(h)=3AJ
2
(h)−3AJ
1
(h).(11)
d(7)(10)(11)ªŒ
I(h)=p
−1
J
−1
(h)+p
0
J
0
(h)+p
1
J
1
(h)+p
2
J
2
(h),(12)
Ù¥p
−1
=e
−1,1
,p
0
=e
0,1
+6Ae
−1,3
,p
1
=e
1,1
−6Ae
−1,3
−3Ae
0,3
,p
2
=e
2,1
+3Ae
0,3
.
2d(9)ª,Œ
J
0
(h)=
1
5A
hJ
1
3
(h)+
2
5
J
1
(h),(13)
J
2
(h)=
2
5A
hJ
1
3
(h)−
1
5
J
1
(h),(14)
J
−1
(h)=(
2
231A
3
h
3
−
1
11
)[
1
5
hJ
1
3
(h)+
2
5
J
1
(h)]+
1
231A
2
h
2
J
2
3
(h).(15)
d(12)(13)(14)(15)ª,Œ
I(h)=α(h)J
1
3
(h)+β(h)J
2
3
(h)+γ(h)J
1
(h),(16)
Ù¥α(h)=
1
5A
h(a
1
h
3
+b
1
),β(h)=a
2
h
2
,γ(h)=a
3
h
3
+b
3
,a
1
=
2
231A
3
,b
1
=−
p
−1
11
+p
0
+p
2
,
a
2
=
p
−1
231A
2
,a
3
=
4p
−1
5·231A
3
,b
3
=−
2p
−1
55
+
2
5
p
0
−
1
5
p
2
.
3.Riccati•§
!¥,̇ïÄAbelÈ©I(h)ÚJ
2
3
(h)ƒm'X,Riccati•§.
©z[10]¥,éuXÚr(9),kXeÚn(Ø.
Ún3.1[10]n>1ž,J(h)÷vXeRaccita•§
B(h)γ
1
(h)I
0
(h)=B(h)γ
0
1
(h)I(h)+S(h),S(h)=E(h)J
0
1
3
(h)+F(h)J
0
2
3
(h),(17)
DOI:10.12677/pm.2023.1361931892nØêÆ
ÓŠ
Ù¥
B(h)=2h
3
−54A
3
=2(h−
1
2
5
)(h
2
+
1
2
5
h+
1
2
10
),
E(h)=γ
1
(h)α
2
(h)−B(h)γ
0
1
(h)α
1
(h),F(h)=γ
1
(h)β
2
(h)−B(h)γ
0
1
(h)β
1
(h),
…α
1
(h)=
2
3
hα(h)−6Aβ(h),β
1
(h)=2hβ(h)−3Aγ(h),γ
1
(h)=hγ(h)−2Aα(h),α
2
(h)=
B(h)α
0
1
(h)+h
2
α
1
(h)+3Ahβ
1
(h)+9A
2
γ
1
(h),β
2
(h)=B(h)β
0
1
(h)−h
2
β
1
(h)−9A
2
α
1
(h)−3Ahγ
1
(h).
Ún3.2[10]n>1ž,W(h)=
S(h)
J
0
2
3
(h)
,KW(h)÷vXeRaccita•§
B(h)E(h)W
0
(h)=−3AhW
2
(h)+D(h)W(h)+G(h),(18)
Ù¥D(h)=B(h)E
0
(h)−2h
2
E(h)−2h
2
F(h),G(h)=B(h)E(h)F
0
(h)−B(h)E
0
(h)F(h)+2h
2
E(h)F(h)+
4A
2
E
2
(h)+h
2
F
2
(h).
Ún3.3[10]h∈(
1
2
5
,+∞)ž,J
m
(
1
2
5
)=0(m=
1
3
,
2
3
,1);J
−1
(h)<0,J
0
m
(h)>0(m=
1
3
,
2
3
,1).
n=3,2,1ž,dÚn3.1Œ
α
1
(h)=h
2
(a
4
h
3
+b
4
),β
1
(h)=a
5
h
3
+b
5
,γ
1
(h)=h(a
6
h
3
+b
6
),(19)
Ù¥a
4
=
2
15A
a
1
,b
4
=
2
15A
b
1
−6Aa
2
,a
5
=2a
2
−2Aa
3
,b
5
=−3Ab
3
,a
6
=a
3
−
1
5
a
1
,b
6
=b
3
−
1
5
b
1
.
α
2
(h)=h(a
7
h
6
+b
7
h
3
+c
7
),β
2
(h)=h
2
(a
8
h
3
+b
8
),
Ù¥a
7
=11a
4
,b
7
=270A
3
a
4
+3Aa
5
+9A
2
a
6
+5b
4
,c
7
=3Ab
5
+9A
2
b
6
−108A
3
b
4
,a
8
=5a
5
−
9A
2
a
4
−3Aa
6
,b
8
=−162A
3
a
5
−9A
2
b
4
−b
5
−3Ab
6
.
E(h)=h
2
(a
9
h
9
+b
9
h
6
+c
9
h
3
+d
9
),F(h)=a
10
h
9
+b
10
h
6
+c
10
h
3
+d
10
,(20)
Ù¥a
9
=a
6
a
7
−8a
4
a
6
,b
9
=216a
4
a
6
A
3
−8a
6
b
4
−2a
4
b
6
+a
7
b
6
+a
6
b
7
,c
9
=216a
6
A
3
b
4
+54a
4
A
3
b
6
+
a
6
c
7
−2b
4
b
6
+b
6
b
7
,d
9
=b
6
c
7
+54A
3
b
4
b
6
,a
10
=a
6
a
8
−8a
5
a
6
,b
10
=216a
5
a
6
A
3
−8a
6
b
5
−2a
5
b
6
+
a
8
b
6
+a
6
b
8
,c
10
=216a
6
A
3
b
5
+54a
5
A
3
b
6
−2b
5
b
6
+b
6
b
8
,d
10
=54A
3
b
5
b
6
.
2dÚn3.2Œ
G(h)=h(e
1
h
21
+e
2
h
18
+e
3
h
15
+e
4
h
12
+e
5
h
9
+e
6
h
6
+e
7
h
3
+e
8
),(21)
Ù¥e
1
=−9a
2
9
A
2
−6a
9
a
10
,e
2
=108a
9
a
10
A
3
−18a
9
A
2
b
9
−3a
2
10
A−12a
9
b
10
,e
3
=−54a
10
A
3
b
9
+
270a
9
A
3
b
10
−18a
9
A
2
c
9
−6a
10
Ab
10
+6a
10
c
9
−18a
9
c
10
−9A
2
b
2
9
−6b
9
b
10
,e
4
=−216a
10
A
3
c
9
+
432a
9
A
3
c
10
−18a
9
A
2
d
9
−6a
10
Ac
10
+12a
10
d
9
−24a
9
d
10
+108A
3
b
9
b
10
−18A
2
b
9
c
9
−3Ab
2
10
−12b
9
c
10
,
e
5
=−54A
3
b
10
c
9
−9A
2
c
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DOI:10.12677/pm.2023.1361931894nØêÆ
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ë•©z
[1]o«£,o•.fzFËA116¯K9ÙïÄyG[J].êÆ?Ð,2010,39(5):513-526.
[2]Han,M.(2013)BifurcationTheoryofLimitCycles.SciencePress,Beijing,310-312.
[3]ë˜.ngHamilton•þ|AbelÈ©[D]:[Æ¬Æ Ø©].®:®ŒÆ,1998.
[4]Li,J.(2003)Hilbert’s16thProblemandBifurcationsofPlanarPolynomialVectorFields.
InternationalJournalofBifurcationandChaos,13,47-106.
https://doi.org/10.1142/S0218127403006352
[5]Horozov,E.andIliev,I.D.(1998)LinearEstimatefortheNumberofZerosofAbelianIntegrals
withCubicHamiltonians.Nonlinearity,11,1521-1537.
https://doi.org/10.1088/0951-7715/11/6/006
[6]Gautier,S.,Gavrilov,L.andIliev,I.D.(2009)PerturbationsofQuadraticCentersofGenus
One.DiscreteandContinuousDynamicalSystems,25,511-535.
https://doi.org/10.3934/dcds.2009.25.511
[7]Zhao,Y.,Li,W.,Li,C.andZhang,Z.(2002)LinearEstimateoftheNumberofZerosof
AbelianIntegralsforQuadraticCentersHavingAlmostAllTheirOrbitsFormedbyCubics.
ScienceinChina,SeriesA:Mathematics,45,964-974.https://doi.org/10.1007/BF02879979
[8]Li,W.,Zhao,Y.,Li,C.andZhang,Z.(2002)AbelianIntegralsforQuadraticCentresHaving
AlmostAllTheirOrbitsFormedbyQuartics.Nonlinearity,15,863-885.
https://doi.org/10.1088/0951-7715/15/3/321
[9]Hong,X.,Xie,S.andChen,L.(2016)EstimatingtheNumberofZerosforAbelianIntegrals
ofQuadraticReversibleCenterswithOrbitsFormedbyHigher-OrderCurves.International
JournalofBifurcationandChaos,26,ArticleID:1650020.
https://doi.org/10.1142/S0218127416500206
DOI:10.12677/pm.2023.1361931895nØêÆ
ÓŠ
[10]Hong,X.,Xie,S.andMa,R.(2015)OntheAbelianIntegralsofQuadraticReversibleCenters
withOrbitsFormedbyGenusOneCurvesofHigherDegree.JournalofMathematicalAnalysis
andApplications,429,924-941.https://doi.org/10.1016/j.jmaa.2015.03.068
[11]Hong,X.,Lu,J.andWang,Y.(2018)UpperBoundsfortheAssociatedNumberofZerosof
AbelianIntegralsforTwoClassesofQuadraticReversibleCentersofGenusOne.Journalof
AppliedAnalysisandComputation,8,1959-1970.https://doi.org/10.11948/2018.1959
[12]Hong,L.,Lu,J.andHong,X.(2020)OntheNumberofZerosofAbelianIntegralsforaClass
ofQuadraticReversibleCentersofGenusOne.JournalofNonlinearModelingandAnalysis,
2,161-171.
[13]Hong,L.,Hong,X.andLu,J.(2020)ALinearEstimationtotheNumberofZerosforAbelian
IntegralsinaKindofQuadraticReversibleCentersofGenusOne.JournalofAppliedAnalysis
andComputation,10,1534-1544.https://doi.org/10.11948/20190247
DOI:10.12677/pm.2023.1361931896nØêÆ

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