XX Cong eso de Ecuaciones Di e enciales y Aplicaciones
X Cong eso de Ma em´
a ica Aplicada
Se illa, 24-28 sep iemb e 2007
(pp. 1–8)
The non–degene a e cen e p oblem in ce ain
amilies o plana di e en ial sys ems.
Jaume Gin´
e, Paz de P ada
Depa amen de Ma em`a ica. Uni e si a de Lleida. A da. Jaume II, 69. 25001 Lleida.
E-mails: [email p o ec ed], [email p o ec ed].
Key wo ds: cen e p oblem, commu a o , isoch onous cen e , ime– e e sible.
Abs ac
This wo k conce ns he non–degene a ed cen e p oblem in ce ain amilies o
di e en ial sys ems in R2. We s udy he exis ence o uni o mly isoch onous cen e s
and he o m o hei commu a o s. We also classi y all cen e s o he amily o he
BiLi´ena d sys ems o deg ee i e.
1 In oduc ion
We conside analy ic sys ems o di e en ial equa ions in he eal plane o he o m:
˙x=−y+P(x, y),˙y=x+Q(x, y),(1)
whe e P(x, y) and Q(x, y) a e eal analy ic unc ions in a neighbo hood o he o igin
wi hou cons an no linea e ms.
2 Uni o mly Isoch onous Cen e s
In his sec ion, we conside he amily o polynomial di e en ial sys ems o he o m:
˙x=−y+xR(x, y),˙y=x+yR(x, y),(2)
wi h R(x, y) = Pn
i=1 Ri(x, y) whe e Riis a homogeneous polynomial o deg ee i. The
cen e s o hese sys ems a e called uni o mly isoch onous cen e s. The o igin, when i is a
cen e , is called uni o mly isoch onous cen e o sys em (2) because in pola coo dina es
(2) akes he o m: ˙ =F( , θ), ˙
θ= 1, see [7, 8].
1
J. Gin´e, P. de P ada
The cen e p oblem o his ype o sys ems has been s udied by se e al au ho s. The
case in which R(x, y) is a homogeneous polynomial o deg ee ihas been s udied in [7]
whe e he ollowing esul is gi en:
Theo em 1. Le R(x, y)be a homogeneous polynomial. Then he o igin is an isoch onous
cen e o sys em (2) i , and only i , one o he ollowing condi ions holds:
(i) sys em (2) has e en deg ee;
(ii) sys em (2) has odd deg ee n= 2m+ 1, and
2m
X
`=0
`Z2π
0
(cos ϕ)2m−`(sin ϕ)`dϕ = 0,
whe e R(x, y) = Pn−1
`=0 `xn−1−`y`.
In he nonhomogeneous class, he i s case ha has been s udied co esponds o
he sys ems wi h R(x, y) = R1+R2, see [6]. In [13] he au ho s s udy he case when
R(x, y) = R1+R3. In [4] he au ho s s udy he case when R(x, y) = R1+R2+R3wi h
R2
1+R2
2+R2
36= 0. The case R(x, y) = R1+R2+R3+R4wi h R46= 0 and only one Rino
equal o ze o, o i= 1,2,3, is s udied in [1]. Sys ems o ype (2) wi h R(x, y) = R2+R4
and R(x, y) = R2+R6ha e been s udied in [20] and [21], espec i ely.
In all he p esen ed cases up o now, he amilies o cen e s a e ime– e e sible, i.e,
he cen e s a e symme ical wi h espec o a s aigh line passing h ough he o igin.
The e o e, he cen e s a e in a ian unde he nex ans o ma ion (modulo a o a ion):
(x, y, )→(x, −y, − ) o (x, y, )→(−x, y, − ).
Hence he na u al ques ion is, a e all he cen e s o amily (2) ime– e e sible? I is
known ha he answe o his ques ion is nega i e. As we will see in he nex sec ion,
he e exis non– e e sible cen e s in he amily o he uni o mly isoch onous sys ems.
2.1 Isoch onous cen e s and commu a o s
A mo e geome ic app oach o di e en ial sys ems o equa ions in he eal plane gi es he
no ion o plana ec o ield.
De ini ion 2. The ec o ield associa ed o he di e en ial sys em (1) is X= (−y+
P(x, y))∂/∂x + (x+Q(x, y))∂/∂y.
The ollowing de ini ion gi es he no ion o commu a o o a ec o ield.
De ini ion 3. Two ec o ields Xand Ycommu e i hei Lie b acke is null, ha is
[X,Y] = DX.Y − DY.X ≡ 0.
The ollowing heo em shows he ela ion be ween he isoch onous cen e p oblem and
he exis ence o ans e sal commu a o , see o ins ance [18].
Theo em 4. A cen e a he o igin o sys em (1), wi h associa ed ec o ield X, is
isoch onous i , and only i , he e exis s an analy ic ec o ield Ysuch ha [X,Y]≡0and
Xand Ya e ans e sal in a punc u ed neighbo hood o he o igin.
2
The non–degene a e cen e p oblem
Fu he mo e, we ha e he ollowing esul p o ed in [17], see also [12].
Theo em 5. The o igin o sys em (1), wi h associa ed ec o ield X, is an isoch onous
cen e i , and only i , he e exis s an analy ic ec o ield o he o m Y= (x+o(x, y))∂/∂x+
(y+o(x, y))∂/∂y such ha [X,Y]≡0.
Hence, he isoch onous cen e p oblem is equi alen o ind a ans e sal commu a o
in a punc u ed neighbo hood o he o igin. The e a e only a ew amilies o polynomial di -
e en ial sys ems in which a comple e classi ica ion o he isoch onous cen e s is known, and
almos all o hem ha e a polynomial commu a o , see o ins ance [5, 10, 15, 16, 18, 19].
Mo eo e , se e al wo ks a e de o ed o ind polynomial commu a o s o di e en amilies
o polynomial di e en ial sys ems, see [10, 18]. Howe e , he e exis polynomial di e -
en ial sys ems wi hou any polynomial commu a o . The i s example o a polynomial
isoch onous cen e wi hou any polynomial commu a o was ound by De lin in [9]. This
example is a qua ic sys em, wi h homogeneous nonlinea pa , whe e an isoch onous
cen e a he o igin and o he s wo non–isoch onous cen e s coexis . The example is:
˙x=−y−x4−4x2y2+y4,˙y=x−4x3y. (3)
The non–exis ence o a polynomial commu a o is based upon he ollowing heo em:
Theo em 6. Le us conside a polynomial sys em (1) wi h an analy ic commu a o de ined
in an open se U. I he e exis s a cen e in U, i is an isoch onous cen e .
P oo . Le Y he analy ic commu a o de ined in Uand X he ec o ield associa ed o
he polynomial sys em. Xand Ya e no ans e sal in he se V= 0 wi h V:= X ∧ Y.
Mo eo e , V(x, y) is in e se in eg a ing ac o . Hence, V= 0 is an in a ian cu e o he
polynomial sys em (1). Le (x0, y0) a cen e o Xin U. Since (x0, y0) is a singula poin
o X, hen, V(x0, y0) = 0. Since (x0, y0) is a cen e in U, (x0, y0) is an isola ed ze o o V.
The e o e, he e exis s a punc u ed neighbo hood o (x0, y0) whe e he ields Xand Ya e
ans e sal and by Theo em 4, (x0, y0) is an isoch onous cen e . Hence, De lin example
canno ha e a polynomial commu a o .
We no e ha , in pa icula , e e y cen e o a sys em wi h a polynomial commu a o
is an isoch onous cen e . Ne e heless, i can exis a ocus singula poin in Uas he
ollowing example shows ˙x=−y+x(x2+y2), ˙y=x+y(x2+y2), which has a ocus a
he o igin and he ollowing polynomial and ans e sal (in a punc u ed neighbo hood o
he o igin) commu a o : ˙x=x(x2+y2), ˙y=y(x2+y2).
3 Uni o mly isoch onous cen e s and commu a o s
We ha e seen ha he isoch onous cen e p oblem is equi alen o ind commu a o s
ans e sal in a punc u ed neighbo hood o he singula poin . In [2], he cha ac e iza ion
o he polynomial commu a o s o uni o mly isoch onous cen e s is s udied. Using hese
esul s, he au ho s classi y he cen e s o he amilies wi h R(x, y) = R1+Rnand he
case wi h R(x, y) = R2+R2nwi h n∈N, see [2, 3]. These esul s exhibi he use ulness
o commu a o s in he classi ica ion o uni o mly isoch onous cen e s. Mo eo e , in all
he amilies o uni o mly isoch onous cen e s s udied in [2], he au ho s ind ha ei he
3
J. Gin´e, P. de P ada
hey a e ime– e e sible o hey ha e a polynomial commu a o . Consequen ly, one asks
whe he his is he gene al ule. In [14], an example wi h a uni o mly isoch onous cen e
a he o igin which is no ime– e e sible and has no polynomial commu a o is ound. The
example is he ollowing:
˙x=−y+x(y3−3xy2+ 2x2y) (1 + x2+y2),
˙y=x+y(y3−3xy2+ 2x2y) (1 + x2+y2).(4)
In addi ion, his sys em is shown o commu e wi h
˙x=x(x2+y2)px2+y2(1 + x2+y2),
˙y=y(x2+y2)px2+y2(1 + x2+y2).(5)
We no e ha his commu a o has no adial linea pa , so Theo em 5 does no
apply. Mo eo e , al hough he ec o ields a e ans e sal in a punc u ed neighbo hood
o he o igin, he commu a o is no analy ic, so we canno use Theo em 4. Hence, his
commu a o does no ensu e he exis ence o a uni o mly isoch onous cen e a he o igin.
In he nex subsec ion we wan o con inue he s udy o he exis ence o polynomial
and analy ic commu a o s o uni o mly isoch onous cen e s s a ed in [2, 3].
3.1 Commu a o s o polynomial sys ems
Fi s , we p esen a esul ha gi es he deg ee o a polynomial commu a o , p o ided ha
his polynomial commu a o exis s, see [3].
Theo em 7. I sys em (2) has a polynomial commu a o , hen his polynomial commu a o
is o he o m:
˙x=xK(x, y),˙y=yK(x, y),
whe e Kis a polynomial o he same deg ee as R.
In ac , he bound o he deg ee o he commu a o is he same o sys ems o he o m:
˙x=−y+P2+P3+· · · +xRn,˙y=x+Q2+Q3+· · · +yRn,
whe e Piand Qia e homogeneous polynomials o deg ee i. This ype o sys ems a e called
in ini y degene a ed sys ems.
Consequen ly, he p oblem o de ec he exis ence o no o a polynomial commu a o
educe o a compu a ion p oblem because he deg ee is ixed gi en a polynomial sys em.
Mo eo e he o m o hese polynomial commu a o s is s udied in [2, 3]. Mo e speci ically,
in [3] a e p o ed he ollowing heo ems:
Theo em 8. Sys em (2), wi h R1=R2=. . . =Rj−1= 0 and Rj6= 0 has a polynomial
commu a o wi h adial lineal pa i , and only i , he e a e α`,β`homogeneous polynomials
o o de `(`≤j, ` di ides o j) e i ying x∂yβ`−y∂xβ`=`α`such ha he sys em eads
o :
˙x=−y+xα`
−1
X
k=j/`−1
akβk
`,˙y=x+yα`
−1
X
k=j/`−1
akβk
`,(6)
4
The non–degene a e cen e p oblem
wi h aka bi a y eal numbe s and = [(n−1)/`]. The commu a o is gi en by
˙x=x+x
−1
X
k=j/`−1
akβk+1
`,˙y=y+y
−1
X
k=j/`−1
akβk+1
`.
By Theo em 5, sys em (6) has a uni o mly isoch onous cen e a he o igin because
he commu a o is polynomial and has adial lineal pa .
Theo em 9. Sys em (2), whe e R(x, y) = Pn−1
j=0 Rj(x, y)wi h Rj(x, y)homogeneous poly-
nomial o deg ee j, has a polynomial commu a o wi h null linea pa i , and only i , i is
o he o m:
˙x=−y+xP2`(x, y)Pm
j=0 aj(x2+y2)j,
˙y=x+yP2`(x, y)Pm
j=0 aj(x2+y2)j,(7)
wi h P2`(x, y)homogeneous polynomial o deg ee 2`, ` ≥0, and aja bi a y eal numbe s.
In his case, he commu a o is gi en by
˙x=x
m
X
j=0
aj(x2+y2)j+`,˙y=y
m
X
j=0
aj(x2+y2)j+`.(8)
The ollowing heo em gi es he condi ions o ha e a uni o mly isoch onous cen e a
he o igin o sys em (7). Mo eo e , when he sys em has a uni o mly isoch onous cen e ,
by Theo em 5, i also admi s an analy ic commu a o wi h adial linea pa . The e o e,
he na u al ques ion is how o ind i . The ollowing heo em also gi es he analy ic
commu a o wi h adial linea pa o he amily o sys ems (7), when i has a uni o mly
isoch onous cen e a he o igin.
Theo em 10. Sys em (7) wi h he condi ion R2π
0P2`(cos ϕ, sin ϕ)dϕ = 0 has a uni o mly
isoch onous cen e a he o igin. Mo eo e , i has an analy ic commu a o wi h adial
linea pa o he o m:
˙x=x(x2+y2)`Pm
j=0 aj(x2+y2)jH−1,
˙y=y(x2+y2)`Pm
j=0 aj(x2+y2)jH−1,(9)
whe e His an analy ic i s in eg al o he o m H= (x2+y2)`+k/(1 + h(x, y)) and akis
he i s non null coe icien o he Pm
j=0 aj(x2+y2)jin (7).
P oo . Taking pola coo dina es, sys em (7) eads o
˙ = 2`+1
m
X
j=0
aj 2jP2`(cos ϕ, sin ϕ),˙ϕ= 1.
In he case R2π
0P2`(cos ϕ, sin ϕ)dϕ = 0, sys em (7) has an analy ic i s in eg al ha in
pola coo dina es akes he o m
H=−2ak 2(`+k)(`+k)
1 + α 2+β 4+· · · + 2ak 2(`+k)(`+k)RP2`(cos ϕ, sin ϕ)dϕ.
Hence, sys em (7) has a uni o mly isoch onous cen e a he o igin. Using his analy ic
i s in eg al we can cons uc he analy ic commu a o (9) wi h adial linea pa .
5

J. Gin´e, P. de P ada
We ema k ha he ec o ields associa ed o sys em (7) and he commu a o (8)
a e always ans e sal in a punc u ed neighbo hood o he o igin. We no e ha i he
commu a o (8) is analy ic in a neighbo hood o he o igin ( ha is, `is a na u al numbe )
hen, using Theo em 4, he o igin o sys em (7) is a uni o mly isoch onous cen e .
In sho , he e exis uni o mly isoch onous cen e s wi hou polynomial commu a o s
bu hey always ha e an analy ic one. Fo ins ance, he Voloki in example (4) is a polyno-
mial sys em o deg ee 6, which has no polynomial commu a o , see [14]. Using Theo em
10, sys em (4) commu es wi h
˙x=x(x2+y2)3
2(1 + x2+y2)H−1,
˙y=y(x2+y2)3
2(1 + x2+y2)H−1,(10)
whe e His he i s in eg al
H=3(x2+y2)3/2
4(−1 + 3x2+ 4x3+ 3y2+ 6xy2+ 3(x2+y2)3
2a c an(px2+y2)).
Sys em (10) p o ides an analy ic commu a o wi h adial lineal pa o sys em (4).
In summa y, he polynomial sys ems which ha e a polynomial commu a o and a
uni o mly isoch onous cen e a he o igin a e de e mined by Theo em 8 o Theo em
10. Mo eo e , he e exis uni o mly isoch onous cen e s o polynomial sys ems which a e
only cha ac e ized by he exis ence o an analy ic commu a o o he o m Y= (x+
o(x, y))∂/∂x + (y+o(x, y))∂/∂y, see Theo em 5.
3.2 Commu a o s o analy ic sys ems
The ollowing heo em is gi en in [3] and es ablishes he o m o an analy ic commu a o
o an analy ic sys em:
Theo em 11. I he analy ic sys em: ˙x=−y+x R(x, y),˙y=x+y R(x, y), wi h
R(0,0) = 0, has a cen e a he o igin, hen he e exis s an analy ic commu a o o he
o m: ˙x=x+x K(x, y),˙y=y+y K(x, y), wi h Kan analy ic unc ion a ound he
o igin wi h K(0,0) = 0.
F om Theo em 11 he commu a i i y condi ion educes o he ollowing pa ial di e -
en ial equa ion:
xµ∂K
∂y −∂H
∂x +H∂K
∂x −K∂H
∂x ¶+yµ−∂K
∂x −∂H
∂y +H∂K
∂y −K∂H
∂y ¶= 0 (11)
A s aigh o wa d gene aliza ion o Theo em 8 o analy ic sys ems is he ollowing:
P oposi ion 12. Conside he sys em
˙x=−y+xa(x, y)g(b(x, y)),˙y=x+ya(x, y)g(b(x, y)) (12)
wi h a(x, y) = (x∂yb(x, y)−y∂xb(x, y))/`,gan a bi a y analy ic unc ion and b(x, y)
a homogeneous polynomial o deg ee `wi h `6= 0. This sys em has a commu a o o he
o m:
˙x=x+xb(x, y)g(b(x, y)),˙y=y+yb(x, y)g(b(x, y)),(13)
and i has an isoch onous cen e a he o igin.
6
The non–degene a e cen e p oblem
P oo . Since b(x, y) is a homogeneous polynomial o deg ee `, we ha e
x∂b(x, y)
∂x +y∂b(x, y)
∂y =`b(x, y),(14)
and aking in o accoun he de i a i es o (14), he Lie b acke o he ec o ields associ-
a ed o sys ems (12) and (13) is null and he claim ollows.
A s aigh o wa d gene aliza ion o Theo em 9 o analy ic sys ems is he ollowing:
P oposi ion 13. The analy ic sys em:
˙x=−y+x (x2+y2)g(x, y),˙y=x+y (x2+y2)g(x, y)
whe e (x2+y2)is an analy ic unc ion and g(x, y)is a homogeneous polynomial o deg ee
2`, has a commu a o o he o m:
˙x=x(x2+y2)` (x2+y2),˙y=y(x2+y2)` (x2+y2)
P oo . The Lie b acke o he ec o ield associa ed o he sys ems gi es:
[X,Y] = (x2+y2)µx∂g(x, y)
∂x +y∂g(x, y)
∂y −2`g(x, y)¶Y
Finally, since g(x, y) is a homogeneous polynomial o deg ee 2`, he claim ollows.
4 BiLi´ena d equa ion
In his sec ion we s udy ano he amily o polynomial sys ems which co esponds:
˙x=−y+F(x),˙y=x+G(y),
whe e F(x) and G(y) a e polynomials wi hou cons an nei he linea e ms. These sys-
ems a e called BiLi´ena d sys ems, see [11]. In his case, he cen e p oblem has been
s udied wi h F(x) and G(y) polynomials un il ou h deg ee and all he cen e s a e ime–
e e sible, see [13]. Fu he mo e, he e a e amilies o cen e s o F(x) and G(y) o a bi-
a y deg ee, see [11]. In he ollowing heo em we classi y all cen e s in which F(x) and
G(y) a e polynomials o deg ee i e.
Theo em 14. Conside he sys em:
˙x=−y+a2x2+a3x3+a4x4+a5x5,˙y=x+b2y2+b3y3+b4y4+b5y5,(15)
whe e aiand bia e eal numbe s. All cen e s a he o igin o sys em (15) a e ime–
e e sible.
Acknowledgemen s
The au ho s a e pa ially suppo ed by a DGICYT g an numbe MTM2005-06098-C02-
02. The i s au ho is also pa ially suppo ed by a CICYT g an numbe 2005SGR 00550,
and by DURSI o Go e nmen o Ca alonia “Dis inci´o de la Gene ali a de Ca alunya pe
a la p omoci´o de la ece ca uni e si `a ia”.
7
J. Gin´e, P. de P ada
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