Publicacions Ma em`a iques, Vol 39 (1995), 355–366.
ABELIAN INTEGRALS OF
QUADRATIC HAMILTONIAN VECTOR FIELDS
WITH AN INVARIANT STRAIGHT LINE*
Chengzhi Li, Jaume Llib e and Zhi en Zhang
Abs ac
We p o e ha he lowes uppe bound o he numbe o isola ed
ze os o he Abelian in eg als associa ed o quad a ic Hamil onian
ec o fields ha ing a cen e and an in a ian s aigh line a e
quad a ic pe u ba ions is one.
1. In oduc ion
Le H(x, y) be a eal polynomial o deg ee n+ 1, and le P(x, y) and
Q(x, y) be eal polynomials o deg ee a mos m. The p oblem o finding
an uppe bound N(n, m) o he numbe o isola ed ze os o he Abelian
in eg als
(1.1) I(h)=Γh
Q(x, y)dx −P(x, y)dy,
whe e Γh a ies in he compac componen s o H−1(h) is called he
weakened 16 h Hilbe p oblem. I was posed by A nold in [1].
The weakened 16 h Hilbe p oblem is closely ela ed o he p oblem
o de e mina ing an uppe bound o he numbe o limi cycles o he
pe u bed Hamil onian sys em
(1.2)ε
dx
d =∂H
∂y +εP(x, y),
dy
d =−∂H
∂x +εQ(x, y),
*The au ho s a e pa ially suppo ed by DGICYT g an s, he fi s and hi d au-
ho s a e also pa ically suppo ed by NSF o China, he second one is also pa ially
suppo ed by a CIRIT g an .
356 C. Li, J. Llib e, Z. Zhang
whe e 0 <|ε|<< 1.The ela ionship be ween bo h p oblems comes om
he ollowing wo ac s:
(1) I I(h∗)=0andI(h∗)= 0, hen he e exis s a hype bolic limi
cycle Lh∗o sys em (1.2)εsuch ha Lh∗→Γh∗as ε→0; and con e sely,
i he e exis s a hype bolic limi cycle Lh∗o sys em (1.2)εsuch ha
Lh∗→Γh∗as ε→0, hen I(h∗)=0.
(2) The o al numbe o isola ed ze os o (1.1) ( aking in o accoun
hei mul iplici y) is an uppe bound o he numbe o limi cycles o
sys em (1.2)ε ending o some pe iodic o bi Γho sys em (1.2)ε=0 when
ε→0.
Kho ansky [16] and Va chenko [24] p o ed independen ly ha
N(n, m) is fini e, bu an explici exp ession o N(n, m) is unknown.
Many au ho s ha e con ibu ed o es ima e he numbe N(n, m) o some
alues o nand m, and o some classes o polynomial unc ions H(x, y),
see o ins ance Bogdano [3] and [4], Pe o [20] and [21], Cushman
and Sande s [10], Dumo ie , Roussa ie and So omayo [13], D achman,
an Gils and Zhang [12], Li and Rousseau [18], Ga ilo and Ho ozo
[14], Li, Llib e and Zhang [17], ...
Aquad a ic Hamil onian ec o field is a ec o field o he o m
(∂H/∂y,−∂H/∂x) whe e H=H(x, y) is a eal polynomial o deg ee 3.
Ou main esul is o show ha N(2,2) = 1 o he quad a ic pe u -
ba ions o he quad a ic Hamil onian ec o fields ha ing an in a ian
s aigh line unde he flow defined by he ec o field. This esul is
p o ed in Sec ion 2. In Sec ion 3 we cha ac e ize he phase po ai s o
he quad a ic Hamil onian ec o fields ha ing an in a ian s aigh line,
we will see ha he e a e six o such phase po ai s.
We wan o hank o A mengol Gasull his commen s on a p elimina y
e sion o his pape .
2. S a emen o he main esul s
Le (∂H/∂y,−∂H/∂x) be a quad a ic Hamil onian ec o field, as
usual we say ha
dx
dy =∂H
∂y ,dy
d =−∂H
∂x ,
Abelian in eg als o quad a ic hamil onian . . 357
is i s associa ed quad a ic Hamil onian sys em, and ice e sa. Fo ab-
b e ia ion we deno e by QH he class o quad a ic Hamil onian sys ems
ha ing a cen e and an in a ian s aigh line.
Lemma 1. E e y sys em in QH can be educed o ano he sys em in
QH o he o m
(2.1)
dx
d =∂H
∂y =2xy,
dy
d =−∂H
∂x =−y2+ (x),
whe e (x)is a polynomial o deg ee 2.
P oo : Le
(2.2) dx
d =∂H1
∂y ,dy
d =−∂H1
∂x ,
be a sys em in QH. By a ansla ion we can pu he o igin o coo dina es
on he in a ian s aigh line o sys em (2.2). Then by a o a ion we
can ans o m he in a ian s aigh line o sys em (2.2) in he y-axis.
Since bo h ans o ma ions a e canonical, hey p ese e he Hamil onian
s uc u e o he sys em, and clea ly hey do no inc ease he deg ee o
he Hamil onian unc ion. Thus sys em (2.2) can be w i en in he o m
(2.3)
dx
d =∂H2
∂y =x(ax +by +c),
dy
d =−∂H2
∂x .
No ice ha b= 0; o he wise ˙x=x(ax+c), and consequen ly sys em (2.3)
would no ha e pe iodic o bi s.
Now conside he change o a iables
(2.4) x=x, by =ax +by +c.
Clea ly
de ∂(x, y)
∂(x, y)=1,
358 C. Li, J. Llib e, Z. Zhang
and he ans o ma ion (2.4) is canonical. Fo simplici y ew i ing (x, y)
ins ead o (x, y) in sys em (2.3) a e doing he change o a iables (2.4),
we ge
(2.5)
dx
d =∂H3
∂y =bxy,
dy
d =−∂H3
∂x .
Since sys em (2.5) belongs o QH, i ollows ha
dx
d =∂H3
∂y =bxy,
dy
d =−∂H3
∂x =−b
2y2+g(x),
whe e g(x) is a polynomial o deg ee 2. Now escaling he a iable y
om y o 2y/b, he lemma ollows.
No ice ha sys em (2.1) is in a ian unde he symme y (x, y, )→
(x, −y,− ) and ha i has x= 0 as an in a ian s aigh line.
Now we p esen wo p elimina y esul s on he Abelian in eg als I(h)
(see Sec ion 1). These esul s will allow us o simpli y he quad a ic
pe u ba ion o a sys em in QH. They a e well-known bu since we
canno find any e e ence o hem, we p o e hem he e.
P oposi ion 2. The ollowing equali y holds
I(h)=ΓhQ(x, y)+∂P(x, y)
∂x dydx.
P oo : F om S okes heo em we ge
ΓhP(x, y)dy +∂P(x, y)
∂x dydx
=ln (Γh)∂P(x, y)
∂x dx ∧dy +∂
∂y ∂P(x, y)
∂x dydy ∧dx
=ln (Γh)∂P(x, y)
∂x −∂P(x, y)
∂x dx ∧dy
=0.
Then om (1.1) he p opesi ion ollows.
F om P oposi ion 2 i ollows immedia ely.
Abelian in eg als o quad a ic hamil onian . . 359
Co olla y 3. The Abelian in eg al I(h)associa ed o he sys em
dx
d =∂H
∂y ,
dy
d =−∂H
∂x +εQ(x, y)+∂P(x, y)
∂x dy,
is iden ical o he Abelian in eg al associa ed o sys em (1.2).
F om P oposi ion 2 and Co olla y 3 we ge easily he ollowing esul .
Co olla y 4. Assume ha he pe u ba ion (P,Q)o sys em (1.2)ε=0
is o deg ee 2. Then he Abelian in eg al I(h)associa ed o sys em (1.2)ε
is iden ical o he Abelian in eg al associa ed o he ollowing sys em
dx
d =∂H
∂y ,
dy
d =−∂H
∂x +ε(µ1y+µ2xy +µ3y2).
Now we will apply Co olla y 4 o a sys em in QH.
Lemma 5. E e y sys em in QH a e a quad a ic pe u ba ion has
he same Abelian in eg al ha he ollowing pe u bed sys em o QH:
(2.6)
dx
d =∂H
∂y =2xy,
dy
d =−∂H
∂x =−y2+ (x)+µ1y+µ2xy,
whe e (x)is a polynomial o deg ee 2.
P oo : By Lemma 1 e e y sys em in QH can be ans o med in he
o m (2.1) wi h Hamil onian unc ion
(2.7) H(x, y)=x(y2+F(x)),
whe e F(x) is a polynomial o deg ee 2 such ha minus he de i a i e
o xF(x) is equal o (x). F om Co olla y 4 he Abelian in eg al o a
Hamil onian sys em (2.1) wi h an a bi a y quad a ic pe u ba ion is
he same ha he Abelian in eg al o he sys em:
(2.8)
dx
d =2xy,
dy
d =−y2+ (x)+µ1y+µ2xy +µ3y2.

360 C. Li, J. Llib e, Z. Zhang
Since o a pe iodic o bi Γho he Hamil onian sys em we ha e Γh∩
{x=0}=∅,i ollows om (2.7) ha
Γh
y2dx =Γhh
x−F(x)dx =0.
So he Abelian in eg al o sys em (2.8) is he same ha he Abelian
in eg al o sys em (2.6).
No ice ha sys em (2.6) only depends on wo pa ame e s.
Now we s a e ou main esul .
Theo em 6. We ha e ha N(2,2) = 1 o he quad a ic pe u ba ion
o quad a ic Hamil onian ec o fields ha ing an in a ian s aigh line.
P oo : We deno e by
Ii(h)=Γh
xiydx
o i=0,1. Then, om Lemma 5, he Abelian in eg al o he quad a ic
pe u ba ion o a sys em in QH is
I(h)=µ1I0(h)+µ2I1(h),
whe e h∈(h0,h
1),and h0and h1co espond o some equilib ium poin
and some homoclinic o he e oclinic loop o he phase pa ai o he
Hamil onian sys em
dx
d =∂H
∂y =2xy,
dx
d =−∂H
∂x =−y2+ (x),
in he Poinca ´e disc.
Since (2.6) is a quad a ic sys em ha ing an in a ian s aigh line, by
he p ope ies o quad a ic sys ems (see [5], [6], [15], [25], [21], [8] and
[9]), sys em (2.6) has a mos one limi cycle, and i i exis s hen i is
hype bolic. So I(h) can ha e a mos one ze o. Hence, N(2,2) = 1.
Abelian in eg als o quad a ic hamil onian . . 361
The numbe o isola ed ze os o he Abelian in eg al (1.1) associa ed
o quad a ic pe u ba ions o a sys em in QH, and he numbe o limi
cycles o he pe u bed sys em in gene al is no he same. I may ex-
is limi cycles o he pe u bed sys em such ha when he quad a ic
pe u ba ion goes o ze o hey end o he equilib ium poin ( he cen e
o he sys em in QH) o o he homoclinic o he e oclinic loop o ming
he bounda y o he cen e in he Poinca ´e disc. Bo h possibili ies has
been confi med by Zoladek in [26] and [27]. Hence, in o de o ob ain
all he limi cycles which can be bi u ca ed om a cen e i is necessa y
o s udy simul aneously h ee cases. The fi s one is o con ol he num-
be o limi cycles which can be bi u ca ed om he equilib ium poin ,
secondly hose which can be bi u ca ed om pe iodic o bi s o he cen e
( o ins ance, h ough Abelian in eg als), and finally hose which can be
bi u ca ed om he homoclinic o he e oclinic loop a he bounda y o
he cen e . Fo mo e de ails see [4], [19], [18] and [22].
3. Classifica ion o he Sys em in QH
We shall use he classifica ion o all he phase po ai s o quad a ic
Hamil onian ec o fields gi en by A es and Llib e in [2] o cha ac e ize
he phase po ai s o he sys ems in QH, i.e. he phase po ai s o he
quad a ic Hamil onian ec o fields ha ing an in a ian s aigh line and
a leas one cen e .
By Lemma 1 e e y sys em in QH can be ans o med o a sys em
(3.1) dx
d =∂H
∂y ,dy
d =−∂H
∂x ,
wi h H(x, y)=x(y2+ax2+bx +c). In o de o classi y he phase
po a i s o sys em (3.1) in QH we dis inguish h ee main cases.
Case 1. a=0.I b= 0 hen sys em (3.1) has no cen e . So we assume
ha b= 0. A e escaling he a iable x he pa ame e bcan be educed
o 1. The e o e H(x, y)=x(y2+x+c) = 0 has wo b anches, he s aigh
line x= 0 and he pa abola y2+x+c=0. I c≥0 hen sys em (3.1) has
no cen e s. So we assume c<0. Then he wo b anches o H(x, y)=0
in e sec a wo saddle poin s o sys em (3.1). The o he singula poin
(−c/2,0) o sys em (3.1) is a cen e . Hence, om [2], sys em (3.1) has
he phase po ai o ype Vulpe 5 (see Figu e 1).
Case 2: a>0. A e escaling he a iable x he pa ame e acan be
aken equal o 1. So H(x, y)=x[y2+(x+b/2)2+c−b2/4].
Subcase 1. c−b2/4<0 . Then H(x, y) = 0 has wo b anches, he
s aigh line x= 0 and he ci cle cen e ed a he poin (−b/2,0) wi h
362 C. Li, J. Llib e, Z. Zhang
adius equal o b2/4−c.I c<0 hen bo h b anches in e sec a wo
saddles o sys em (3.1). The e o e, om [2], sys em (3.1) has he phase
po ai o ype Valpe 3 (see Figu e 1). I c= 0 he s aigh line x=0
is angen o he ce cle y2+(x+b/2)2=b2/4, and om [2] sys em (3.1)
has he pha e pa ai o ype Vulpe 2 (see Figu e 1). I c>0 he wo
b anches o H(x, y) = 0 do no in e sec , hen om [2] sys em (3.1) has
he phase po ai o ype Vulpe 2 (see Figu e 1).
Subcase 2. c−b2/4≥0.Then H(x, y) = 0 has a unique b anch,
he s aigh line x=0. I b2−3c≤0 hen sys em (3.1) has no cen e s.
The e o e we assume b2−3c>0. Now sys em (3.1) has exac ly wo
singula poin s o coo dina es ((−b±√b2−3c)/3,0), a cen e and a
saddle. The e o e, om [2] he phase po ai o sys em (3.1) is o ype
Vulpe 2 (see Figu e 1).
Case 3. a<0. A e escaling he a iable x he pa ame e acan be
aken equal o −1. So H(x, y)=x[y2−(x−b/2)2+c+b2/4].
Subcase 1. c+b2/4=0.Then H(x, y) = 0 has h ee b anches, he
s aigh lines x=0,y =x−b/2 and y=x+b/2. Each wo o hem
in e sec s a a saddle. The e o e, om [2], sys em (3.1) has he phase
po ai o ype Vulpe 10 (see Figu e 1).
Subcase 2. c+b2/4= 0. Then H(x, y) = 0 has h ee b anches, he
s aigh line x= 0 and he wo b anches o he hype bola y2−(x−b/2)2=
−(c+b2/4). I b2+3c≤0 hen sys em (3.1) has no cen e s. So we assume
ha b2+3c>0. I c+b2/4<0 hen x= 0 in e sec s he wo b anches
o he hype bola, a wo saddles. The e o e, om [2] i ollows ha
sys em (3.1) has he phase po ai o ype Vulpe 9 (see Figu e 1). I
c+b2/4>0 hen x= 0 in e sec s only one b anch o he hype bola, a
wo saddles. Hence, om [2], sys em (3.1) has he phase po ai o ype
Vulpe 8 (see Figu e 1).
Thus we ha e ob ained six diffe en opological phase po ai s o he
sys ems in QH. We ema k ha he unique o hese six phase po ai s
which can be ealized o a quad a ic Hamil onian sys em wi hou ha ing
an in a ian s aigh line is he phase po ai o Vulpe 2. This is due
o he p ope y ha i a quad a ic sys em has an o bi wi h α-limi
one saddle and ω-limi ano he saddle, hen his o bi is con ained in
an in a ian s aigh line. This esul is due o Dong [11], see also Ye
Yanqian and o he s [25]. An imp o emen o his esul is due o Chicone
and Sha e , see Theo em 2.8 o [7].
We also ema k ha Vulpe 2 wi hou in a ian s aigh line is e-
la ed wi h he Bogdano -Takens bi u ca ion es ic ed o quad a ic ec-
o fields.
Abelian in eg als o quad a ic hamil onian . . 363
a=0
Vulpe 5
c−b2/4<0, c<0
Vulpe 3
c−b2/4<0, c=0
Vulpe 2
c−b2/4<0, c>0
Vulpe 2
a>0
c+b2/4=0
Vulpe 10
c+b2/4<0
Vulpe 9
c+b2/4>0
Vulpe 8
a<0
Figu e 1: The six diffe en opological phase po ai s o
he quad a ic Hamil onian ec o fields ha ing
an in a ian s aigh line and a leas one cen e .