Table Of ContentON A COMPUTER-AIDED APPROACH TO THE
COMPUTATION OF ABELIAN INTEGRALS
9
0
0 TOMASJOHNSON,WARWICKTUCKER
2
r
p
A
Abstra
t. An a
urate method to
ompute en
losures of Abelian integrals
2 is developed. This allows for an a
urate des
ription of the phase portraits
2 ofplanarpolynomialsystemsthat are perturbations ofHamiltoniansystems.
Asanexample,itisappliedtothestudyofbifur
ationsoflimit
y
lesarising
] froma
ubi
perturbation ofanellipti
Hamiltonianofdegree four.
S
D
1. Introdu
tion
.
h
t Nonlinear ordinary di(cid:27)erential equations are one of the most
ommon models
a
usedin anyappli
ationofmathemati
almodelling. Inthis paperwestudyfamilies
m
of su
h equations
[ x˙ = −H +ǫf(x,y)
y
2 (1) y˙ = Hx+ǫg(x,y),
(cid:26)
v
ǫ
7 depending on a small parameter .
6 A fundamental question about su
h systems is to determine the number and
8 ǫ→0
lo
ation of limit
y
les bifur
ating from it as .
2
In general, the question about the maximal number of limit
y
les, and their
.
9
lo
ation, of a polynomial planar ve
tor (cid:28)eld is the se
ond part of Hilbert's 16th
0
problem, whi
h is unsolved even for polynomials of degree 2. For an overview of
8
0 the progressthat has been made to solvethis problem we refer to [20℄. Results for
: the degree 2
ase, and a general introdu
tion to the bifur
ation theory of planar
v
polynomial ve
tor (cid:28)elds
an be found in [31℄. What is known, is that any given
i
X polynomial ve
tor(cid:28)eld
an haveonly a(cid:28)nite number of limit
y
les; this is proved
r in [11, 19℄.
a
A restri
tedversionofHilbert's 16thproblem, knownastheweak, orsometimes
the tangential, or the in(cid:28)nitesimal, Hilbert's 16th problem, asks for the number of
limit
y
les that
an bifur
ate from a perturbation of a Hamiltonian system, see
e.g. [3℄. The weak Hilbert's 16th problem has been solved for the degree 2
ase,
see [2℄.
Spe
iaHl(
xa,sxe˙s)=ofxH˙2a+mhil(txo)nian systems are those
oming from a one dimensional
system, 2 ,whi
hwestudyintheexamplegiveninSe
tion4.1. If
f =0 g(x,x˙)=g(x)x˙
one,inaddition,assumesthat ,and ,(1)isknownasaLienard
dH
equation. Su
h equations have been thoroughly studied, and the
ase where ,
g g
and have degree 3 has been solved, see [6, 7, 8, 9℄. We study general of degree
3; the set-up of the problem is given in Se
tion 4.1.
Inthispaperwepresentarigorous,
omputer-aidedapproa
hto(cid:28)ndlimit
y
les
of planarpolynomial ve
tor(cid:28)elds. A di(cid:27)erent
omputer-aided approa
hwas intro-
du
edby Malo in his PhD-thesis [22℄, (alsodes
ribed in [15, 16℄) whi
h isbased on
1
2 TOMASJOHNSON,WARWICKTUCKER
the
on
ept of a rotated ve
tor (cid:28)eld, as introdu
ed in [5℄. Our approa
h is
om-
pletely di(cid:27)erent: we develop a method to rigorously
ompute what is known as an
Abelianintegral. A brief introdu
tionto Abelian integralsisin
luded in Se
tion 2.
The
on
ept of a
omputer-aided proof in analysis is based on te
hniques to rigor-
ouslyen
losetheresultofanumeri
al
omputation. Abasisforsu
hapro
edureis
intervalanalysis,introdu
edbyMoorein [23℄. By
al
ulatingwithsetsratherthan
(cid:29)oating points, it is possible to obtain guaranteed results on a
omputer, enabling
automated proofs for
ontinuous problems.
We emphasize that the methods developed in this paper are neither restri
ted
f g
to any spe
i(cid:28)
degree of the polynomial fun
tions , and , nor to the stru
ture
H
of the polynomial Hamiltonian . It
an be used to
ompute Abelian integrals
of any polynomial perturbation from any family of
ompa
t level
urves, ovals,
of a polynomial Hamiltonian. The method
an be used as a
omputational tool
to a
urately des
ribe the phase portraits of a family of planar systems. In the
f =0 dH
example given in this paper, however, we restri
t to the
ase when , and
g
and have degree 3.
2. Abelian integrals
A
lassi
al method to prove tΓhe⊂exHis−te1n(h
e) of limit
y
les bifur
ating hfrom a
h
family of ovals of a Hamiltonian, , depending
ontinuously on , is to
study Abelian integrals, or, more generally, the Melnikov fun
tion, see e.g. [3, 14℄.
Some
aution,however,must be taken regardingthe
orresponden
ebetween limit
y
les and Abelian integrals, see e.g. [10℄. Given a Hamiltonian system and a
perturbation,
x˙ = −H +ǫf(x,y)
y
(2) y˙ = Hx+ǫg(x,y),
(cid:26)
the Abelian integral is de(cid:28)ned as
I(h)= f(x,y)dy−g(x,y)dx.
(3) ZΓh
In this paper all perturbations are polynomial. The most important property of
P
Abelian integrals is des
ribed by the Poin
aré-Pontryagin theorem. Let be the
H
return map de(cid:28)ned on some se
tion transversalto the ovals of , parametrised by
h H h (a,b)
thevalues of , where istakenfromsomeboundedinterval . We
onsider
d(h) = P(h)−h
the displa
ement fun
tion . The theorem by Poin
aré-Pontryagin
states that
d(h)=ǫ(I(h)+ǫφ(h,ǫ)), as ǫ→0,
(4)
φ(h,ǫ)
where is analyti
and uniformly bounded on a
ompa
t neighbourhood of
ǫ=0, h∈(a,b)
.
3. Computer-aided
omputation of Abelian integrals
3.1. Computer-aidedproofs. Toprovemathemati
alstatementsona
omputer,
weneedanarithmeti
whi
hgivesguaranteedresults. Many
omputer-aidedproofs,
in
luding the results in this paper, are based on interval analysis, e.g. [12, 17,
32℄. Interval analysis yields rigorous results for
ontinuous problems, taking both
dis
retisation and rounding errors into a
ount. For a thorough introdu
tion to
interval analysis we refer to [1, 23, 24, 25, 27℄.
COMPUTER-AIDED COMPUTATION OF ABELIAN INTEGRALS 3
3.2. Outline of the approa
h. The main idea of this paper is to develop a very
a
urate,validatedmethodto en
losethevalue ofageneralAbelianintegral. Su
h
I(h) Γ
a method enables us to sample values of . If we
an (cid:28)nd two ovals h1, and
Γ
h2, su
h that
I(h1)I(h2)<0,
(5)
h∗ ∈(h1,h2), I(h∗)=0
then there exists su
h that .
P
ǫ
Sin
e , the return map of the perturbed ve
tor (cid:28)eld, is analyti
and non-
onstant, it has isolated (cid:28)xed points. Thus, we have proved the existen
e of (at
Γh∗
least) one limit
y
le bifur
ating from .
3.3. Computing the integrals. To
ompute the Abelian integral(3) ofthe form
(16), we apply Stokes theorem to get
I(h)= dw,
(6)
ZDh
D Γ
h h
where denotesthe interiorofanoval . The reasonwhywepreferto
al
ulate
surfa
e integrals, rather than
ontour integrals, is that we
annot represent the
H
ovalsof exa
tly. We
anonly (cid:28)nd a
overofthe ovals,and theareaofthis
over
yields the un
ertainty of our
al
ulations, automati
ally handled by the interval
arithmeti
. If wehad
hosen to
ompute
ontourintegrals,all ofour
omputations
wouldhavebeen subje
ted to thoseerrors,sin
ewewouldalwaysintegrateoveran
unknown lo
ation. When
al
ulating surfa
e integrals, however, the e(cid:27)e
t of the
un
ertainty of the lo
ation of the ovals only
ontributes on a very small portion of
D D dω
h h
the total area of . Note that, inside it is possible to integrate exa
tly,
that is, there are no trun
ation errors.
The a
tual
omputation of the integrals is performed in four steps; (cid:28)rst we (cid:28)nd
atrappingregionfortheinterestingfamilyofovals,se
ondweadaptivelysplitthis
region into three parts, one that
overs the oval, one representing the inside and
onerepresentingtheoutside,thirdwe
hangethe
oordinatesontheboxes
overing
dω
the oval in order to minimise the area of the
over, fourth we integrate on the
boxes representing the inside and the
over of the oval.
The (cid:28)rst step is simple, sin
e we only
onsider ovals that are situated inside a
homo- or hetero
lini
orbit. A short bran
h(cid:21)and(cid:21)bound algorithm qui
kly (cid:28)nds a
box en
losing the homo- or hetero
lini
orbit; this box is our initial domain used
for the main part of the program.
In the se
ond step (cid:21) the adaptive splitting of the domain (cid:21) we perform a series
B
of tests to determine whether a box interse
ts the oval, is inside it, or outside
B
it. We start by evaluating the Hamiltonian on using monotoni
ity and
entral
forms; sin
e the Hamiltonians we study are su(cid:30)
iently simple, we implement the
H <h B
derivativessymboli
ally. Three
aseso
ur: if ,then isinsidetheoval,and
B H >h B
we label as su
h. If then is outside the oval and we ignore it. Finally,
h∈H
if , then we try to perform the
hange of variables as des
ribed below. If the
B
hangeofvariablespro
edurefails, and the sizeof isgreaterthansomestopping
B
toleran
e, minsize, then we split the box into four parts and re(cid:21)examine them
B
separately. If the size of is smaller than minsize,then it is labelled fail. If the
B
hange of variables pro
edure works, then we label as on.
4 TOMASJOHNSON,WARWICKTUCKER
2
3 1
4
Figure 1. The labelling of boxes interse
ting an oval.
The third, and most
ompli
ated,part of ourprogramis the
hange ofvariables
b∈B B
in the boxes that interse
t the oval. Let be the midpoint of . Compute
u=∇H(b),
v
and
hoose su
h that
u⊥v v1 ≥0.
and
Using the labelling illbu+strtavtedt ∈inRFigure 1, let right and left be thepsides inq-
terse
ted by the line , . Denote the interse
tion points by , and ,
respe
tively. The possible
on(cid:28)gurations of an interse
tion of the oval with a box
H
are illustrated in Figure 2. The restri
tion of to the sides right and left, re-
spe
tively, are one-dimensional fun
tions, and the lo
ation of the interse
tions
an
be approximated, and their uniqueness proved, using the interval Newton method
p q
[23℄ initialised from the points , and , respe
tively. Let,
accuracy=minsize/10.
p , p q , q
up down up down
De(cid:28)ne the points on the right-side and the points on the
p q
left-side at the distan
e a
ura
y from and , respe
tively, as illustrated in
Figure 3.
q
p q p p
q p
q
p p q
q
q
p
q p
Figure 2. The possible
on(cid:28)gurations of the interse
tion of an
oval and a box.
If the following
onditions hold, then the oval is inside the tube illustrated in
Figure3, andwe
an
hange
oordinatesto getasmallbox,whi
hisguaranteedto
B
ontain the segment of the oval passing through . This small box represents the
error
aused by the unknown lo
ation of the oval.
COMPUTER-AIDED COMPUTATION OF ABELIAN INTEGRALS 5
PSfrag repla
ements p
up
p
q
up
p
q down
q
down
Figure 3. Constru
ting a small, lo
al en
losure of the oval.
Condition 3.1.
sign(H(p )−h) = sign(H(q )−h)
up up
= −sign(H(p )−h) .
down
= −sign(H(q )−h)
down
l l p q p
up down up up down
q Let , and denote theHli′ne segmentsbetween and , and and
down
, respe
tively. Denote by di(cid:27)erentiation with respe
t to the parametrisa-
l l
up down
tion of the line , and , respe
tively.
Condition 3.2. ′
0∈/ (H(l )−h) or 0∈/ H (l )
up up
′
0∈/ (H(l )−h) or 0∈/ H (l ).
down down
B
Let otherside1,and otherside2be the two other sides of the box , that is,
otherside1∪otherside2∪right∪left={1,2,3,4}.
Condition 3.3.
Γ ∩otherside1=∅ and Γ ∩otherside2=∅,
h h
We en
lose the segment of the oval inside of the box between two straightlines:
p , p , q , q
up down up down
Condition3.1guaranteesthatthepoints and areondi(cid:27)erent
l
up
sides of the oval as in Figure 3, Condition 3.2 guarantees that the lines and
l
down
donotinterse
ttheoval,andCondition3.3guaranteesthattheovaldoesnot
p q
ross the other sides of the box. Re
all that the uniqueness of and is proved
as they are approximated. Hen
e, we have proved that the segment of the oval
rossing the box has exa
tly two interse
tions with the boundary of the box, and
l l
up down
that it is
on(cid:28)ned to the region between and .
If (3.1), (3.2), and (3.3), hold, then we set a
ura
y=a
ura
y/2, re-
al
ulate
p ,p ,q , q
up down up down
and , and try to verify (3.1), (3.2), and (3.3). This pro
edure
B
is iterated until (3.1), or (3.2) do not hold. Finally, we label as on.
The fourth and (cid:28)nal part of ourintegration algorithm,is the a
tual integration.
The integration is done separately for the boxes that are labelled, inside, fail,
and on.
B
If is inside we
ompute
xiyjdx∧dy = sup(B1)i+1 − inf(B1)i+1 sup(B2)j+1 − inf(B2)j+1 .
(7) i+1 i+1 j+1 j+1
ZB (cid:18) (cid:19)(cid:18) (cid:19)
B B
If is labelled fail, we know that might interse
t the oval, that is, we have
neither been able to prove interse
tion, nor non-interse
tion. Therefore, we must
B
in
lude any possible result; the integral over is
al
ulated as the interval hull of
6 TOMASJOHNSON,WARWICKTUCKER
0
B˜a⊂ndBthe largest, and smallest, respe
tively, result of (7)
al
ulated on a subbox
.
Boxes labelled fail
ause large over-estimations. Fortunately su
h boxes are
5%
rare, typi
ally less than of the on-boxes, see Se
tion 4. If minsize is taken
su(cid:30)
iently small, the e(cid:27)e
t of the fail-boxesis negligible.
B
u
PSfrag repla
ements
T
u
P
T
l
B
l
Figure 4. The
hange of variables splitting.
Theboxesthatarelabelledon,aresplitinto(cid:28)veparts,asillustratedinFigure4.
T ,T B,B B
l u l u
By
onstru
tion, none of the triangles, , or boxes in the splitting of
H
interse
ttheoval,thusitsu(cid:30)
estoevaluate inonepointofea
h,andhen
ethey
B,B
l u
an all be labelled as inside or outside. The boxes are then treated as
above,thatis, if they arelabelledinsidetheyareintegrateda
ordingto(7), and
if they arelabelledoutsidethey arenegle
ted. A trianglelabelledoutsideisalso
negle
ted, the integrals on triangles labelled inside are en
losed by the formula
xiyjdx∧dy ∈(cid:3)Ti(cid:3)Tj|T|,
1 2
(8)
ZT
(cid:3)T T |T| T
where is the box hull of , and is the area of(cid:3)T. This gives a reasonably
narrow en
losure of the integral, sin
e the width of is typi
ally small. The
P
parallelepiped, , whi
h
overs the segment of the oval, remains to be studied.
P
When weintegrateover , thesameproblemasin thefail
aseo
urs;wedo not
know how mu
h of the parallelepiped to in
lude. Therefore, we have to take the
hull of all possible out
omes. Hen
e, the integrals are
omputed as
xiyjdx∧dy ∈ 0,(cid:3)Pi(cid:3)Pj|P| ,
1 2
(9) Hull
ZP (cid:16) (cid:17)
(cid:3)P P |P| P
where is the box hull of and is the area of .
The value of the Abelian integral is en
losedby summing overall the
omputed
integrals that are labelled as either inside, fail, or on.
I(h) ∈ ( )+ ( )
B∈inside 7 T∈inside 8
+ (0,( ))
(10) PB∈fail Hull P7
+ ( )
PP∈on 9
P
4. Computational results
In this se
tion we apply the methods developed in Se
tion 3.3 to an ellipti
al
Hamiltonian of degree four, des
ribed in Se
tion 4.1. The main idea is to inte-
grate monomial forms at some points, and then to spe
ify the
oe(cid:30)
ients of the
COMPUTER-AIDED COMPUTATION OF ABELIAN INTEGRALS 7
ω I(h)
perturbation su
h that is zero at the sampled points. Therefore, let
I (h)= xiyjdx∧dy.
ij
(11)
ZDh
h
We sample at some number of -values, uniformly distributed between the saddle
loopsandthesingularity. Fromthese
al
ulationswededu
e
andidate
oe(cid:30)
ients.
ω I (h)
ij
Given some
andidate
oe(cid:30)
ients of the form , we
al
ulate the , at
I (h)
ij
intermediateovals. Ifthelinear
ombinationofthe hasvalidatedsign
hanges
between the sample points we aredone: it has been proved that the
orresponding
ǫ→0
perturbation yields bifur
ations with the given number of limit
y
les as .
All
omputations wereperformed on a Intel Xeon 2.0Ghz, 64bit pro
essorwith
7970MbofRAM.The programwas
ompiledwithg
,version3.4.6. Thesoftware
forintervalarithmeti
wasprovidedbytheC-XSCpa
kage,version2.1.1,see[4,18℄.
4.1. Example - bifur
ations from a (cid:28)gure eight loop. We study the ellipti
Hamiltonian of degree 4 with a (cid:28)gure eight loop, given by
y2 x4 1−λ λ
H = + + x3− x2,
(12) 2 4 3 2
λ∈(0,1)
wHh=ere− 1 (2λ+,1)see [9℄.HTh=e−
o1rrλes3p(λon+di2n)g di(cid:27)erential system has two
entres, at
12 , and 12 , that are surrounded by a (cid:28)gure eight
H = 0 λ λ = 1
loop, lo
ated at , see Figure 5. As grows the right loop grows; is a
λ = 0.95
symmetri
(cid:28)gure eight loop. We
hoose to study ; a motivation why we
λ
want large is given below. The Hamiltonian (12)
orresponds to the di(cid:27)erential
system,
x˙ = −H = −y
y
(13) y˙ = Hx = x3+(1−λ)x2−λx.
(cid:26)
We are interested in limit
y
les bifur
ating from the periodi
solutions of (13),
orresponding to integral
urves of (12). The
losed level-
urves of (12) are
alled
ovals. In aseriesofpapers [6, 7, 8, 9℄, Dumortier and Li study
ubi
perturbations
of ellipti
Hamiltonians
orresponding to Lienard equations. That is,
x¨+ǫ(α+βx+γx2)x˙ +ax3+bx2+cx=0.
(14)
For the ellipti
Hamiltonians of degree four with
ompa
t ovals, there are (cid:28)ve
di(cid:27)erent
lasses of phase portraits, see e.g [3℄. They are, the trun
ated pendulum,
the saddle loop. the global
entre, the
uspidal loop, and tδhey3(cid:28)gure-eight loop.
Compared to the Lienard
ase, we add a fourth term, 3 , to the perturbation,
andexplorewhatkindofbifur
ationswe
anprovetoexist. Westudytheperturbed
system,
x˙ = −y
(15) ( y˙ = x3+(1−λ)x2−λx+ǫ (α+βx+γx2)y+δy33 .
(cid:16) (cid:17)
The 1-form asso
iated with this perturbation is
y3
ω =− (α+βx+γx2)y+δ dx.
(16) 3
(cid:18) (cid:19)
For
omputational e(cid:30)
ien
y we primarily study its exterior derivative,
dω = (α+βx+γx2)+δy2 dx∧dy.
(17)
(cid:0) (cid:1)
8 TOMASJOHNSON,WARWICKTUCKER
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0−.18. 5 −1 −0.5 0 0.5 1 1.5
Figure 5. Theellipti
Hamiltonianofdegree4witha(cid:28)gureeight
loop, studied in se
tion 4.1.
In [29℄ Petrov proves that when restri
ting to one family of ovals, surrounding
one of the two
entres, the spa
e of Abelian integrals has dimension 4, and that
thespa
ehasthe Chebyshev property, thatis, the number ofzerosofafun
tion in
this spa
e is less than the dimension of the spa
e. He also proves that this bound
is sharp. To
onstru
t an example with more than three limit
y
les surrounding
either of the two
entres, we
an therefore not simply use the Chebyshev property
of the spa
e of Abelian integrals.
Our heuristi
argument to guess parameters is the following: we start by in-
tegrating at 100 uniformly distributed ovals, in ea
h eye of the loop. We do this
with moderate a
ura
y, whi
h gives a fast and su(cid:30)
iently pre
ise result. Sin
e
we have
hosen to study a (cid:28)gure eight loop that is not far from being symmetri
,
it is reasonable to assume that the two bran
hes behave similarly, whi
h makes it
probable that we should be able to determine
oe(cid:30)
ients so that ea
h bran
h has
two zeros. To determine su
h zeros, we solve the following linear system:
(18) I0l0(−0.0362) I1l0(−0.0362) I2l0(−0.0362) I0l2(−0.0362) α 1
26 II00ll00((−−00..11280182)) II11ll00((−−00..11280182)) II22ll00((−−00..11280182)) II00ll22((−−00..11280182)) 3726 βγ 37=26 −11 37
64 I0r0(−0.1054) I1r0(−0.1054) I2r0(−0.1054) I0r2(−0.1054) 7564 δ 75 64 −1 75
Il (h) Ir(h)
where ij , and ij , denote the monomialAbelian integrals
al
ulatedon the
left and right ovals, respe
tively.
α=438.4905 β =−25.2469 γ =−452.7899
Thisgivestheapproximatesolution , , ,
δ =−741.0341
and , whi
h we use as our perturbation. The graph of the resulting
fun
tion is given in Figure 7, whi
h appears to have 4 zeros, illustrated in Figure
6. This, of
ourse, has to be proved.
To prove that the perturbation
onstru
ted above has 4 zeros, we pro
ede as in
thepreviousexamples,and
omputeen
losuresoftheAbelianintegralatintermedi-
I(−0.0121) I(−0.0846) I(−0.1933)
ateovals. Ontheleftbran
hwe
al
ulate , ,and ,
I(−0.0105) I(−0.0738) I(−0.1686)
andontherightbran
hwe
ompute , ,and . The
result is given in Tables 1, and 2.
COMPUTER-AIDED COMPUTATION OF ABELIAN INTEGRALS 9
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0−.18. 5 −1 −0.5 0 0.5 1 1.5
Figure 6. The limit
y
les from Example 4.1; unstable limit
y-
lesaredashed. Note that weonlyprovetheexisten
eofthe limit
y
les, their lo
ations as drawn in the (cid:28)gure are the lo
ations of
the ovals they bifur
ate from.
20
15
10
5
0
−−05.25 −0.2 −0.15 −0.1 −0.05 0
Figure 7. Thetwobran
hesoftheAbelianIntegralforthe(cid:28)gure
eight loop.
Il(h) Ir(h)
Finally, we
ompute , and at the intermediate ovals,
Il(−0.0121) = [+8.698,+9.290],
Il(−0.0846) = [−2.204,−1.780],
Il(−0.1933) = [+0.9121,+1.119],
.
(19) Ir(−0.0105) = [+11.56,+12.10],
Ir(−0.0738) = [−1.181,−0.7959],
Ir(−0.1686) = [+0.2095,+0.3847]
Hen
e,the system with thegiven perturbationhasfourlimit
y
les, oneattra
ting
andonerepellinginsideea
hloop,seeFigures6and8. Therun-timeoftheprogram
was, for the left (right) bran
h, 82 (78) se
onds, a total of 1182 (1166) boxes were
used to
overthe 3 ovals, 82 (56) of these belong to the fail
lass.
To prove that the unstable separatri
es of the saddle are attra
ted to a limit
I
yo(
hle)en
losing the (cid:28)gure eight loop, as inhdi
>at0ed in Figure 8, we (cid:28)rst
al
ulate
, the outer Abelian integral, for some values with low a
ura
y to (cid:28)nd
10 TOMASJOHNSON,WARWICKTUCKER
h Il Il Il Il
00 10 20 02
-0.0121 [1.206,1.207℄ [-1.034,-1.033℄ [0.9945,0.9951℄ [0.1290,0.1293℄
-0.0846 [0.7661,0.7665℄ [-0.7073,-0.7068℄ [0.6902,0.6907℄ [0.05829,0.05839℄
-0.1933 [0.2219,0.2222℄ [-0.2178,-0.2175℄ [0.2160,0.2164℄ [0.005326,0.005340℄
Table1. The
omputeden
losuresfortheleftbran
hofthe(cid:28)gure
eight loop.
h Ir Ir Ir Ir
00 10 20 02
-0.0105 [1.077,1.078℄ [0.8773,0.8778℄ [0.8033,0.8039℄ [0.1006,0.1008℄
-0.0738 [0.6846,0.6851℄ [0.6002,0.6006℄ [0.5573,0.5577℄ [0.04545,0.04553℄
-0.1686 [0.1984,0.1987℄ [0.1848,0.1850℄ [0.1744,0.1747℄ [0.004154,0.004164℄
Table 2. The
omputed en
losures for the right bran
h of the
(cid:28)gure eight loop.
h Io Io Io Io
00 10 20 02
0.09 [3.576,3.587℄ [-0.1843,-0.1709℄ [2.560,2.575℄ [0.5307,0.5376℄
0.11 [3.776,3.786℄ [-0.1862,-0.1740℄ [2.708,2.724℄ [0.6044,0.6109℄
Table 3. The
omputed en
losures for the outside of the (cid:28)gure
eight loop.
anindi
aHtio=n0o.f1asign
hange. ItappearsItho(a0t.0a9l)imit
yI
ol(e0b.1if1u)r
ates fromanoval
lose to . Therefore, we
ompute , and ,
Io(0.09) = [+8.715,+24.83],
(20) Io(0.11) = [−25.37,−9.821].
These
al
ulationsverifythattheperturbedsystemhasanattra
tinglimit
y
le
bifur
atingfromanovaloutsidethe(cid:28)gureeightloop. Therun-timeoftheprogram
was 39 se
onds, a total of 496 boxes were used to
over the 2 ovals, 52 of these
belong to the fail
lass.
5. Con
lusions
We have presented a method to rigorously
al
ulate Abelian integrals. The
method
an be applied to study any polynomial perturbation of a polynomial
Hamiltonian ve
tor (cid:28)eld. As an appli
ation, we have applied the method to an
ellipti
Hamiltonian of degree 4.
The method
an be used in several ways: either one
an use it to verify that a
spe
i(cid:28)
perturbationguessedbysomeothermethodindeedhasa
ertainnumberof
zeros,or one
an use it as in Se
tion 4.1 to sample and plot the monomial Abelian
integrals. In the latter
ase, if a good
hoi
e of parameters
an be made from the
approximateknowledgeofthemonomialAbelianintegrals,thenone
anre-usethe
programto verify that guess, as is done in Se
tion 4.1.
A major
hallenge is to devi
e a method whi
h
an be used to guess what per-
turbationstoinvestigate. Onesu
hmethodthatappearsin theliteratureisthatof
adete
tion fun
tion, as used in e.g. [33℄. Another problem, whi
h we haveignored
in this paper, is that typi
ally when one has a Hamiltonian depending on param-
eters, the maximal number of limit
y
les that
an bifur
ate from one member of
this family, will only appear for some spe
ial values of the parameters. It would
