Homoclinic orbits to invariant tori in Hamiltonian systems
Abstract
We consider a perturbation of an integrable Hamiltonian system which possesses invariant tori with coincident whiskers (like some rotators and a pendulum). Our goal is to measure the splitting distance between the perturbed whiskers, putting emphasis on the detection of their intersections, which give rise to homoclinic orbits to the perturbed tori. A geometric method is presented which takes into account the Lagrangian properties of the whiskers. In this way, the splitting distance is the gradient of a splitting potential. In the regular case (also known as a priori-unstable: the Lyapunov exponents of the whiskered tori remain fixed), the splitting potential is well- approximated by a Melnikov potential. This method is designed as a first step in the study of the singular case (also known as a priori-stable: the Lyapunov exponents of the whiskered tori approach to zero when the perturbation tends to zero).
Full text
HOMOCLINIC ORBITS TO INVARIANT TORI IN
HAMILTONIAN SYSTEMS
AMADEU DELSHAMS
y
AND
PERE GUTI
ERREZ
z
Abs ac .
We conside a p e u ba ion o an in eg able Hamil onian sys em which
p ossesses in a ian o i wi h coinciden whiske s (like some o a o s and a p endulum).
Ou goal is o measu e he spli ing dis ance b e ween he p e u b ed whiske s, pu ing
emphasis on he de ec ion o hei in e sec ions, which gi e ise o homoclinic o bi s
o he p e u b ed o i. A geome ic me ho d is p esen ed which akes in o accoun he
Lag angian p op e ies o he whiske s. In his way, he spli ing dis ance is he g adi-
en o a spli ing po en ial. In he egula case (also known as a p io i-uns able: he
Lyapuno exp onen s o he whiske ed o i emain xed), he spli ing p o en ial is well-
app oxima ed by a Melniko p o en ial. This me ho d is designed as a s s ep in he
s udy o he singula case (also known as a p io i-s able: he Lyapuno exponen s o he
whiske ed o i app oach o ze o when he p e u ba ion ends o ze o).
Key wo ds.
Hamil onian sys ems, KAM and Nekho oshe heo y, whiske ed o i,
spli ing o sepa a ices, A nold diusion.
AMS(MOS) sub jec classica ions.
58F05, 34C37, 58F36, 34C30.
1. In o duc ion and esul s.
1.1. Nea ly-in eg able Hamil onians.
The Hamil onian dynami-
cal sys ems ha a e close o in eg able ones appea in a na u al way as
mo dels o a wide class o eal sys ems, and hus cons i u e a e y ac i e
eld o esea ch in classical mechanics. The b eha io o hese sys ems is a
om being comple ely unde s oo d, and one o he mos ele an ques ions
is he s abili yo ins abili yo hei a jec o ies. This p oblem emains
unsol ed o sys ems wi h mo e han 2 deg ees o eedom.
The unp e u b ed ^ole is played by a (comple ely) in eg able Hamil-
onian wi h
n
deg ees o eedom. The Liou ille{A nold heo em (see
o ins ance 3]) es ablishes, unde ce ain hypo heses, he exis ence on
some egion o he phase space o canonical
ac ion{angle a iables
(
' I
)=
(
'
1
:::'
n
I
1
:::I
n
)
2
T
n
G
T
n
R
n
, in which he Hamil onian
only dep ends on he ac ion a iables:
h
(
I
). The asso cia ed Hamil onian
equa ions o a a jec o y (
'
(
)
I
(
)) a e
_
'
=
!
(
I
)
_
I
=0
whe e
!
=
@
I
h
. Hence he dynamics is e y simple: e e y
n
-dimensional
o us
I
= cons is in a ian , wi h linea ow
'
(
)=
'
(0) +
!
(
I
)
,and hus
This wo k was suppo ed in pa by he EC g an ERBCHRXCT940460.
y
Depa amen deMa ema ica Aplicada I, Uni e si a Poli ecnica de Ca alunya, Dia-
gonal 647, 08028 Ba celona, e-mail:
[email p o ec ed]
. Also supp o ed in pa by he
Spanish g an DGICYT PB94{0215 and he Ca alan g an CIRIT 1996SGR{00105.
z
Depa amen deMa ema ica Aplicada I I, Uni e si a Poli ecnica de Ca alunya, Pau
Ga gallo 5, 08028 Ba celona, e-mail:
[email p o ec ed]
.
1
2
AMADEU DELSHAMS AND PERE GUTI
ERREZ
all a jec o ies a e s able. The mo ion on a o us is called quasip e io dic,
wi h asso cia ed
equencies
gi en by he ec o
!
(
I
)=(
!
1
(
I
)
:::!
n
(
I
)).
E e y
n
-dimensional in a ian o us can b e non esonan o esonan ,
acco ding o whe he i s equencies a e a ionally indep enden o no . A
non esonan o us is densely lled byany o i s a jec o ies. On he o he
hand, a esonan o us is olia ed in o a amily o lowe dimensional o i.
A
nea ly-in eg able
Hamil onian can b e w i en in he o m
H
(
' I
)=
h
(
I
)+
"
(
' I
)
(1.1)
whe e
"
is a small p e u ba ion pa ame e . Then he Hamil onian equa ions
a e
_
'
=
!
(
I
)+
"@
I
(
' I
)
_
I
=
;
"@
'
(
' I
)
:
Fo
"
6
= 0, he dynamics can be e y complica ed: he e can exis , in
p inciple, chao ic a jec o ies o e en uns able (in he sense ha hey can
wande e y a om hei ini ial condi ions). No e ha he a ia ion o
he ac ions
j
I
(
)
;
I
(0)
j
could g owslowly bu unb oundedly.
Al hough he e may exis uns able a jec o ies, wo e y ele an e-
sul s on s abili yha e b een es ablished o nea ly-in eg able Hamil onian
sys ems. These a e he
KAM heo em
and he
Nekho oshe heo em
. We
a e going o gi e a b ie desc ip ion o bo h o hem, pu ing emphasis on
he ac ha hey lead o wo die en no ions o s abili y.
1.2. KAM heo em.
Wecansay ha KAM (Kolmogo o {A nold{
Mose ) heo y is conce ned wi h he p ese a ion o quasip e io dic mo ions
unde small p e u ba ions: he heo em s a es, unde a sui able nondegen-
e acy condi ion, ha mos o he
n
-dimensional in a ian o i
I
=cons
o he in eg able Hamil onian
h
su i ein (1.1), wi h some de o ma ion,
o
j
"
j
small enough. In ac , KAM heo em gua an ees he p ese a ion o
he o i ha ing sucien ly" non esonan equencies, o which he inu-
ence o he
smal l di iso s
h
k !
(
I
)
i
can b e o e come. This is exp essed by
means o a
Diophan ine condi ion
on he equency ec o
!
(
I
): o some
cons an s
and
,
jh
k !
(
I
)
ij
j
k
j
;
8
k
2
Z
n
n
0
g
whe e
j
k
j
=
P
n
j
=1
j
k
j
j
. The ec o s sa is ying his condi ion o gi en
>n
;
1 and
>
0llaCan o ian se o ela i e measu e 1
;
O
(
)in
R
n
.
In his way, one ob ains
pe pe ual s abili y
in he p e u b ed Hamil onian,
bu only o ini ial condi ions on he p e u b ed o
KAM o i
( he su i ing
ones), which o maCan o ian se no con aining any op en subse al hough
i s measu e is la ge.
The KAM heo em was disco e ed" in he 50's by Kolmogo o 30],
and cons i u ed one o he mos ele an ad ances ab ou s abili yin nea ly-
in eg able Hamil onian sys ems. In his s heo em, Kolmogo o es ab-
lished he p ese a ion o only one xed Diophan ine o us (see also 4]).
HOMOCLINIC ORBITS IN HAMILTONIAN SYSTEMS
3
Some yea s la e , A nold 1]p o ed, in he analy ic case, he exis ence o
a la ge amily o in a ian o i, gi ing an es ima e o he measu e o hei
complemen a y se (see also 48]). Also, Mose 42]p o ed an analogous
heo em, in he die en iable case, o a ea-p ese ing maps sa is ying a
wis condi ion. As gene al e e ences on KAM heo y, see 9,11].
The nondegene acy condi ion equi ed in KAM heo em is imp osed on
he equency map
!
=
@
I
h
o ensu e ha a la ge se o ac ions
I
ha e
Diophan ine equencies
!
(
I
). The e a e wo usual yp es o condi ions:
de (
@
I
!
(
I
))
6
=0
de
@
I
!
(
I
)
!
(
I
)
!
(
I
)
>
0
!
6
=0
( o all
I
2
G
), ha can b e called, esp ec i ely,
Kolmogo o nondegene acy
and
isoene ge ic nondegene acy.
The second one is also called A nold non-
degene acy. Unde any o hese condi ions, o a gi en
">
0 hein a ian
o i whose equencies sa is y he Diophan ine condi ion wi h
> n
;
1
and
=
O
(
p
"
) a e p ese ed, and he measu e o he complemen is small:
O
(
p
"
) (see 48]).
Al hough he wo e sions (Kolmogo o and isoene ge ic) o KAM he-
o em a e known since he wo ks by Kolmogo o , A nold and Mose , he e
a e only a ew published p o o s o he iso ene ge ic e sion. Comple e p o o s
a e gi en in 23,10], indi ec ly om he Kolmogo o e sion. A di ec p o o
o he iso ene ge ic KAM heo em, wi hou using he Kolmogo o e sion,
can b e ound in 19].
I has o be p oin ed ou ha he iso ene ge ic condi ion is mo e signi-
can om he poin o iew o he s abili y, since i ensu es he exis ence o
a la ge amily o KAM o i on each ene gy le el
H
=cons . Fo
n
=2, one
deduces s abili ye en o a jec o ies which do no lie on hese o i, be-
cause hese 2-dimensional in a ian o i sepa a e he 3-dimensional ene gy
le els. This easoning do es no hold o
n>
2, and one canno gua an ee
s abili y. So in his case he e could exis uns able a jec o ies, al hough
i seems clea ha he KAM o i should cons i u e s onge ba ie s o
ins abili y.
1.3. Nekho oshe heo em and eec i e s abili y.
The o he
ele an esul o b e commen ed he e is Nekho oshe heo em, which leads
o he concep o
eec i e s abili y.
Nekho oshe heo em 44], s s a ed
in 1977, es ablishes o all he a jec o ies o he sys em (1.1) ha he
a ia ion o he ac ion a iables emains small o a e y long ime, ex-
p onen ially la ge wi h espec o he pa ame e
" >
0. Fo e e y ini ial
condi ion (
'
(0)
I
(0)) one has an es ima e o he ype
j
I
(
)
;
I
(0)
j
0
"
b
o
j
j
T
0
exp
(
"
0
="
)
a
g
:
The cons an s
a b >
0 a e called
s abili y exponen s.
Fo he alidi y o Nekho oshe heo em one imp oses a
s eepness con-
di ion
(see 44]) on he in eg able Hamil onian
h
. This is a e y gene al
4
AMADEU DELSHAMS AND PERE GUTI
ERREZ
condi ion, a oiding he quick escap e o a jec o ies along ce ain di ec ions
ela ed o esonances o he equency ec o
!
(
I
)=
@
I
h
(
I
). The simples
case is ha o
quasicon ex
unc ions: one says
h
o b e quasicon ex i , o
any
I
2
G
and
2
R
n
,
h
!
(
I
)
i
=0 =
)
@
2
I
h
(
I
)
6
=0
:
Fo a p e u ba ion o a quasicon ex Hamil onian, he es ima e o Nekho o-
she heo em holds wi h he s abili y exp onen s
a
=
b
=
1
2
n
:
The p o o o he heo em wi h hese exp onen s has been gi en in 37,
49]. Howe e , Chi iko 14] had p edic ed se e al yea s b e o e ha he
exp onen
a
=1
=
2
n
should be op imal, a ac ha was o e lo oked un il i
was explained in Lo chak's su ey 34] (see also 35]), whe e he exp onen
a
=1
=
(2
n
+1) was al eady ob ained.
Compa ing Nekho oshe heo em o KAM heo em, we see ha p e -
p e ual s abili y has b een eplaced by ni e ime s abili y, bu on he o he
hand he es ima es a e alid o all a jec o ies in he phase space. Since
(despi e he heo e ical imp o ance o KAM heo em) i is no possible
o know whe he a gi en a jec o y lies on a KAM o us, he eec i e
s abili y es ima es a e o ob ious in e es om he poin o iew o he
applica ions. Some ecen esul s a e in ending o ll he gap be ween
KAM and Nekho oshe heo ems. These esul s conce n he s ickiness"
o KAM o i 46], as well as he ema kable esul s ab ou sup e exp onen ial
s abili y" 40], o he exis ence o quasi-in a ian o i" 19].
1.4. A nold diusion: whiske ed o i, spli ing, and ansi-
ion chains.
No hing is said in KAM heo em ab ou he s abili yo he
a jec o ies close o unp e u b ed esonan in a ian o i. Nekho oshe he-
o em do es no exclude he exis ence o uns able a jec o ies, bu p edic s
o hem an exp onen ially long s abili y ime. This ex emely slow phe-
nomenon o ins abili yis called
A nold diusion,
and a s desc ip ion o
i was gi en in 1964 by A nold 2] by means o his amous example. As
no iced in sec ion 1.2, diusion can only ake place o mo e han 2 deg ees
o eedom.
The
A nold's example
p op osed in 2] is a nonau onomous Hamil o-
nian, p e io dic in he ime a iable
: in canonical a iables (
x y '
1
I
1
)
2
T
R
T
R
,
H
(
x y '
1
I
1
)=
1
2
;
y
2
+
I
2
1
+
"
(cos
x
;
1) (1 +
(sin
'
1
+cos
))
wi h
"
and
as wo indep enden pa ame e s. Taking
as a new angu-
la a iable, he Hamil onian becomes he ollowing 3-deg ees-o - eedom
HOMOCLINIC ORBITS IN HAMILTONIAN SYSTEMS
5
Hamil onian: in canonical a iables (
x y ' I
)
2
T
R
T
2
R
2
,
H
(
x y ' I
)=
1
2
;
y
2
+
I
2
1
+
I
2
(1.2)
+
"
(cos
x
;
1) (1 +
(sin
'
1
+cos
'
2
))
:
We shall igno e he a iable
I
2
which do es no ake pa in he Hamil onian
equa ions, conside ing a 5-dimensional phase space.
The equency ec o
!
(
y I
1
) = (
y I
1
1) has, on
y
= 0, a
single
esonance
(double o a ional
I
1
), and o
"
= 0 he 3-dimensional o i
y I
1
= cons a e olia ed in o 2-dimensional in a ian o i. On he eso-
nance, KAM heo em canno b e applied.
A s p e u ba i e s ep is o conside
">
0bu
= 0, s ill ha ing
an in eg able Hamil onian (1.2) whichisan
uncoupled
sys em o med by
one p endulum ( a iables (
x y
)) and 2 o o s ( a iables (
'
1
'
2
I
1
)). The
3-dimensional o i on he esonance
y
= 0 ha e been des oyed, bu a
1-pa ame e amily o 2-dimensional o i
T
I
1
s ill emains along he eso-
nance:
x
=
y
=0,
I
1
=cons , (
'
1
'
2
)
2
T
2
, wi h asso cia ed equencies
(
I
1
1). These o i come om he hype b olic equilib ium p oin
x
=
y
=0
o he p endulum, and hence hey a e
hype bolic in a ian o i,
also called
whiske ed o i,
wi h asso cia ed 3-dimensional
s able
and
uns able whiske s
(in a ian mani olds)
W
I
1
(
"
0), inhe i ed om he
sepa a ices
o he p en-
dulum. Since hese sepa a ices cons i u e a homo clinic connec ion gi en
by he equa ion
y
2
=
2+
"
(cos
x
;
1) = 0, i u ns ou ha o
">
0,
=0,
he whiske s coincide:
W
+
I
1
(
"
0) =
W
;
I
1
(
"
0). No ice ha he cha ac e is-
ic exponen s o he o igin o he p endulum (and he e o e he Lyapuno
exp onen s o he whiske ed o i
T
I
1
) a e
p
"
, and he e o e he whiske ed
o i a e
weakly hype bolic
o
j
"
j
small. No ice also ha he e a e ellip ic
in a ian o i on
x
=
,
y
=0, which come om he ellip ic poin o he
p endulum.
Fo
">
0and
6
= 0, he Hamil onian (1.2) is no longe in eg able. The
same
o i
T
I
1
a e s ill whiske ed o i, indep enden ly o
, bu hei s able
and uns able whiske s
W
I
1
(
"
) do dep end on
and a e no exp ec ed o
coincide o
6
=0. Indeed, A nold 2] applied he Poinca e's p e u ba i e
me ho d 47,
x
19] o de ec ing
spli ing
o sepa a ices and showed ha ,
o
small enough: 0
<
j
j
<
0
(
"
), he whiske s do no coincide and
in e sec ans e sely. The in e sec ions a e ob ained as ze os o a dis ance
unc ion b e ween he whiske s o he Hamil onian (1.2). The maximum o
his dis ance can b e es ima ed as
exp
;
;
1
=
p
"
+
O
;
2
(1.3)
whe e
exp (
;
1
=
p
"
) is a bound o some in eg als p o ided byPoinca e's
me ho d, which a e usually known as
Poinca e{Melniko in eg als
(o , mo e
b iey,
Melniko in eg als
). An imp o an ea u e is ha he Melniko
in eg al is exp onen ially small in
"
, as a esul o he weak hyp e b olici yo
6
AMADEU DELSHAMS AND PERE GUTI
ERREZ
he unp e u b ed whiske s ( he p owe 1
=
2o
"
in he exp onen ial is due o
he ac ha he Hamil onian (1.2) has only a ni e numb e o ha monics
in
'
), and is he dominan e m in (1.3) only i
is chosen exp onen ially
small wi h esp ec o
"
:
=
o
;
exp
;
;
1
=
p
"
:
The in e sec ions ab o e be ween whiske s a e called
homoclinic
be-
cause hey in ol e only one o us. These homo clinic in e sec ions can be
ex ended o
he e oclinic
in e sec ions be ween he uns able whiske o a
gi en o us
T
I
1
and s able whiske s o o he o i
T
I
0
1
in a neighb o ho o d, as
long as hey a e chosen exp onen ially close wi h esp ec o
"
(see (1.3)):
j
I
0
1
;
I
1
j
exp
;
;
1
=
p
"
:
I is hen p ossible o cons uc a
ansi ion chain
along he esonance
y
=0.
This no ion was in o duced in 2], and means a ni e sequence o o i
T
I
(0)
1
:::
T
I
(
N
)
1
ha ing
non esonan
equencies, wi h
I
(
N
)
1
;
I
(0)
1
=
O
(1),
and such ha he uns able whiske o each o us in e sec s ans e sely
he s able whiske o he nex one. Besides, i was asse ed in 2] ha
he o i in he chain a e
ansi ion o i
: o e e y o us, a bi a ily small
neighb o ho o ds o p oin s in i s s able and uns able whiske s a e connec ed
by a jec o ies o he Hamil onian. F om hese ac s, A nold concluded
ha diusion can ake place along neighb o ho o ds o all he successi e o i
o a ansi ion chain.
A ea u e ha makes A nold's example e y pa icula is he ac
ha he p e u ba ion and i s de i a i es anish on he whiske ed o i
T
I
1
.
This implies ha all hese o i su i ein he pe u ba ion o
6
=0,
emaining e en unchanged. This is e y con enien o he pu pose o
nding ansi ion chains o whiske ed o i, since i makes p ossible o choose
wo o hem as close as necessa y.
Fo a gene al pe u ba ion, he su i al o all he o i asso cia ed o
a single esonance will no b e ue, and an adap ed e sion o KAM he-
o em (see sec ion 1.7 o e e ences) has o be applied. This
hype bolic
KAM heo em
only ensu es he p ese a ion o hose whiske ed o i wi h
Diophan ine equencies, cons i u ing a Can o ian se nea he esonance.
Thus, he se o su i ing whiske ed o i has a la ge ela i e measu e bu
also many gaps", which a e due o he p esence o double esonances.
These gaps and he ac ha he spli ing is expec ed o b e exp onen ially
small can make he uns able whiske o a o us wi h equency on his gi en
se no o in e sec he s able whiske o he nex " su i ing o us, and
p e en s he cons uc ion o a ansi ion chain in his way.
A gene aliza ion o A nold's example o he case o an a bi a y num-
be o deg ees o eedom, was in oduced by Lochak 33, 34] in o de o
p oin ou he main dicul ies conce ning he de ec ion o he diusion. In
HOMOCLINIC ORBITS IN HAMILTONIAN SYSTEMS
7
canonical a iables (
x y ' I
)
2
T
R
T
n
R
n
,
Lochak's example
is:
H
(
x y ' I
)=
1
2
;
y
2
+
h
I I
i
+
"
(cos
x
;
1) +
"F
(
x '
)
(1.4)
whe e
F
is an analy ic unc ion on a complex neighbo hood o
T
n
+1
. As
anew ea u e (see 34, V
x
2]), he ac ha he unc ion
F
is allowed o
ha e a bi a ily high ha monics in he angles
'
b ings ou he p oblem o
he small di iso s inside he Melniko in eg als, as an essen ial ing edien
in o de o decide which he ele an ha monics a e in he s udy o hese
in eg als. I u ns ou ha he Melniko in eg als can b e es ima ed only
o Diophan ine whiske ed o i, and hey a e again exp onen ially small in
"
. Thei asymp o ic beha io is no longe exp (
;
1
=
p
"
), bu dep ends on
he a i hme ic p ope ies o he equencies o he in a ian o i, as was
no iced in 34, page 119], whe e he connec ion be ween he size o he
spli ing and he sp eed o diusion was s discussed. (See 53,17,18,52]
o some explici examples.)
Recen ly, i has b ecome a p oin o g ea in e es he ques ion o he
alidi y o he p edic ions o he spli ing gi en by he Melniko in eg als
o a p owe -like ela ion
=
"
p
, wi h
p>
0. Un o una ely, his is no an
easy p oblem b ecause o he exp onen ially small cha ac e o hese in eg als
wi h esp ec o
"
.
We ema k ha A nold's and Lo chak's examples a e imp o an no
only as sp ecic examples designed o de ec A nold diusion bu , mainly,
b ecause hey a e sp ecial cases o he gene al model nea a single esonance
o a nea ly in eg able Hamil onian sys em. We susp end he e his ep o o
esul s ela ed o A nold diusion o deduce such mo del. In passing, he
ac ual kind o ela ion
=
"
p
needed o he eal applica ions will b ecome
anspa en .
1.5. One s ep o no mal o m nea a single esonance.
Nea
a single esonance, unde some gene ic hyp o heses, we a e going o ca y
ou
one
s ep o ( esonan ) no mal o m p o cedu e and y o ob ain a new
exp ession o he Hamil onian ha gene alizes he examples conside ed in
sec ion 1.4. To eacha
hype bolic no mal o m
,we will ha e o imp ose some
nondegene acy condi ions on he p e u ba ion
, since no hype b olici ycan
be ob ained om he unp e u b ed Hamil onian
h
.Essen ially,we ollow
Eliasson 25] and Niede man 45] (see also 51]).
To simpli y he no a ion, he numbe o deg ees o eedom in (1.1)
will b e, om now on,
n
+ 1 ins ead o
n
.Thus, he angle{ac ion a iables
a e (
' I
)
2
T
n
+1
R
n
+1
.
Ha ing selec ed an ac ion
I
= 0, he unp e u b ed Hamil onian
h
in (1.1) can b e w i en (dis ega ding he addi i e cons an ) as:
h
(
I
)=
h
I
i
+
1
2
h
QI I
i
+
O
3
(
I
)
:
8
AMADEU DELSHAMS AND PERE GUTI
ERREZ
Assume, o he selec ed ac ion, ha he asso cia ed equency ec o
=
@
I
h
(0)
2
R
n
+1
has a
single esonance
(
h
k
i
= 0 o a ce ain
k
2
Z
n
+1
n
0
g
and
h
k
i 6
=0 o any
k
2
Z
n
+1
no co-linea o
k
). By a
classical algeb aic esul , we can assume
o he o m
=(0
!
)
whe e
!
2
R
n
is non esonan . In ac , we shall assume a Diophan ine
condi ion on
!
.
Rew i e (
' I
)
2
T
n
+1
R
n
+1
as (
x y ' I
)
2
T
R
T
n
R
n
,and
he ma ix
Q
=
@
2
I
h
(0) as
2
q
>
q Q
whe e weha epu
2
>
0 in o de o x ideas,
q
2
R
n
, and he new ma ix
Q
is
n
n
. We will assume
= 1! his can b e achie ed eplacing
y
,
I
by
y=
,
I=
(changing in his way he ime scale by a ac o
), and ew i ing
!
=
,
q=
2
,
Q=
2
as
!
,
q
,
Q
esp ec i ely, and edening also he unc ion
. Then, we can w i e ou Hamil onian in he o m
H
(
x y ' I
)=
h
(
y I
)+
"
(
x y ' I
)
h
(
y I
)=
h
!
I
i
+
y
2
2
+
h
q I
i
y
+
1
2
h
QI I
i
+
O
3
(
y I
)
:
Wenow p e o m
one
s ep o esonan no mal o m p ocedu e: ollowing
he Lie me hod, we seek o unc ions
S
(
x '
) and
R
(
x y ' I
)=
O
(
y I
)
such ha
S h
g
+
V
+
R
=
(1.5)
whe e
V
(
x
) is he p e io dic unc ion ob ained bya e aging wi h esp ec o
he angles
'
:
V
(
x
)=
(
x
0
0) =
1
(2
)
n
Z
T
n
(
x
0
'
0)d
' x
2
T
:
The cons uc ion o
S
and
R
is easily ca ied ou : one s sol es he
equa ion
h
!
@
'
S
i
+
V
=
(
0
0)
wi h he help o s anda d small di iso s es ima es, and hen one akes
R
simply by ing equa ion (1.5). The ime-1 symplec ic ow " o he
gene a ing Hamil onian
"S
leads o
H
"=
H
+
H "S
g
+
O
;
"
2
=
h
+
"
(
V
+
R
)+
O
;
"
2
=
H
0
+
H
1
HOMOCLINIC ORBITS IN HAMILTONIAN SYSTEMS
9
wi h
H
0
(
x y I
!
"
)=
h
!
I
i
+
y
2
2
+
"V
(
x
)+
h
q I
i
y
+
1
2
h
QI I
i
H
1
(
x y ' I
!
"
)=
"R
(
x y ' I
)+
O
3
(
y I
)+
O
;
"
2
:
This exp ession gene alizes ha o Lo chak's example (1.4), wi h
H
0
( he
unca ed no mal o m) playing he ^ole o he in eg able Hamil onian, and
H
1
b eing he p e u ba ion o some size
"
whe e
is going o b e de e mined
in e ms o
"
.
I can b e assumed, excep o degene a e cases, ha he unc ion
V
(
x
)
has a unique and nondegene a e maximum
x
0
!we deno e
2
=
;
V
00
(
x
0
)
>
0. Then, o
">
0, he 1-deg ee-o - eedom Hamil onian
P
(
x y
!
"
)=
y
2
2
+
"V
(
x
)
has a hype b olic poin in (
x
0
0), wi h (homo clinic) sepa a ices. The
case
" <
0 is analogous, p o ided one conside s a minimum ins ead o
a maximum. Then, he Hamil onian
H
0
has whiske ed o i wi h coinciden
whiske s asso cia ed o his hyp e b olic p oin . No e ha
H
0
cons i u es a
Hamil onian si ua ed b e ween he unp e u b ed Hamil onian
h
and he p e -
u b ed one
H
,which p ossesses hyp e b olic in a ian o i bu hei whiske s
s ill coincide. No e also ha , in gene al,
H
0
is no an uncoupled Hamil o-
nian b ecause o he
coupling e m
h
q I
i
y
. This seems an addi ional compli-
ca ion, bu i canno b e a oided i one s a s wi h an a bi a y unp e u b ed
Hamil onian
h
.
The Lyapuno exp onen s o he hyp e b olic p oin o
P
a e
p
"
,which
end o ze o o
"
!
0
+
. Toha e xed Lyapuno exp onen s, we can p e o m
he ollowing (non-canonical) linea change: we eplace
y
,
I
by
p
"y
,
p
"I
.
The new sys em is s ill Hamil onian i we di ide he Hamil onian by
"
(making in his waya change o ime scale by a ac o
p
"
). We ob ain o
H
=
H
0
+
H
1
he exp essions:
H
0
=
h
! I
i
+
y
2
2
+
V
(
x
)+
h
q I
i
y
+
1
2
h
QI I
i
(1.6)
H
1
=
R
;
x
p
"y '
p
"I
+
1
"
O
3
;
p
"y
p
"I
+
O
(
"
)=
O
(
)
(1.7)
whe e
!
=
!
p
"
=
p
":
We close his sec ion by ema king ha an essen ial p oin in he ap-
p oach desc ib ed in his wo k is o ha eanin eg able
H
0
. A simila p o-
cedu e could ha e b een ca ied ou o double esonances (o highe mul-
iplici y) ins ead o single ones, bu hen
H
0
would no be, in gene al,
in eg able.
16
AMADEU DELSHAMS AND PERE GUTI
ERREZ
neighb o ho o d o he o igin is a well-known Mose 's esul 41]on hecon-
e gence o he Bi kho no mal o m in a neighbo hood o a hype b olic
equilib ium p oin (see also 13,
x
A3]).
We deno e
G
0
,
G
, he Hamil onians
H
0
,
H
, exp essed in he hype b olic
a iables:
G
0
(
u I
)=
H
0
$=
h
! I
i;
u
+
g
2
(
u
)+
1
2
D
b
QI I
E
(3.3)
G
(
u ' I
!
)=
H
$=
H
0
$+
H
1
$=
G
0
+
G
1
(3.4)
whe e
b
Q
=
Q
;
qq
>
:
Wo king in hese hyp e b olic a iables, he hype b olic KAM heo em will
p o ide us a ans o ma ion
b
" o no mal o m, and we will deno e
e
G
=
G
b
"
he new Hamil onian.
In o de o o mula e quan i a i e s a emen s, wein o duce some no-
a ions. As a neighbo ho o d o he whiske ed o us, we dene o
s >
0
he complex domain
B
s
=
(
u ' I
):
j
u
j
j
j
s
j
I
j
j
Im
'
j
g
:
Fo a unc ion
analy ic on some b ounded domain
D
and con inuous on
i s b ounda y,we conside he no m
j
j
D
=sup
D
j
(
u ' I
)
j
:
The no a ion
1
2
means ha
1
;
2
= cons (so he asso cia ed Hamil-
onian equa ions a e he same). In gene al, ou unc ions depend also on
as an addi ional pa ame e ! he wo d cons an " will no exclude dep en-
dence on
.
A s a emen o he hyp e b olic KAM heo em o be applied in his
wo k is gi en b elow. In ac his is a s anda d hype b olic KAM heo em,
wi h some mino changes due o ha one is seeking o an exac symplec ic
ans o ma ion o no mal o m. The exac ness is imp o an in o de o
de ec in e sec ions be ween he whiske s, wi h he help o a gene a ing
unc ion, as weshow in sec ion 4. The p o o is a enemen o Eliasson's
p o o (see 20]).
Theo em 3.1 (
hyp e b olic KAM heo em
).
Le
G
=
G
0
+
G
1
as
in (3.3{3.4), analy ic on
B
s
.Assume he equency ec o
!
sa ises
he Diophan ine condi ion (2.6) o some
>n
;
1
and
>
0
.Assume
also he nondegene acy condi ions
6
=0
de
b
Q
6
=0
:
Le
0
< <s
,
0
< <
and
0
< <
gi en, and dene
0
=
4
+4
4
C
1
:
(3.5)
HOMOCLINIC ORBITS IN HAMILTONIAN SYSTEMS
17
Then, o
j
j
0
he e exis an exac symplec ic ans o ma ion
b
" :
B
s
;
;
;
;!
B
s
(depending on
), and cons an s
a
=
a
(
)
,
b
=
b
(
)
such ha
e
G
=
G
b
"
akes he o m
e
G
h
! I
;
a
(
)
i;
b
(
)
u
+
b
R
(
u ' I
!
)
b
R
=
O
2
(
u I
;
a
(
))
:
The ans o ma ion
b
"
and he cons an s
a
,
b
a e analy ic on hei domain
wi h espec o
(
u ' I
)
and
. Fo any
j
j
0
he ol lowing es ima es
hold:
b
"
;
id
B
s
;
;
;
C
2
2
2
+2
j
j
j
a
j
j
b
;
1
j
C
3
+1
j
j
:
The cons an s
C
j
do no depend on
,
,
!
.
No ice ha he h eshold gi en in (3.5) is p op o ional o
4
,asin45],
die ing om he one gi en in 51], p op o ional o
2
.This seems due o
he die en me ho ds o p o o ha ollow he Kolmogo o 's and A nold's
app oaches o KAM heo em.
An addi ional ema k is ha he alidi y o his e sion o KAM he-
o em in he singula case can b e es ablished by ega ding how he in ol ed
pa ame e s (mainly he h eshold
0
) depend on
!
.In his way one can
ew i e
!
as
!
=
p
"
wi hou aec ing, o
"
!
0
+
, he smallness condi ion
and he es ima es (o a he making hem b e e ).
The nex s a emen is a mo e ened e sion o Eliasson's esul 25,
p. 65]. F om he hype b olic a iables
w
, his esul comes back o he
o iginal a iables
z
h ough he ans o ma ion $ in oduced in (3.1{3.2).
The ans o ma ion o no mal o m is hen " = $
b
"
$
;
1
and he
new Hamil onian
e
H
=
e
G
$
;
1
=
H
". This s a egy o exp essing he
KAM heo em in he o iginal a iables will be e y use ul o ou pu p oses.
Conce ning he domains, no ice ha , o any
s >
0,
$
1
(
B
s
)=
(
x y ' I
):
j
x
j
s
j
y
+
h
q I
ij
s
j
I
j
j
Im(
'
;
qx
)
j
g
:
Theo em 3.2.
Le
H
=
H
0
+
H
1
as desc ibed in (2.1{2.3), analy ic
on
$
1
(
B
s
)
. Assume he equency ec o
!
sa ises he Diophan ine
condi ion (2.6) o some
>n
;
1
and
>
0
. Assume also he nondegen-
e acy condi ion (2.5). Le
0
< <
gi en, and dene
0
=
4
+4
4
C
1
:
Then, o
j
j
0
and some
0
<<
1
=
2
he e exis an exac symplec ic
ans o ma ion
":$
1
;
B
s =
2
;
;!
$
1
(
B
s
)
(depending on
), and
18
AMADEU DELSHAMS AND PERE GUTI
ERREZ
cons an s
a
=
a
(
)
,
b
=
b
(
)
such ha
e
H
=
H
"
akes he o m
e
H
h
! I
;
a
(
)
i
+
b
(
)
P
(
x y
+
h
q I
i
)+
R
(
x y ' I
!
)
R
=
O
2
(
P
(
x y
+
h
q I
i
)
I
;
a
(
))
:
The ans o ma ion
"
and he cons an s
a
,
b
a e analy ic on hei domain
wi h espec o
(
x y ' I
)
and
. Fo any
j
j
0
he ol lowing es ima es
hold:
j
"
;
id
j
1
(
B
s =
2
;
)
C
2
2
2
+2
j
j
j
a
j
j
b
;
1
j
C
3
+1
j
j
:
The cons an s
C
j
do no depend on
,
,
!
.
The mos imp o an poin abou heo em 3.2 is, in ou opinion, he ac
ha he lo cal no mal o m
e
H
is pu in e ms o
P
(
x y
+
h
q I
i
). Using his
exp ession, i is cons uc ed in 25] a global" Hamil onian (in
x
)ha ing
exac ly he same hype b olic o us and i s (lo cal) whiske s as
e
H
,aswell as
he dynamics on hem. These whiske s a e global (and coinciden ) in his
new Hamil onian. We a e going o use in ensi ely his ea u e in sec ion 4.
F om now on, in o de o keep a mo e eadable no a ion, we shall
usually omi he
-dep endence o ou unc ions.
I is clea ha he Hamil onian
e
H
has a hype b olic in a ian o us o
equencies
!
. This o us and i s asso cia ed local whiske s can b e pa ame-
e ized as ollows:
e
T
=
e
T
(
): ~
z
(
'
)=(0
;h
q a
i
'a
)
'
2
T
n
W
lo c
=
W
lo c
(
): ~
z
(
'
)=(
x
0
(
b
)
y
0
(
b
)
;h
q a
i
'
+
0
(
b
)+
! a
)
'
2
T
n
0
wi h a sui able
0
=
0
(
s
). We henha e, o he o iginal Hamil onian
H
,a
hyp e b olic o us and i s asso cia ed lo cal s able and uns able whiske s ( o
j
j
0
):
T
=
T
(
):
z
(
'
)="(~
z
(
'
))
'
2
T
n
W
lo c
=
W
lo c
(
):
z
(
'
)="(~
z
(
'
))
'
2
T
n
0
:
These pa ame e iza ions o he whiske s
W
lo c
can be ex ended o u -
he alues o
in a na u al way, as a jec o ies asso cia ed o ou Hamil-
onian
H
. We deno e
W
=
W
(
) he ex ended o
global whiske s,
and
ou aim is o measu e he dis ance b e ween hem.
4. The spli ing p o en ial and he spli ing unc ion.
As men-
ioned in he p e ious sec ion, he sp ecial o mula ion o heo em 3.2 allows
HOMOCLINIC ORBITS IN HAMILTONIAN SYSTEMS
19
us o ca y ou a mo e global con ol on he p e u b ed whiske s. Wein o-
duce as in 25] he ollowing in eg able Hamil onian:
N
(
x y I
!
)=
h
! I
;
a
i
+
bP
(
x y
+
h
q a
i
)+
b
h
q I
;
a
i
(
y
+
h
q a
i
)
+
b
2
h
Q
(
I
;
a
)
I
;
a
i
( ecall ha
a
and
b
dep end on
, al hough his is no made explici ). This
Hamil onian is dened globally in he a iable
x
2
T
, and has he same
hyp e b olic o us
e
T
and i s whiske s
W
lo c
, and he dynamics on hem, as
he lo cal no mal o m
e
H
. The only die ence is ha he pa ame e iza ion
~
z
(
'
) (o he whiske s in
e
H
)cannow b e dened o any
2
R
!in hisway
he lo cal whiske s
W
lo c
can be ex ended o a (unique) global homo clinic
whiske o
N
. Following 25], we a e going o ake ad an age o he ac
ha he a icial" Hamil onian
N
is in eg able.
We assume ou s a ing Hamil onian
H
in (2.1) analy ic on a complex
domain
D
s
=
(
x y ' I
):
j
Im
x
j
j
y
j
s
j
I
j
j
Im
'
j
g
wi h
s
big enough in o de o con ain a neighb o ho o d o he (global) un-
p e u b ed whiske s. We deno e &
0
,&
,&
N
he ime-
ows o he Hamil-
onians
H
0
,
H
,
N
esp ec i ely.
We choose wo poin s
;
x
0
y
0
,
;
x
1
y
1
on he p osi i e(
y
0
y
1
>
0)
homo clinic o bi o he p endulum
P
,wi h
x
0
>
and
x
1
<
, and such ha
he asso cia ed
'
-sec ions
;
x
j
y
j
;h
q a
i
T
n
a
,
j
=0
1, a e con ained in
he domain o he no mal o m
e
H
. Conside
T >
0such ha
&
T
N
;
x
1
y
1
;h
q a
i
T
n
a
=
;
x
0
y
0
;h
q a
i
T
n
a
:
Compa ing he ows o
N
and
H
we can dene, in aneighbo hood ' o
"
;
x
0
y
0
;h
q a
i
T
n
a
, he map
(=&
T
"
&
;
T
N
"
;
1
:
This map is exac symplec ic, and akes
W
+
lo c
' (a subse o he lo cal s able
whiske ) in o
W
;
( he global uns able whiske ). Besides, i is easy o check
ha he map ( gi es a co esp ondence b e ween ou pa ame e iza ions o
he whiske s:
(
;
z
+
(
'
)
=
z
;
(
'
)
(4.1)
o any
,
'
such ha
z
+
(
'
)
2
'. The e o e, he die ence (
;
id
cons i u es a measu e o he spli ing on
W
+
lo c
'. We s ess ha a
o mula like (4.1) canno b e exp ec ed ou side
W
+
lo c
, since we need he ac
ha he Hamil onians
e
H
and
N
ha e he same dynamics on he whiske s.
20
AMADEU DELSHAMS AND PERE GUTI
ERREZ
No e also ha (4.1) ells us ha he map ( do es no dep end on he p oin s
;
x
0
y
0
,
;
x
1
y
1
chosen. Wha do es dep end s ongly on hese p oin s is he
neighb o ho o d ' whe e ( is dened.
To con inue,i iscon enien o exp ess he map ( in he no mal o m
hyp e b olic a iables
w
=(
u ' I
)in o duced in sec ion 3, in which he
lo cal whiske s b ecome co o dina e planes, and he global uns able whiske
canbeseenasag aphico e he lo cal s able one. Recall ha
e
G
=
e
H
$
is ou lo cal no mal o m exp essed in he hyp e b olic a iables. The lo cal
whiske s o
e
G
a e gi en by
c
W
+
lo c
=$
;
1
W
+
lo c
=
=0
I
=
a
g
c
W
;
lo c
=$
;
1
W
;
lo c
=
u
=0
I
=
a
g
:
The exac symplec ic map
b
(=("
$)
;
1
(
("
$)
is dened in he neighbo hood
b
'=("
$)
;
1
(') o
;
u
0
0
T
n
a
, whe e
u
0
>
0 is dened by
;
x
0
y
0
=
;
u
0
0
. I is adequa e o dene
c
W
;
=
b
(
c
W
+
lo c
b
'
as an in a ian mani old o
e
G
, which is he equi alen in he hype b olic
a iables o (a piece o ) he global uns able whiske
W
;
.
In he hyp e b olic a iables, he p oblem o measu ing he spli ing has
a simple o mula ion, since
c
W
;
can be seen as a g aphic o e he local
whiske
c
W
+
lo c
. Le us pa ame e ize:
c
W
+
lo c
: ^
w
+
(
'
)=$
;
1
(~
z
(
'
)) = (
u
0
(
b
)
0
'
;
q
+
! a
)
c
W
;
: ^
w
;
(
'
)=("
$)
;
1
;
z
;
(
'
)
whe e
u
0
(
) is dened by
(
u
0
(
)
0) =(
x
0
(
)
y
0
(
)). In comp onen s, we
will w i e ^
w
=
^
u
^
^
'
^
I
. Then he spli ing is gi en by he
die ence
^
I
;
;
^
I
+
=
^
I
;
;
a
. We ema k ha we do no need o conside
he die ence ^
;
;
^
+
=^
;
b ecause
^
I
;
(
'
)=
a
implies ha ^
;
(
'
)=0!
his is ela ed o he ac ha he whiske s a e con ained in ene gy le els
o
e
G
.
The exac ness o he symplec ic map
b
( and he ac ha
b
(
;
id =
O
(
),
imply ha o
small enough he e exis s a
gene a ing unc ion
;
u
)
'
)
I
such ha
b
(:
;
)
u
)
)
'
)
I
7!
(
u ' I
)isgi en by
u
=)
u
;
@
)
=
;
@
u
'
= )
'
;
@
I
)
I
=
I
;
@
'
:
HOMOCLINIC ORBITS IN HAMILTONIAN SYSTEMS
21
We dene he
spli ing po en ial
as he ollowing p e io dic unc ion:
L
(
'
)=
L
(
0
'
)=
(
u
0
(
b
0
)
0
'
;
q
+
!
0
a
)
'
2
T
n
wi h
0
such ha ^
w
+
(
0
T
n
)
b
'. The gene a ing unc ion
is de e mined
up o an addi i e cons an , ha can b e chosen o ensu e ha
L
has ze o
a e age. No e ha
L
dep ends on
and on he choice o
0
. The g adien
M
(
'
) =
@
'
L
(
'
) will be called he ( ec o )
spli ing unc ion,
and he
ollowing heo em s a es ha , on a sui able
'
-sec ion, he unc ion
M
gi es he spli ing dis ance.
Theo em 4.1.
The e exis unc ions
(
'
)=
O
(
)
and
*(
'
)=
'
+
O
(
)
,pe iodic in
'
, such ha , o al l
'
2
T
n
,
M
(
'
)=
^
I
;
(
0
+
(
'
)
*(
'
))
;
a:
Fo ou xed
0
, no e ha
^
I
;
(
0
'
)
;
a
=
@
'
;
^
u
;
(
0
'
)
0
^
'
;
(
0
'
)
a
:
(4.2)
One can hink ha his should be he mos na u al elec ion o wha we
call spli ing unc ion". Ne e heless, i do es no come om o mula (4.2)
ha his is he g adien o a scala unc ion, al hough his obs uc ion has
been easily o e come wi h he change o a iables gi en by he p e io dic
unc ions
and *.
An imp o an consequence o heo em 4.1 is he exis ence o a leas
n
+1 eec i e in e sec ions be ween he whiske s
W
, gi ing ise o a
leas
n
+ 1 a jec o ies biasymp o ic o he in a ian o us
T
.These in e -
sec ions a e ob ained as c i ical p oin s o a unc ion on
T
n
, and he mini-
mum numb e o hem comes om Lyus e nik{Schni elman heo y (see 15,
sec . 2.12]). This cons i u es he main esul con ained in 25]. No e also
ha , in nondegene a e cases, he numbe o in e sec ions becomes a leas
2
n
, as one deduces om Mo se heo y.
5. Fi s o de app oxima ions o he spli ing: Melniko in-
eg als.
We now use Poinca e{Melniko heo y o gi ea s o de ap-
p oxima ion (in
) o he spli ing po en ial in o duced in he p e ious
sec ion. We dene he (scala )
Melniko po en ial
and he ( ec o )
Mel-
niko unc ion
as, esp ec i ely, he ollowing p e io dic unc ions:
L
(
'
)=
;
Z
1
;1
;
H
1
;
H
1
;
H
0
g
(
z
0
(
'
)) d
(5.1)
M
(
'
)=
@
'
L
(
'
)=
;
Z
1
;1
@
'
(
H
1
;
H
0
g
)] (
z
0
(
'
)) d
:
(5.2)
In hese o mulas,
(
x y ' I
)=
(
'
) is he (ze o a e age) unc ion sol ing
he small di iso s equa ion
h
! @
'
i
+
H
1
(0
0
0) =
H
1
(0
0
0)
:
(5.3)
22
AMADEU DELSHAMS AND PERE GUTI
ERREZ
I is no dicul o check ha he in eg als in (5.1{5.2) a e absolu ely
con e gen (in con as wi h 29,50, 57], whe e condi ionally con e gen
Melniko in eg als a e in o duced o
n
2). This absolu e con e gence
is due o he inco p o a ion o he unc ion
(
'
) (in o duced byT esche
in 56]), which is closely ela ed o he shi sue ed by he pe u bed
whiske ed o us
T
wi h espec o he unp e u b ed o us
T
0
in applying
KAM heo em. Indeed, w i ing he pa ame e iza ion o he o us
T
in
comp onen s:
z
=(
x
y
'
I
), he ollowing lemma gi es a s o de
app oxima ion o
I
(
'
).
Lemma 5.1.
The
I
-componen o
z
(
'
)
sa ises
I
(
'
)=
(cons
;
@
'
(
'
)) +
O
;
2
:
We no e ha
L
(
'
), as dened in (5.1), has ze o a e age (
L
= 0), and
ha
L
(
'
) and
M
(
'
) do no dep end on
.
We s ess ha ou o mula o
L
(
'
) is qui e compac , and use ul in he
coupled case (
q
6
= 0), as weshow in sec ion 6. I is also wo h ema king
ha (5.1) i is equi alen o he o mula app ea ing in 13,
x
4], bu much
simple .
Wenowshow some al e na i e exp essions o
L
(
'
)and
M
(
'
). Using
ha
Z
2
1
H
0
g
(
z
0
(
'
)) d
=
(
z
0
(
2
'
))
;
(
z
0
(
1
'
))(5.4)
o any
1
,
2
, and applying (2.7), wege
L
(
'
)= lim
T
!1
"
;
Z
T
;
T
;
H
1
;
H
1
(
z
0
(
'
))d
+
(
'
+
q
+
!T
)
;
(
'
;
q
;
!T
)
#
:
Taking a
'
-de i a i e, weob ain
M
(
'
)= lim
T
!1
"
;
Z
T
;
T
@
'
H
1
(
z
0
(
'
))d
+
@
'
(
'
+
q
+
!T
)
;
@
'
(
'
;
q
;
!T
)
#
:
In he uncoupled case
q
=0,we can p o ide simple (and p e haps mo e
classical) exp essions. P o ceeding like in (5.4) and using ha
h
! @
'
i
=
H
0
g
(0
0
0), weha e
(
'
+
!
2
)
;
(
'
+
!
1
)=
Z
2
1
H
0
g
(0
0
'
+
!
0)d
HOMOCLINIC ORBITS IN HAMILTONIAN SYSTEMS
23
=
Z
2
1
;
H
1
;
H
1
(0
0
'
+
!
0)d
:
Thus, o he sp ecial case
q
=0 we ob ain he ollowing exp essions (com-
pa e wi h o mula (2.15) o 21]):
L
(
'
)=
;
Z
1
;1
;
H
1
;
H
1
(
z
0
(
'
))
;
;
H
1
;
H
1
(0
0
'
+
!
0)
d
M
(
'
)=
;
Z
1
;1
@
'
H
1
(
z
0
(
'
))
;
@
'
H
1
(0
0
'
+
!
0)]d
:
Coming again o he gene al case, he ollowing s anda d lemma shows
ha a s o de app oxima ion o he die ence
I
;
(
0
)
;
I
+
(
0
)isgi en
by he Melniko unc ion
M
(
'
) (we deno e
z
=(
x
y
'
I
)in he
pa ame e iza ions o he whiske s
W
). In ac , an analogous exp ession
is gi en in 56] o
F
0
(
z
;
(
0
))
;
F
0
(
z
+
(
0
)), whe e
F
0
is anygi en s
in eg al o he unp e u b ed Hamil onian
H
0
. Ou s a emen conce ns he
case
F
0
=
I
j
,
j
=1
:::n
.
Lemma 5.2.
Fo any xed
0
2
R
,
I
;
(
0
'
)
;
I
+
(
0
'
)=
M
(
'
)+
O
(
2
)
:
(5.5)
Finally, we see ha he spli ing p o en ial
L
(
'
) in oduced in heo-
em 4.1 can b e app oxima ed by he Melniko p o en ial
L
(
'
). This equi es
o show ha he app oxima ion (5.5), exp essed in he o iginal a iables,
emains ue a e changing o he no mal o m a iables in which he spli -
ing p o en ial
L
(
'
) had o b e dened.
Theo em 5.1.
Fo he spli ing unc ion and he spli ing po en ial
in oducedin heo em 4.1, one has:
L
(
'
)=
L
(
'
)+
O
;
2
M
(
'
)=
M
(
'
)+
O
;
2
:
The p o o shows ha he Melniko unc ion
M
(
'
) emains alid as a
s o de app oxima ion o he spli ing unc ion
M
(
'
), in spi e o he
ac ha
M
(
'
)was dened in heo em 4.1 using he no mal o m a iables.
We ema k ha he s o de app oxima ions o
M
(
'
) and
L
(
'
)do
no dependon
0
a s o de in
bu hey do a highe o de s.
Since b o h he spli ing po en ial and he Melniko p o en ial a e de-
ned on
T
n
, a di ec applica ion o Mo se heo y implies he exis ence,
o
j
j
small enough, o a leas 2
n
ans e se homo clinic o bi s o he
whiske ed o us
T
, as long as he Melniko po en ial is a Mo se unc ion,
ha is, all i s c i ical p oin s a e nondegene a e (a gene ic p op e y).
24
AMADEU DELSHAMS AND PERE GUTI
ERREZ
6. A compu able example in he coupled case.
As an example,
we conside a p e u ba ion o a coupled in eg able Hamil onian (
q
6
= 0
in (2.2)), wi h
n
+ 1 deg ees o eedom. In he in eg able pa , wecho ose
he classical p endulum:
V
(
x
) = cos
x
;
1. In he p e u ba ion, we conside
a unc ion only dep ending on
'
:
H
1
(
'
)=
X
k
2
Z
n
h
k
e
i
h
k'
i
:
We use he ollowing well-known o mulas o he homo clinic a jec o y o
he s anda d p endulum:
x
0
(
) = 4 a c an
e
y
0
(
)=
2
cosh
0
(
)=
q
(
x
0
(
)
;
)
z
0
(
'
)=(
x
0
(
)
y
0
(
)
'
+
0
(
)+
!
0)
:
The solu ion
(
'
) o equa ion (5.3) is simply
(
'
)=
;
X
k
6
=0
ih
k
h
k !
i
e
i
h
k'
i
and he Melniko po en ial is gi en by i s Fou ie se ies
L
(
'
)=
X
k
6
=0
L
k
e
i
h
k'
i
whe e each co ecien is gi en by he ollowing in eg al:
L
k
=
h
k
h
k q
i
e
;
i
h
kq
i
h
k !
i
Z
1
;1
e
i
h
k!
i
e
i
h
kq
i
x
0
(
)
y
0
(
)d
:
Then, o he Melniko unc ion
M
(
'
)=
P
k
6
=0
M
k
e
i
h
k'
i
, i is clea ha
M
k
=
ik L
k
.
Fo he sake o simplici y,we assume ha
q
is hal -in ege " (i.e. 2
q
2
Z
n
). In his way, we can easily compu e he Melniko in eg als using
esidue heo y since all he singula i ies o he unc ions in ol ed a e p oles.
Roughly,wege :
j
L
k
jj
h
k
j
e
2
h
k!
i
j
sinh (
h
k !
i
)
j
jh
k !
ij
jh
k
2
q
ij;
1
:
I is e y in e es ing o ske ch an analysis o his exp ession in he case
o
as
equencies. Thus, we conside
!
=
!
=
p
"
wi h
!
Diophan ine,
and in o duce
=
=
p
"
in (2.6), o some
n
;
1. We also assume
exp onen ially dec easing co ecien s o he p e u ba ion, like
j
h
k
j
e
;j
k
j
o e e y
k
( he pa ame e
is he wid h o analy ici yo
H
1
(
'
)). Then
HOMOCLINIC ORBITS IN HAMILTONIAN SYSTEMS
25
wecanmake he ollowing es ima e: he dominan ha monic
L
k
e
i
h
k'
i
in
he Fou ie se ies o
L
(
'
)isgi en o he indexes
k
such ha
jh
k !
ij
j
k
j
;
j
k
j
2
p
"
1
=
(
+1)
(assuming ha he co esp onding co ecien s
h
k
a e non anishing), and
we deduce ha he Melniko po en ial
L
(
'
) is exp onen ially small in
"
:
;
log
L
=
O
;
"
;
1
=
2(
+1)
,aswell as i s g adien
M
(
'
).
A mo e p ecise asymp o ic b eha io can b e ob ained o he case o
wo
o a o s (
n
= 2 in (2.2)), since hen one can apply he heo y o con inued
ac ions o he a io o equencies
!
2
=!
1
. Fo ins ance, in he case o a
quad a ic
numbe
!
2
=!
1
like he `golden mean' (1 +
p
5)
=
2, one can nd,
as in 17], ha log
L
;
c
(log
"
)
"
;
1
=
4
, whe e
c
(
u
)is a p osi i e p e io dic
unc ion wi h pe iod 2log(
!
2
=!
1
). A di ec applica ion o Theo em 5.1
ensu es ha he spli ing dis ance is as p edic ed by he Melniko unc ion,
bu equi es
=
o
;
exp
;
;
c
(log
"
)
"
;
1
=
4
.
A jus ica ion o he case
=
"
p
o some
p>
0 is immedia e as long
as one has a signican enemen o Theo em 5.1:
L
(
'
;
0
!
=
p
"
)=
L
(
'
;
0
!
=
p
"
)+
O
;
2
"
;
p
o
'
,
0
on he
complex
s ip
j
Im
'
j
;
"
1
=
4
,
j
Im
0
j
=
2
;
"
1
=
4
. Such
kind o esul is a byp oduc o an ex ension heo em (see, o ins ance, 17,
52]), which is cu en ly b eing esea ched by he au ho s.
Acknowledgmen s.
We a e indeb ed o P. Lo chak o his c i ical
commen s and e iew, and o L. Niede man o se e al discussions. One o
he au ho s (A.D.) is e y g a e ul o he hospi ali y om he Ins i u e o
Ma hema ics and I s Applica ions in Minneap olis du ing he p epa a ion
o his manusc ip .
REFERENCES
1]
V. A nold
,
P oo o a heo em o A.N. Kolmogo o on he in a iance o quasi-
pe iodic mo ions unde smal l pe u ba ions o he Hamil onian
, Russian Ma h.
Su eys, 18 (1963), pp. 9{36.
2] ,
Ins abili y o dynamical sys ems wi h se e al deg ees o eedom
,So ie
Ma h. Dokl., 5 (1964), pp. 581{585.
3]
V. A nold, V. Kozlo , and A. Neish ad
,
Ma hema ical aspec s o classical
and celes ial mechanics
, in Dynamical sys ems I I I, V. A nold, ed., ol. 3 o
Encyclopaedia Ma h. Sci., Sp inge -Ve lag, Be lin{Heidelb e g, 1988.
4]
G. Bene in, L. Galgani, A. Gio gilli, and J.-M. S elcyn
,
Ap oo o Kol-
mogo o 's heo em on in a ian o i using canonical ans o ma ions dened
by he Lie me hod
,IlNuo o Cimen o B, 79 (1984), pp. 201{223.
5]
P. Be na d
,
Pe u ba ion d'un hamil onien pa iel lemen hype bolique
, C. R.
Acad. Sci. Pa is Se . I Ma h., 323 (1996), pp. 189{194.
6]
U. Bessi
,
Anapp oach o A nold's diusion h ough he calculus o a ia ions
,
Nonlinea Anal., 26 (1996), pp. 1115{1135.