POINCAR

E-MELNIKOV-ARNOLD METHOD FOR TWIST MAPS
AMADEU DELSHAMS AND RAFAEL RAMREZ-ROS
1. In o duc ion
A gene al heo y o p e u ba ions o an in eg able plana map wi h a sepa a ix o a
hyp e b olic xed p oin has been de elop ed in a p e ious lec u e 5]. The spli ing o he
p e u b ed in a ian cu es was measu ed, in  s o de wi h esp ec o he pa ame e o
p e u ba ion, by means o a p e io dic
Melniko unc ion
M
dened on he unp e u b ed
sepa a ix. In he case o plana wis maps,
M
has ze o mean and he e o e he e exis s a
p e io dic unc ion
L
(called he
Melniko po en ial
) such ha
M
=
L
0
. Consequen ly,i
L
is no iden ically cons an ( esp ec i ely, has non-degene a e c i ical p oin s), he sepa a ix
spli s ( esp ec i ely, he p e u b ed cu es c oss ans e sely).
The aim o his lec u e is o p esen a simila heo y o mo e dimensions. The na u al
ame is o conside wis maps on co angen bundles. Once he sui able deni ion o
unp e u b ed sepa a ix has b een in o duced (a non- i ial p oblem in he high-dimensional
case), a scala unc ion
L
can be dened on i , in such a way ha
L
e ies he same
p op e ies han in he plana case. The de i a ion o
L
is easily ela ed o a ia ional
p inciples, and he p op e y o b eing a scala unc ion ins ead o a ec o ial unc ion like
he classical Melniko unc ion, makes i mo e use ul o compu a ions and geome ical
unde s anding. E en mo e, i allows he applica ion o Mo se heo y o es ablish he
minimal numb e o ans e se homo clinic o bi s.
The esul s o be p esen ed in his lec u e a e alid o exac symplec ic maps on
a bi a y exac symplec ic mani olds, ha is, he wis cha ac e is no essen ial. Weha e
es ic ed ou sel es o wis maps only o simplici y. Full de ails o he ideas p esen ed
he e a e con ained in 4], whe e ano he mo e gene al si ua ion (i.e., he exac symplec ic
case) is s udied. Rela ed ideas can b e ound in 1, 15, 14, 8, 7].
2. The maps
A
wis map
F
is a map om a connec ed subse
U
o he co angen bundle o a mani old
M
(which can be non-compac ) in o
U
, which comes equipp ed wi h a wis gene a ing
unc ion
L
:
MM !
R
ha sa ises
F

(
y
d
x
)
;
y
d
x
=
Y
d
X
;
y
d
x
= d
L
(
x X
)

(
X Y
)=
F
(
x y
)

whe e (
x y
) a e any co angen co o dina es on
T

M
, ha is,
x
a e co o dina es on
M
,
ex ended o co o dina es (
x y
)in heob ious way. The symplec ic o m
!
0
on
T

M
eads
2
as
!
0
= d
x
^
d
y
in co angen co o dina es. This can also be w i en in a co o dina e ee
manne . Gi en
L
, one can e ie e he map (a leas implici ly) om
y
=
;
@
1
L
(
x X
), and
Y
=
@
2
L
(
x X
). This can b e done globally (i.e.,
U
=
T

M
) only when
M
is dieomo phic
o ab e o
T

M
, o example when
M
is he co e ing space o
T
n
o a mani old o
cons an nega i e cu a u e.
Finally, le us deno e by

:
T

M!M
he canonical p o jec ion.
3. The heo y
Assume now ha we a e gi en a smo o h wis dieomo phism
F
0
on he co angen bundle
T

M
. Le
L
0
be i s wis gene a ing unc ion. We assume ha he e exis s a
hype bolic
xedpoin
z
1
0
o
F
0
,such ha i s
n
-dimensional (uns able and s able) in a ian mani olds
W
u

s
0
a e
doubled
, ha is, hey coincide:
W
:=
W
u
0
=
W
s
0
.
In he plana case, he
sepa a ix
consis s o he in e sec ion o he in a ian cu es,
excep o he hyp e b olic xed p oin , which is he only p oin whe e suchin a ian mani olds
a e no submani olds o he co angen bundle. In he high-dimensional case, he si ua ion
is mo e complica ed. We can conside h ee op ologies on he se
W
: he one induced by
he inclusion
W 
T

M
, and he wo ones induced by he inclusions
W  W
u

s
0
. We
 s dene he
bi u ca ion se
 o his p oblem as he subse o
W
o med by he p oin s
such ha he h ee op ologies do no coincide. Then, he
sepa a ix
 is dened as i s
complemen a y in
W
, i.e.,
:=
Wn

:
Wi h his deni ion, i u ns ou ha 
is
a doubly asymp o ic exac submani old o
T

M
,in a ian by
F
0
.
Nex , conside a p e u b ed wis map
F
"
, and le
L
"
=
L
0
+
"
L
1
+
O
(
"
2
) b e he wis
gene a ing unc ion o
F
"
.Fo 0
<
j
"
j
1, he e exis s a hyp e b olic xed p oin
z
1
"
o
F
"
,
close o
z
1
0
, and i is no es ic i e o no malize he wis gene a ing unc ion by imp osing
L
"
(
x
1
"
x
1
"
)=0, whe e
x
1
"
=

(
z
1
"
). In pa icula ,
L
1
(
x
1
0
x
1
0
) = 0, whe e
x
1
0
=

(
z
1
0
).
Wenow dene he
Melniko po en ial
in he same way as o plana wis maps 5]:
L
:
;!
R
 L
(
z
)=
X
k
2
Z
L
1
(
x
k
x
k
+1
)
 x
k
=

(
z
k
)
 z
k
=
F
k
0
(
z
)
 z
2

:
(1)
The Melniko heo y is based on he ollowing p op e ies o he Melniko p o en ial 4]:
;
L
:
!
R
is well-dened, smo o h and in a ian unde he ac ion o he unp e u b ed
map:
L

F
0
=
L
. Consequen ly,
L
canbedenedon he
educedsepa a ix


=
=F
0
.
;
The die en ial o he Melniko po en ial
M
= d
L
(called he
Melniko unc ion
),
measu es, in  s o de in
"
, he dis ance b e ween he p e u b ed in a ian mani olds,
and is also dened on he educed sepa a ix 

.
;
I
L
6
cons an , hen he p e u b ed in a ian mani olds
W
u

s
"
spli o 0
<
j
"
j 
1,
i.e., hey do no coincide.
;
I
L
has a c i ical p oin a
z
=
z
0
hen, o 0
<
j
"
j
1,
W
u

s
"
in e sec ans e sally
on a homo clinic p oin nea
z
0
.
;
I he unp e u b ed in a ian mani olds a e
comple ely doubled
(i.e., =
z
1
0
g
), he
educed sepa a ix is a
compac
n
-dimensional mani old wi hou b ounda y. Ac ually,
i

deno es he sign o he p o duc o he eigen alues wi h mo dulus g ea e ha one
o
DF
0
(
z
1
0
), and
S
n
;
1
s ands o he uni sphe e o
R
n
, hen 

is homeomo phic o
3
S

S
n
;
1
o

= +, and i is somewha mo e complica ed o

=
;
( o mo e de ails,
see 4]).
In his si ua ion, Mo se heo y applied o he Melniko po en ial, hough as a unc ion
o e 

,gi es he minimal numb e o ans e se homo clinic o bi s, unde condi ions
o gene ic p osi ion.
;
The e exis s a
a ia ional p inciple
,in an analogous way o he one o he plana
case 9, 6], which es ablishes ha he homo clinic o bi s o a wis map wi h wis
gene a ing unc ion
L
a e he ex emals o he
homoclinic ac ion
W

O
]:=
X
k
2
Z
L
(
x
k
x
k
+1
)

O
=(
x
k
)
k
2
Z

and a
homoclinic a ea
can b e dened o e e y pai o homo clinic o bi s
O
=(
x
k
)
k
2
Z
,
O
0
= (
x
0
k
)
k
2
Z
,and is gi en by he die ence o homo clinic ac ions 
W

O

O
0
] =
W

O
0
]
;
W

O
]. In e ms o he Melniko p o en ial, he e is also a nice exp ession o
he homo clinic a ea:

W

O

O
0
]=
"
;
L
(
z
0
0
)
;
L
(
z
0
)

+
O
(
"
2
)
:
We nish his su ey o esul s wi h wo ema ks ab ou die en , bu ela ed, se ings.
1. Rega ding Hamil onian ows, le
H
"
:
T

M
R
!
R
b e a ime-p e io dic Hamil onian
o p e io d
T
,and
F
"
= 
T
"
unde he condi ions o his sec ion, whe e 
"
(
z
) is he
solu ion o he asso cia ed Hamil onian equa ions wi h ini ial condi ion
z
a
=0. I
H
=
H
0
+
"H
1
+
O
(
"
2
), one can see 4] ha he Melniko po en ial akes he o m
(al eady known o Poinca e)
L
(
z
)=
;
Z
R
H
1
(
0
(
z
)

)d

whe e
H
1
is de e mined by imp osing
H
1
(
0
(
z
1
0
)

)

0, o simply
H
1
(
z
1
0

)

0, i
H
0
is au onomous.
2. In ela ion wi h a non-symplec ic se ing, le us assume now ha
M
=
R
n
, ha is,
T

M
=
R
2
n
. In ha case, using a die en poin o iew 13, 2], one can assume
ha he unp e u b ed map
F
0
:
R
2
n
!
R
2
n
p ossesses
n
indep enden  s in eg als
H
1
:::H
n
on he sepa a ix  (no necessa ily in in olu ion, since his concep e-
qui es a symplec ic s uc u e), and conside a p e u ba ion
F
=
F
0
+
"F
1
+
O
(
"
2
)
(no necessa ily symplec ic). Then, he Melniko ec o ial unc ion
M
:
!
R
n
can
b e w i en as (compa e wi h he plana case in 5]):
M
=(
M
1
::: M
n
)
>
 M
j
(
z
)=
X
k
2
Z
h
H
j
(
z
k
+1
)
F
1
(
z
k
)
i
 z
k
=
F
k
0
(
z
)
 z
2

:
4. The example
Le us conside
cen al
s anda d-like maps on
R
2
n
=
T

R
n
, ha is,
F
0
(
x y
)=(
y
;
x
+
V
0
(
y
)) o
L
0
(
x X
)=
;h
x X
i
+
V
0
(
X
) (2)
whe e
V
0
(
x
)=
V
c
(
k
x
k
2
) o some unc ion
V
c
:0

1
)
!
R
. Then, he angula momen a"
A
ij
(
x y
)=
x
i
y
j
;
x
j
y
i
a e  s in eg als and he (
n
+ 1)-dimensional mani old in
R
2
n
o ze o
angula momen a is
A
n
+1
0
:=
(
x y
):
A
ij
(
x y
)=0
g
=
(
qa pa
):
a
2
S
n
;
1

(
q p
)
2
R
2
g
.
4
We nowin o duce he
educed map
in
A
n
+1
0
o
F
, as he plana s anda d-like map
:
R
2
!
R
2
dened by
(
q p
)=(
p
;
q
+2
V
0
c
(
p
2
)
p
). We no e ha
(
q p
)=(
Q P
)
()
F
(
qa pa
)=(
Qa P a
)

8
(
q p
)
2
R
2
a
2
S
n
;
1

(3)
so ha he non- i ial dynamics on he sepa a ix is induced by he educed map.
To akead an age o he esul s o plana wis maps in he lec u e 5], wein o duce
now he McLachlan map 10] as he cen al s anda d-like map wi h po en ial
V
0
(
y
) =

ln(1 +
k
y
k
2
),
>
1. I has he exp ession
F
0
(
x y
)=

y
;
x
+
2
y
1+
k
y
k
2
!
>
1
:
(4)
I s educed map is no hing else bu he McMillan map 11] whose sepa a ix ; = ;
+
has
he ollowing na u al pa ame e iza ion 3]:
;=
z
0
(
)=(
q
0
(
)
p
0
(
))
g
 q
0
(
)=
p
0
(
;
h
)
 p
0
(
) = sinh
h
sech

whe e
h>
0 is de e mined by he equa ion
cosh
h
=
:
Now, i is easy o check ha
1. The o igin is a hyp e b olic xed p oin o
F
0
, and Sp ec
DF
0
(0)] =
e

h
g
.
2. The in a ian mani olds o
F
0
a e comple ely doubled, and he sepa a ix is gi en by
=
(
qa pa
):(
q p
)
2
;
a
2
S
n
;
1
g
:
3. The unc ion
z
0
:
R

S
n
;
1
;!
gi en by
z
0
(
 a
)=(
p
0
(
;
h
)
a p
0
(
)
a
)
 p
0
(
)=sinh
h
sech

(5)
is a
na u al pa ame e iza ion
o he sepa a ix, i.e.,
z
0
is a dieomo phism ha sa ises
F
0
(
z
0
(
 a
)) =
z
0
(
+
h a
), o
2
R
and
a
2
S
n
;
1
.
As exp ec ed, wenow conside a gene al p e u ba ion o (4) ha p ese es he s anda d
cha ac e , i.e.,
F
"
(
x y
)=

y
;
x
+
2
y
1+
k
y
k
2
+
"
V
(
y
)
!
>
1
"
2
R

(6)
wi h
V
:
R
n
!
R
de e mined by imp osing
V
(0) = 0. The gene a ing unc ion o
F
"
ha
anishes a he o igin is
L
"
=
L
0
+
"
L
1
, whe e
L
0
(
x X
)=
;h
x X
i
+

ln(1 +
k
X
k
2
) and
L
1
(
x X
)=
V
(
X
). The Melniko po en ial is simply
L
:
R

S
n
;
1
!
R
 L
(
 a
)=
X
k
2
Z
V
(
p
0
(
+
hk
)
a
)
 p
0
(
)=
sinh
h
cosh
:
(7)
Since
L
is
h
-p e io dic in
( his is he in a iance o he Melniko po en ial unde he
ac ion o he unp e u b ed map), wecan conside
dened mo dulo
h
, i.e.,
L
dened o e
he educed sepa a ix
S
1

S
n
;
1
.
Rep ea ing he a gumen s o he case o he plana wis maps 5], we see ha i
V
is a
non-cons an eal en i e unc ion, hen
V
(
p
0
(
)
a
) has he same isola ed singula i ies in he
complex a iable
as
p
0
(
), and i is no di cul o check ha hey emain as singula i ies
o he Melniko p o en ial, whichmus b e non-cons an . In his waywe ha e es ablished
he ollowing esul .
5
Theo em 1
I
V
is a non-cons an eal en i e unc ion, hen he pe u bed in a ian man-
i olds o he s anda d-like map (6) spli , o
0
<
j
"
j
1
.
Re e ences
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in Ad . So . Ma h.
2. T.C. Boun is, A. Go iely, and M. Kollmann. A Melniko ec o o
n
-dimensional mappings.
Phys.
Le . A
, 206:38{48, 1995.
3. A. Delshams and R. Ram ez-Ros. Poinca e-Melniko -A nold me ho d o analy ic plana maps.
Nonlinea i y
, 9(1):1{26, 1996.
4. A. Delshams and R. Ram ez-Ros. Melniko po en ial o exac symplec ic maps. P ep in , May
1996. To app ea in
Comm. Ma h. Phys.
5. A. Delshams and R. Ram ez-Ros and T.M. Sea a. Spli ing o sepa a ices in Hamil onian sys ems
and symplec ic maps. In Simo 12].
6. R.W. Eas on. T anspo h ough chaos.
Nonlinea i y
, 4:583{590, 1991.
7. H.E. Lomel. Saddle connec ions and he e o clinic o bi s o s anda d maps.
Nonlinea i y
, 9:649{668,
1996.
8. H.E. Lomel. Applica ions o he Melniko me ho d o wis maps in highe dimensions using he
a ia ional app oach. P ep in , Uni . o Minneso a, 1993. To appea in E go dic Theo y Dynamical
Sys ems.
9. R.S. MacKay, J.D. Meiss, and I.C. Pe ci al. T ansp o in Hamil onian sys ems.
Phys. D
, 13:55{81,
1984.
10. R.I. McLachlan. In eg able ou {dimensional maps o s anda d yp e.
Phys. Le . A
, 177:211{214,
1994.
11. E.M. McMillan. A p oblem in he s abili y o p e iodic sys ems. In E. B i in and H. Odabasi, edi o s,
Topics in mode n physics, a ibu e o E.V. Condon
, pages 219{244. Colo ado Asso c. Uni . P ess,
Boulde , CO, 1971.
12. C. Simo, edi o .
Hamil onian Sys ems wi h Th ee o Mo eDeg ees o F eedom
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