Singular splitting of separatrices for the perturbed McMillan map
Full text
SINGULAR SPLITTING OF SEPARATRICES
FOR THE PERTURBED MCMILLAN MAP
AMADEU DELSHAMS y RAFAEL RAM
IREZ-ROS
Depa amen deMa ema ica Aplicada I, Uni e si a Poli ecnica de Ca alunya,
Diagonal 647, 08028 Ba celona (Espa ~na).
The p oblem
In his lec u e, we conside he ollowing amily o plana s anda d-likemaps
F
(
x y
)=
y
;
x
+
2
0
y
1+
y
2
+
"V
0
(
y
)
!
whe e
V
(
y
)=
P
n
1
V
n
y
2
n
is an e en en i e unc ion.
P o ided ha
0
+
V
1
" >
1, he o igin
O
= (0
0) is a hyp e b olic xed poin wi h
Sp ec d
F
(
O
)] =
n
e
h
o
, and i s
cha ac e is ic exponen
h>
0isgi en bycosh
h
=
0
+
V
1
"
.
Fo
"
=0,
F
is an in eg able map (called
McMil lan map
), whose s able and uns able
in a ian cu es o he o igin coincide, gi ing ise o a
sepa a ix
.Thus, he map
F
can
be conside ed as a p e u ba ion o he McMillan map,
"
b eing he
pe u ba ion s eng h
.
These wo pa ame e s,
h>
0and
"
, will b e conside ed he in insic pa ame e s o he
map
F
unde s udy.
Ou goal is o show ha o
"
6
= 0 and o a gene al p e u ba ion, he s able and
uns able in a ian cu es o he p e u b ed map in e sec ans e sally along exac ly wo
p ima y homoclinic o bi s
in he s quad an in pa icula , he unp e u b ed sepa a ix
spli s. The e m p ima y means ha he homo clinic o bi s p e sis o all
"
small enough.
The pieces o he p e u b ed in a ian cu es b e ween wo consecu i e homo clinic p oin s
enclose a egion called
lobe
.Ou measu e o he spli ing size will be he a ea o his
lob e. This
lobe a ea
is a homo clinic symplec ic in a ian , ha is, i do es no dep end on
he symplec ic co o dina es used, and all he lob es ha e he same a ea. Lob e a eas also
measu e he ux along he homo clinic angle, which is ela ed o he s udy o ansp o .
Bo h pa ame e s,
h >
0and
"
, will be small enough", bu he exac in e p e a ion
o his sen ence is c ucial o unde s anding he die en kinds o esul s o b e p esen ed.
Sp ecically,we a e going o deal wi h he ollowing si ua ions:
1.
The egula case
: xed
h>
0, and
"
!
0.
2.
The singula case
:
h
!
0
+
. In i s u n his case sub di ides in wo sub-cases:
(a)
The non-pe u ba i e case:
"
xed and
h
!
0
+
.
(b)
The pe u ba i e case:
"
=
O
(
h
p
)and
h
!
0
+
, o some
p>
0.
The analy ical esul s he e p esen ed a e exp essed in e ms o he
Melniko po en ial
o he p oblem, which gi es
explici
o mulae o ou map. This is he eason o ou
choice o he p e u b ed McMillan map as a mo del, ins ead o mo e celeb a ed maps like
he Henon map o he Taylo -Chi iko map. (See 6] o esul s conce ning hose maps.)
The name singula " o he case
h
!
0
+
,is due o he ac ha he lob e a eas a e
exp onen ially small in
h
. The measu e o suchsmall quan i ies equi es a e y ca e ul
ea men , b o h om a nume ical and an analy ical p oin o iew.
The mo del
The amily o s anda d-like maps unde s udy is gi en by
F
(
x y
)=(
y
;
x
+
U
0
(
y
))
U
(
y
)=
0
log(1 +
y
2
)+
"V
(
y
)
(1)
whe e
V
(
y
)=
P
n
1
V
n
y
2
n
is an e en en i e unc ion. Fo
:=
0
+
"V
1
>
1
he o igin
O
= (0
0) is a hyp e b olic xed poin wi h Sp ec d
F
(
O
)] =
n
e
h
o
,whe e he
cha ac e is ic exponen
h>
0is de e mined by cosh
h
=
.
We will conside he cha ac e is ic exp onen
h
and he
pe u ba ion s eng h
"
as he
in insic pa ame e s o ou mo del. Acco dingly, o e e y
h >
0and e e y eal
"
, we
ew i e he map (1)in he o m
F
(
x y
)= (
y
;
x
+
U
0
(
y
))
U
(
y
)=
U
0
(
y
)+
"U
1
(
y
)
U
0
(
y
)=
log(1 +
y
2
)
U
1
(
y
)=
V
(
y
)
;
V
1
log(1 +
y
2
)
:
(2)
F om now on, he subsc ip 0" will deno e an unp e u b ed quan i y, ha is,
"
=0, and
he ollowing no a ions will b e used wi hou u he commen :
= cosh
h
=sinh
h:
Se ing
"
= 0 in (2), we ob ain he so-called
McMil lan map
F
0
(
x y
)=(
y
;
x
+
U
0
0
(
y
)) =
y
;
x
+
2
y
1+
y
2
!
which is an
in eg able
map, wi h a p olynomial s in eg al gi en by
I
0
(
x y
)=
x
2
;
2
xy
+
y
2
+
x
2
y
2
:
The phase space asso cia ed o
F
0
is a he simple, since i is olia ed by he le el cu es
o he s in eg al
I
0
, which a e symme ic wi h esp ec o he o igin. As
>
1, he ze o
le el o
I
0
is a lemnisca e, whose lo ops a e
sepa a ices
o he o igin. F om nowon,we will
concen a e on he sepa a ix in he quad an
x y >
0
g
,whichcan be pa ame e ized
by
z
0
(
)=(
x
0
(
)
y
0
(
)) = (
0
(
;
h=
2)
0
(
+
h=
2))
0
(
)=
sech
:
(3)
This pa ame e iza ion is called
na u al
since
F
0
(
z
0
(
)) =
z
0
(
+
h
), a ac ha can be
checked simply by no ing ha
0
(
) is a homo clinic solu ion o he die ence equa ion
0
(
+
h
)+
0
(
;
h
)=
U
0
0
(
0
(
))
:
A na u al pa ame e iza ion is unique excep o a ansla ion in he indep enden a i-
able. To de e mine i ,i iswo h lo oking a he e e so s o he map.
Indeed, he in olu ion
R
+
0
(
x y
):=(
y x
)isa
e e so
o he McMillan map
F
0
, ha is,
F
;
1
0
=
R
+
0
F
0
R
+
0
.The sepa a ix is
R
+
0
-symme ic, i.e.,
R
+
0
=, and in e sec s ans-
e sely he xed se
C
+
0
:=
z
:
R
+
0
z
=
z
g
o
R
+
0
in one p oin
z
+
0
. The pa ame e iza ion (3)
o has been chosen o sa is y
z
0
(0) =
z
+
0
.
Mo eo e , he in olu ion
R
;
0
:=
F
0
R
+
is ano he e e so o
F
0
.The sepa a ix
is also
R
;
0
-symme ic and in e sec s ans e sely he xed se
C
;
0
o
R
;
0
in one p oin
z
;
0
, and i u ns ou ha
z
0
(
h=
2) =
z
;
0
.The asso cia ed o bi s
O
+
0
:=
z
0
(
nh
) :
n
2
Z
g
,
O
;
0
:=
z
0
(
h=
2+
nh
):
n
2
Z
g
, a e called
symme ic
homo clinic o bi s, since
R
0
O
0
=
O
0
.
Fo
"
6
= 0, he phase p o ai o he exac map (2) lo oks mo e in ica e. The o igin is
a hyp e b olic xed p oin wi h he
same
cha ac e is ic exp onen
h
, since he p e u ba ion
"U
0
1
(
y
) =
O
(
y
3
) do es no no con ain linea e ms a he o igin. We deno e by
W
u
s
i s
uns able and s able in a ian cu es wi h esp ec o
F
. Since he map (2) is odd, he
in a ian cu es a e symme ic wi h esp ec o he o igin, so ha we concen a e only on
he p osi i e quad an
x y >
0
g
.
By he o m o he p e u ba ion,
R
+
:=
R
+
0
is also a e e so o
F
, as well as he
in olu ion
R
;
:=
FR
+
, which is gi en by
R
;
(
x y
) = (
x
;
y
+
U
0
(
x
)). Thei xed se s
C
=
z
:
R
z
=
z
g
a e imp o an b ecause
R
W
u
=
W
s
. Consequen ly, any poin in
he in e sec ion
C
W
u
is a homo clinic p oin , and gi es ise o a symme ic homo clinic
o bi .
Since he sepa a ix in e sec s ans e sely he unp e u b ed cu e
C
0
a he p oin
z
0
, he e exis s a poin
z
=
z
0
+
O
(
"
)
2
C
W
u
and, he e o e, he e exis a leas
wo symme ic homo clinic o bi s on he quad an
x y >
0
g
, o
j
"
j
small enough. They
a e called
p ima y
since hey exis o a bi a y small
j
"
j
.
The Melniko heo y
Wenow ecall some p e u ba i e esul s 1, 2] o de ec he exis ence o ans e se p ima y
homo clinic o bi s o exac maps. Fo simplici y, we will assume ha all he ob jec s a e
smo o h and we shall es ic he discussion o maps on he plane wi h he usual symplec ic
s uc u e: he a ea.
Gi en he symplec ic o m
!
= d
x
^
d
y
on he plane
R
2
, a map
F
:
R
2
!
R
2
is called
exac
i he e exis s some unc ion
S
:
R
2
!
R
such ha
F
(
y
d
x
)
;
y
d
x
= d
S
. The
unc ion
S
is called he
gene a ing unc ion
o
F
and, excep o an addi i e cons an , i
is uniquely de e mined.
Le
F
0
:
R
2
!
R
2
be an in eg able exac dieomo phism wi h a sepa a ix o a
hyp e b olic xed p oin
z
1
. Nex , conside a amily o exac dieomo phisms
F
"
=
F
0
+
"F
1
+
O
(
"
2
), as a gene al p e u ba ion o he si ua ion ab o e, and le
S
"
=
S
0
+
"S
1
+
O
(
"
2
)
be he gene a ing unc ion o
F
"
.
Wein o duce he
Melniko po en ial
o he p oblem as he smo o h eal- alued unc ion
L
:
!
R
L
(
z
)=
X
n
2
Z
b
S
1
(
z
n
)
z
n
=
F
0
n
(
z
)
z
2
(4)
whe e
b
S
1
:
R
2
!
R
is dened by
b
S
1
=
S
1
;
y
d
x
(
F
0
)
F
1
]. (In comp onen s, w i ing
F
0
=(
X
0
Y
0
),
F
1
=(
X
1
Y
1
),
b
S
1
is simply gi en by
b
S
1
=
S
1
;
Y
0
X
1
.) In o de o ge an
absolu ely con e gen se ies (4),
b
S
1
is de e mined by imp osing
b
S
1
(
z
1
)=0.
The die en ial o
L
is a geome ical ob jec whichgi es he
O
(
"
)-dis ance b e ween he
p e u b ed in a ian cu es
W
u
s
. Mo e p ecisely, le (
e
)be some
co angen coo dina es
adap ed o | ha is, in hese co o dina es he sepa a ix is gi en lo cally by
e
= 0
g
and he symplec ic o m
!
eads as d
^
d
e
|and le
e
=
E
u
s
(
)
g
be apa o
W
u
s
.
(Le us ecall ha co angen co o dina es can be dened in neighb o ho o ds o Lag angian
sub-mani olds.) Then, in 2]i isshown ha
E
u
(
)
;
E
s
(
)=
"L
0
(
)+
O
(
"
2
)
and ha he cons uc ion ab o e do es no dep end on he co angen co o dina es used.
The ollowing heo em is as aigh o wa d co olla y o his geome ic cons uc ion.
Theo em 1
Unde he abo e no a ions and hypo heses, he non-degene a e c i ical poin s
o
L
a e associa ed o pe u bed ans e se homoclinic o bi s. Mo eo e , when al l he
c i ical poin s o
L
a e non-degene a e, al l he p ima y homoclinic o bi s a ising om
a e ound in his way. Final ly, i
z
and
z
0
a e consecu i e (in he in e nal o de o
he sepa a ix) non-degene a e c i ical poin s o
L
, hei associa ed pe u bed homoclinic
o bi s de e mine a lobe wi h a ea
A
=
"
L
(
z
)
;
L
(
z
0
)] +
O
(
"
2
)
:
The egula case
We a e now eady o apply he heo y ab o e o ou mo del. Along his sec ion, he
cha ac e is ic exp onen
h>
0willbe conside ed
xed
,and hen
"
!
0.
I is wo h no ing ha he knowledge o he na u al pa ame e iza ion (3) o he un-
p e u b ed sepa a ix will be he c ucial poin o compu e explici ly he Melniko po-
en ial (4).
The map
F
=
F
0
+
"F
1
+
O
(
"
2
)gi en in (2) is exac wi h gene a ing unc ion
S
(
x y
)=
;
xy
+
U
0
(
y
)+
"U
1
(
y
). W i ing i s exp ession in comp onen s
F
0
=(
X
0
Y
0
),
F
1
=(
X
1
Y
1
),
i u ns ou ha
X
1
=0, and consequen ly
b
S
1
(
x y
)=
S
1
(
x y
)=
U
1
(
y
).
The pa ame e iza ion (3) allows us o w i e he Melniko p o en ial (4) o ou p oblem
as
L
(
):=
L
(
z
0
(
)) =
X
n
2
Z
U
1
(
y
0
(
+
hn
)) =
X
n
2
Z
(
+
hn
)
;
g
(
+
hn
)]
whe e
(
):=
V
(
0
(
+
h=
2)) and
g
(
):=
V
1
log (1 +
0
(
+
h=
2)
2
).
We a e now con on ed o he compu a ion o
L
(
). Le
P
n
2
Z
n
(
h
)
2
n
b e he Lau en
expansion a ound
=0 o he unc ion
7!
(
;
h=
2+
i
=
2
;
i
h
), and
0
(
h
):=8
X
n
1
(2
)
2
n
;
1
(2
n
;
1)!
;
n
(
h
)=8
b
V
(2
)+
O
(
h
2
)
(5)
whe e
b
V
(
):=
P
n
1
V
n
2
n
;
1
=
(2
n
;
1)! is he so-called Bo el ans o m o
V
(
y
). I
V
(
y
)is
a p olynomial,
0
(
h
) can b e explici ly compu ed in a ni e numb e o s eps. Fo ins ance,
0
(
h
)=
(
8
2
2
h
;
2
o
V
0
(
y
)=
y
8
3
2
4
h
;
2
1 +
2
h
;
2
] o
V
0
(
y
)=
y
3
:
I u ns ou ha
0
(
h
) is an e en en i e unc ion such ha
L
(
)=cons an +
e
;
2
=h
cos(2
=h
)
h
;
0
(
h
)
=
2+
O
(
e
;
2
2
=h
)
i
:
(6)
We e e o 3] o he de ails.
F om he o mula (5), i is clea ha i
b
V
(2
)
6
= 0 and
h
is small enough, he se o
c i ical p oin s o he Melniko p o en ial (6) is
h
Z
=
2. All o hem a e non-degene a e, and
pa ame e ize he wo unp e u b ed, symme ic, p ima y homo clinic o bi s
O
0
.Now, he
ollowing esul is a co olla y o heo em 1.
Theo em 2
Assume ha
b
V
(2
)
6
=0
. Then, o any smal l enough (bu xed) cha ac e -
is ic exponen
h >
0
, he e exis s a posi i e cons an
"
=
"
(
h
)
such ha he map (2)
has exac ly wo ans e se, symme ic, p ima y homoclinic o bi s
O
in he quad an
x y >
0
g
, o
0
<
j
"
j
<"
. These o bi s de e mine a lobe wi h a ea
A
=
"A
Mel
+
O
(
"
2
)
,
whe e he s o de in
"
app oxima ion
A
Mel
is gi en by
A
Mel
=
L
(
h=
2)
;
L
(0) =
e
;
2
=h
h
0
(
h
)+
O
(
e
;
2
2
=h
)
i
:
We no e ha
"A
Mel
is he dominan e m o he Melniko o mula o he lob e a ea
A
only i
j
"
j
< "
(
h
) =
o
(exp(
;
2
=h
)). O he wise, in he case
"
=
O
(
h
p
), he Melniko
heo y as desc ib ed is no use ul, since i only gi es he e y coa se es ima e
A
=
O
(
h
2
p
),
and no he desi ed exp onen ially small asymp o ic b eha io .
The singula non-p e u ba i e case
The limi
h
!
0
+
in (2) is highly singula , since all he in e es ing dynamics is con ained
in a
O
(
h
) neighbo hood o he o igin, which b ecomes a pa ab olic poin o he map o
h
=0. To see clea ly his beha io , we p e o m he ollowing linea change o a iables:
z
=
Cw C
=
h
;
1
=
2
1
=
2
1
=
2
;
1
=
2
!
z
=(
x y
)
w
=(
u
)
ha is, we diagonalize he linea pa o (2) a he o igin and we scale by a ac o
h
.
Then,
C
;
1
(
F
(
Cw
)) =
w
+
hX
0
(
w
)+
O
(
h
2
), whe e
X
0
(
u
)=
u
;
(
u
+
)
3
;
+
(
u
+
)
3
=1
;
(
V
1
+2
V
2
)
"
(7)
is a Hamil onian ec o eld, wi h asso cia ed Hamil onian
H
0
(
u
)=
u
;
(
u
+
)
4
=
4
:
(8)
This shows clea ly ha
C
;
1
FC
is
O
(
h
)-close o he iden i y, and ha , a e he change
o a iables
z
=
Cw
, he map (2) asymp o es o he Hamil onian ow asso cia ed o he
ec o eld (7) when
h
!
0
+
. When such si ua ion akes place, i is known ha he map
has homo clinic p oin s o i s (weakly) hyp e b olic xed p oin o
h
!
0
+
, i and only i he
limi Hamil onian ow has a homo clinic o bi o i s hyp e b olic equilib ium p oin .
F om he exp ession (8), we see ha he ze o le el
H
0
(
u
)= 0
g
con ains homo clinic
connec ions o he o igin i and only i
>
0, i.e., i and only i
(
V
1
+2
V
2
)
"<
1
:
(9)
Assuming
>
0, he homo clinic o bi o he Hamil onian (8) is gi en by
w
0
(
)=
;
1
=
2
cosh
;
sinh
2 cosh
2
cosh
+ sinh
2 cosh
2
!
which is analy ic on he s ip
2
C
:
j=
j
<d
:=
=
2
g
.In his si ua ion, i is also well-
known 5] ha he spli ing size is
O
(exp(
;
=h
)), o all
<
2
d
=
2
. We summa ize
his esul in he ollowing heo em.
Theo em 3
Fo any eal
"
e i ying (9), and any
2
(0
2
)
, he eexis s a cons an
N
=
N
(
"
)
0
such ha he a ea o he lobe be ween he in a ian cu es o he
map (2) sa ises:
j
A
j
N
e
;
=h
(
"
xed
h
!
0
+
)
:
The singula p e u ba i e case
The p e ious heo em gi es only an upp e b ound o he lob e a ea and no an asymp o ic
one ( he cons an
N
(
"
) can blow up when
!
2
). On he o he hand, i do es no
exclude he case
A
= 0, ha is, i canno de ec eec i e spli ing o sepa a ices. In he
p e u ba i ecase
"
=
O
(
h
p
), o
p>
6, he ollowing esul gi es an asymp o ic exp ession
o he lob e a ea in e ms o he Melniko p o en ial, and es ablishes ans e sal spli ing
o sepa a ices.
Theo em 4
Assume ha
"
=
O
(
h
p
)
,
p >
6
. Then, i
b
V
(2
)
6
= 0
, he e exis s
h
>
0
such ha he map (2) has exac ly wo ans e se, symme ic, p ima y homoclinic o bi s
in he s quad an , o al l
0
<h<h
. Mo eo e , hey enclose a lobe wi h a ea
A
=
"
e
;
2
=h
h
8
b
V
(2
)+
O
(
h
2
)
i
(
h
!
0
+
)
:
I
b
V
(2
)=0
, he e may exis mo e p ima y homoclinic o bi s, bu he a eao anylobe
is
O
(
"h
2
e
;
2
=h
)
.
The p o o o his heo em is con ained in 3, 4]. I is based on he s udy o he
p e u b ed in a ian cu es o he maps (1) o complex alues o he disc e ime
, as
close as p ossible o he singula i ies o he unp e u b ed na u al pa ame e iza ion
z
0
(
)
gi en in (3). This app oachwas sugges ed by V.I. Lazu kin se e al yea s ago, o he case
o he Taylo -Chi iko map.
To he b es o ou knowledge, his heo em is he s analy ical esul ab ou asymp-
o ics o singula sepa a ix spli ing o a map wi h a comple e and igo ous p o o .
Anume ical s udy o singula cases can b e ound in 4]. The nume ical esul s sugges
ha he lob e a ea
A
is gi en by
A
=
"
e
;
2
=h
h
"
(
h
)+
O
(
e
;
2
2
=h
)
i
(
"
xed
h
!
0
+
)
whe e
"
(
h
)isane en Ge ey-1 unc ion such ha he adius o con e gence o i s Bo el
ans o m is 2
2
, and
"
(
h
)=
0
(
h
)+
O
(
"
), uni o mly in
h
2
0
1].
We nish his lec u e by ema king ha he nume ical compu a ion o he lob e a eas
o singula cases equi es he use o an exp ensi e mul iple-p ecision a i hme ic and o
expand he in a ian cu es
W
u
s
up o an op imal o de , whichis e y la ge, see 4].
Re e ences
1] A. Delshams and R. Ram" ez-Ros, Poinca e-Melniko -A nold me ho d o analy ic
plana maps",
Nonlinea i y
,
9
, (1996), 1{26.
2] A. Delshams and R. Ram" ez-Ros, Melniko p o en ial o exac symplec ic maps",
o app ea in
Comm. Ma h. Phys.
, (1997).
3] A. Delshams and R. Ram" ez-Ros, Exp onen ially small spli ing o sepa a ices o
p e u b ed in eg able s anda d-like maps", o app ea in
J. Nonli. Sci.
, (1997).
4] A. Delshams and R. Ram" ez-Ros, Singula sepa a ix spli ing and Melniko
me ho d: An exp e imen al s udy", submi ed o
Expe imen . Ma h.
, (1997).
5] E. Fon ich and C. Simo, The spli ing o sepa a ices o analy ic dieomo phisms",
E god. Th. & Dynam. Sys.
,
10
,(1990), 295{318.
6] V.G. Ge eich, V.F. Lazu kin and M.B. Tabano , Exp onen ially small spli ing in
Hamil onian sys ems",
Chaos
,
1
, (1991), 137{142.