Table Of ContentExponentially small splitting and Arnold di(cid:11)usion for multiple
time s
ale systems
Mi
hela Pro
esi
Abstra
t
We
onsider the
lass of Hamiltonians:
1Pn(cid:0)1 2 1 2 p2 2 Pn
2 j=1 Ij + 2"In+ 2 +"[(
osq(cid:0)1)(cid:0)b (
os2q(cid:0)1)℄+"(cid:22)f(q) i=1sin( i);
1 iq
where 0 (cid:20) b < 2, and the perturbing fun
tion f(q) is a rational fun
tion of e . We prove
upper and lower bounds on the splitting for su
h
lass of systems, in regions of the phase spa
e
hara
terizedbyonefastfrequen
y. FinallyusinganappropriateNormalFormtheoremweprove
theexisten
e of
hains of hetero
lini
interse
tions.
Contents
1 Presentation of the model and main Theorems. 1
2 Perturbative
onstru
tion of the homo
lini
traje
tories 5
t
2.1 Whisker
al
ulus, the \primitive" = . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 The re
ursive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Proofs of the Theorems 9
3.1 The formal linear equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Lower bounds on the Melnikov term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Hetero
lini
interse
tion for systems with one fast frequen
y . . . . . . . . . . . . . . . 13
4 Tree representation 16
4.1 De(cid:12)nitions of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Admissible trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 Values of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 Tree identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4.1 Mark adding fun
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4.2 Fruit adding fun
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4.3 Changing the (cid:12)rst node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.5 Upper bounds on the values of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A Appendix 31
A.1 Proof of proposition 4.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
A.2 Normal form theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
A.3 Proof of Lemma 4.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1 Presentation of the model and main Theorems.
Thegeneralsettingofthispaperistheproblemofhomo
lini
splittingandArnol'ddi(cid:11)usionina-priori
stable systems with three or more relevant time s
ales. The general strategy is the one proposed in
[A2℄and[CG℄andinparti
ulartheappli
ationtoa-prioristablesystemsproposedin[G1℄andfurther
2 1 Presentation of the model and main Theorems.
developed in [GGM1℄. More pre
isely we
onsider a
lass of
lose to integrable n degrees of freedom
Hamiltonian systems for whi
h one
an prove the existen
e of n(cid:0)1 dimensional unstable KAM tori
togetherwith their stable and unstable manifolds. We use a perturbative diagrammati
onstru
tion
(proposedanddevelopedin[C℄,[G1℄and[GGM1℄)toproveupperboundsontheanglesofinterse
tion
ofthestableandunstablemanifoldsofaKAMtorus(homo
lini
splitting). Su
hboundsaregenerally
exponentially small in the perturbation parameter and depend on the
hosen torus and in parti
ular
onthenumberoffastdegreesoffreedom. Forsystemswithonefastdegreeoffreedomweproveaswell
lowerbounds onthe homo
lini
splitting through the me
hanismof Melnikovdominan
e. Finally for
su
hsystemsweprovetheexisten
eof\long"
hainsofhetero
lini
interse
tions;namelyweprodu
e
a list of unstable KAM tori T1:::;Th su
h that T1;Th are at distan
es of order one in the a
tion
variables and the unstable manifold of ea
h Ti interse
ts the stable manifold of Ti+1. This paper is
a generalization of the results of [G1℄,[GGM1℄,[GGM2℄, therefore in proving our
laims we will rely
heavily on intermediate results proved in the latter papers whi
h we will not prove again.
Consider the
lass of Hamiltonians:
1nX(cid:0)1I~2+ 1"I~n2+ p~2 +"[(
osq~(cid:0)1)(cid:0) 1(cid:0)
2(
os2q~(cid:0)1)℄+"(cid:22)f(q~)Xn sin ~i; (1.1)
2 2 2 4
j=1 i=1
where the pairs I~ 2 Rn; ~ 2 Tn and p~ 2 R;q~ 2 T are
onjugate a
tion-angle
oordinates, 0 <
(cid:20) 1,f(q~) is odd and analyti
on the torus and (cid:22)," are small parameters. We will
onsider them
P
independent and then prove that one
an prove Arnold Di(cid:11)usion for (cid:22)(cid:20)" , for an appropriate P.
This
lass of Hamiltonians is a model for a near to integrable system
lose to a simple resonan
e
where the dependen
e on the hyperboli
variables is not through the standard pendulum, but still
maintainsvariousqualitativepropertiesofthependulum. Namelywehavea\generalizedpendulum":
2 2
p~ 1(cid:0)
+"[(
osq~(cid:0)1)(cid:0) (
os2q~(cid:0)1)℄
2 4
p
whi
h has an unstable (cid:12)xed point in p~=q~=0 with Lyapunov exponent (cid:21)=
".
Generally one re-s
ales the time and a
tion variables so that the Lyapunov exponent is one:
I(t) = I~(
ppt"); (t)= ~( pt ); p(t) = p~(
ppt"); q(t)=q~( pt ): (1.2)
"
"
"
"
Su
h res
aling sends Hamiltonian 1.1 in:
(I;A(")I) p2 1 1(cid:0)
2 Xn
2 + 2 +
2[(
osq(cid:0)1)(cid:0) 4 (
os2q(cid:0)1)℄+(cid:22)f(q) sin( i) (1.3)
i=1
where A(") is the diagonal matrix with eigenvalues ai = 1 for i = 1;:::;n(cid:0)1 and an = ". So from
nowonwewillworkonHamiltonian1.3andturnba
ktoHamiltonian1.1onlytoprovetheexisten
e
of hetero
lini
hains. The system (1.3) is integrable for (cid:22) = 0. It represents a list of n un
oupled
rotators and a generalized pendulum (depending on the parameter
). We will denote the frequen
y
of the rotators (whi
h determines the initial data I(0)) by ! so that:
(cid:0)1
I(t)=I(0)=A !; (t)= (0)+!t:
The initial data are
hosen in an appropriate domain (physi
ally interesting in the variables I~) so
thatthereareatleastthree
hara
teristi
ordersofmagnitudeforthe frequen
iesof theunperturbed
system.
3
De(cid:12)nition 1.1. In frequen
y spa
e we (cid:12)rst
onsider the ellipsoid:
(cid:8) Xn (cid:9)
n 2
(cid:6):= x2R : xi=ai =2E
i=1
1
where E is an order one
onstant E (cid:24)O"(1).
For notational
onvenien
e we split the frequen
y ! in two ve
torial
omponents: ! =(p!1";"(cid:11)!2) with
m n(cid:0)m 1
!1 2R , !2 2R , and 0(cid:20)(cid:11)(cid:20) 2. Finally, given two suitable order one
onstants R;r (cid:24)O"(1),
we
onsider the region:
p p p
n (cid:11) (cid:11)
(cid:10)(cid:17)f!2R : "! 2(cid:6) ;r <j!1;ij<R and r <j!2j<R ; " j!2;ij(cid:21) "; " j!2;n(cid:0)mj(cid:24) "g:
We have
hosen the generalized pendulum so that its dynami
s on the separatrix is parti
ularly
2
simple , namely:
1 iq(t) sinh((cid:6)t)+i
q(t)=2ar
otg( sinh((cid:6)t)); e = : (1.4)
sinh((cid:6)t)(cid:0)i
There areat least three
hara
teristi
time s
alesO"("(cid:0)21), O"("(cid:11)), O"(p") (
oming from the degen-
erate variable In) and 1 whi
h is the Lyapunov exponent of the unperturbed pendulum.
m
Wewill
all 1;(cid:1)(cid:1)(cid:1) ; m thefastvariablesandwewillsometimesdenotethemas F 2T . Conversely
n(cid:0)m
we will
all m+1;(cid:1)(cid:1)(cid:1) ; n the slow variables S 2T .
Theperturbingfun
tionisatrigonometri
polynomialofdegreeoneintherotators andarational
3 iq
fun
tion in e . We have de
oupled the dependen
e of and q only to simplify the
omputations.
n
For ea
h ! 2R the unperturbed system has an unstable (cid:12)xed torus :
(cid:0)1
p(t)=q(t)=0; I(t)=I(0)=A !; (t)= (0)+!t:
Thestableandunstablemanifoldsofsu
htori
oin
ideand
anbeexpressedasgraphsontheangles.
1
De(cid:12)nition 1.2. Given any (cid:13) 2R, "<(cid:13) (cid:20)O("2) and a (cid:12)xed (cid:28) >n(cid:0)1, we de(cid:12)ne the set
n o
(cid:13) n
(cid:10)(cid:13) (cid:17) ! 2(cid:10):j!(cid:1)lj> jlj(cid:28) 8l2Z =f0g
of (cid:13);(cid:28) Diophantine ve
tors in (cid:10). Now we
onsider
(cid:3) 1 1
(cid:10)(cid:13) (cid:17)(cid:10)(cid:13) (cid:2)((cid:0) ; )
2 2
(cid:3)
and for all (!;(cid:26))2(cid:10)(cid:13) we set !(cid:26) =(1+(cid:26))!.
(cid:3) n (cid:13)
For all (!;(cid:26))2(cid:10)(cid:13) and for all l2Z =f0g j!(cid:26)(cid:1)lj> 2jlj(cid:28).
!2(cid:10)(cid:13) implies that !1 and !2 are Diophantine as well; we will
all (cid:28)F and (cid:28)S their exponents.
2
KAM like theorems (see [CG℄,[C℄) imply that there exists (cid:22)0(";(cid:13)) (cid:24) " su
h that if j(cid:22)j (cid:20) (cid:22)0
(cid:3)
and if (!;(cid:26))2(cid:10)(cid:13), there exists one and only one n-dimensional H(cid:22)-invariant unstable torus T(cid:22)(!;(cid:26))
n
whose Hamiltonian (cid:13)ow is analyti
ally
onjugated to the (cid:13)ow T 3 # !#+!(cid:26)t. Moreover one
an
(cid:6)
parameterizethestableandunstablemanifoldsofT(cid:22)(!;(cid:26))byfun
tionsI (!;';q;(cid:22)), analyti
inthe
n 3 3
last three arguments, with ';q 2T (cid:2)[(cid:0) (cid:25); (cid:25)℄. Namely given
2 2
(cid:0) (cid:1)
(cid:6) (cid:6) (cid:6)
z (!;';q;(cid:22))= I (!;';q;(cid:22));p (!;';q;(cid:22));';q ;
1nowandinthefollowingwewillsaya(")(cid:24)O"(f("))if lim"!0+ fa(("")) =L6=0.
2The motion on the separatrix
an be easily obtained by dire
t
omputation; the mainfeature is that the motion
ontheseparatrixissu
hthateiq(t) isarationalfun
tionofet. Hereweare
onsideringthesimplest
lassofexamples,
whi
h
ontainsthestandardpendulum
=1.
3A
tuallyitissuÆ
ientthatthesingularityoff( (t);q(t)),whi
hisnearesttotherealaxisispolarandisolated.
4 1 Presentation of the model and main Theorems.
4
where the pendulum a
tion is derived by energy
onservation, the traje
tory :
(cid:26)
t +
(cid:8)Hz (!;';q;(cid:22)) if t>0
z(!;';q;(cid:22);t)= t (cid:0)
(cid:8)Hz (!;';q;(cid:22)) if t<0
tends exponentially to a quasi-periodi
fun
tion of frequen
y !.
5
Remark 1.3. We have introdu
ed the variable (cid:26) in order to (cid:12)x the energy of the perturbed system ,
namely given a list of !i 2(cid:10)(cid:13) one
an (cid:12)nd (cid:26)(!i;(cid:22)) su
h that all the
orresponding whiskered tori are
on the same energy surfa
e, see for instan
e [C℄.
De(cid:12)nition 1.4. We will study the di(cid:11)eren
e between the stable and unstable manifolds on an hyper-
plane transverse to the (cid:13)ow (a Poin
ar(cid:19)e se
tion), we
hoose the hyper-plane q =(cid:25) and
onsequently
drop the dependen
e on q. We
all
0 1 (cid:0) (cid:0) + (cid:1)
Gj(';!)= aj Ij(';!;0 )(cid:0)Ij(';!;0 )
2
0
the splitting ve
tor and prove that Gj('=0;!)=0. A measure of the transversality is
0 0
(cid:1)ij ='jGi(')j'=0
alled splitting matrix.
We will prove the following theorems:
0 6
Theorem 1. The splitting matrix (cid:1) satis(cid:12)es the formal power series relation :
0 0
(cid:1) (cid:24)AD B
0
where A;B are
lose to identity matri
es and D is the \holomorphi
part" of the splitting matrix;
namely its entries are expressed as integrals over R of analyti
fun
tions. Moreover the formal power
7
series involved are all asymptoti
This statement was posed as a
onje
ture in [G2℄ Paragraph3.
Corollary 1.5. The pre
eding Theorem implies that Hamiltonian (1.3), in regions of the a
tion vari-
ables
orrespondingtom6=0fasttimes
ales,hasexponentiallysmallupperboundsonthedeterminant
of the splitting matrix:
jdet(cid:1)0j(cid:20)Ce(cid:0)"
b ; withb= 1 ;
2m
provided that (cid:22)<"1+2mn .
Noti
e that Theorem 1
an be proved for mu
h more general systems than model (1.3).
Theorem2. Consider Hamiltonian 1.3 inregions of thea
tion variables
orresponding tom=1fast
variables and for perturbing fun
tions f(q) su
h that the pole f(q(t))
losest to the imaginary axis,
say t(cid:22), is su
h that jIm t(cid:22)j=d(cid:20)ar
sin
. Setting (cid:22)(cid:20)"P with P =p=2+8+4n where p is the degree
of the pole of f(q(t)) in t(cid:22)we prove that:
C1"(cid:0)p1e(cid:0)djp!"1j (cid:20)jdet(cid:1)0j(cid:20)C2"(cid:0)p2e(cid:0)djp!"1j
where C1;C2;p1;p2 are appropriate order one
onstants.
4(cid:8)tH istheevolutionattimetoftheHamiltonian(cid:13)ow1.3.
5The(cid:12)nalgoalisto(cid:12)ndhetero
lini
interse
tionsonthe(cid:12)xedenergysurfa
e,andso\Arnolddi(cid:11)usion",butinthe
followingse
tionswewilldis
ussonlyhomo
lini
interse
tionsandsowewilldroptheparameter(cid:26)
6wedenoteformalpowerseriesidentitieswiththesymbolA(cid:24)B
P
7Aformalpower series (cid:22)nan(") isasymptoti
iffor allq>0 there exists Q>0 su
h that foralln(cid:20)"(cid:0)q then
an(")(cid:20)"(cid:0)Qn.
5
Corollary 1.6. Under the
onditions of Theorem 2 the Hamiltonian (1.1) has hetero
lini
hains,
1 N 8
namely a set of N (cid:21) 1 traje
tories z (t);:::;z (t) together with N + 1 di(cid:11)erent minimal sets
T0;:::;TN su
h that for all 1(cid:20)i(cid:20)N
i i
lim dist (z (t);Ti(cid:0)1)=0= lim dist (z (t);Ti):
t!(cid:0)1 t!1
Moreover one
an
onstru
t su
h
hains between tori T(!a;(cid:22)), T(!b;(cid:22)) su
h that !a;!b 2 (cid:10)(cid:22) (cid:26) (cid:10)(cid:13)
and
j"(cid:0)12(!na (cid:0)!nb)j(cid:24)O"(1):
Thete
hniquesusedforprovingtheTheoremsarethoseproposedin[G1℄anddevelopedin[GGM1℄
for partially iso
hronous three time s
ale systems with three degrees of freedom. In this paper, par-
ti
ularattentionisgiventotheformalizationofthetreeexpansionsandofthe\Dysonequation"and
relative
an
ellations proposed in[GGM1℄. This enables us to extend Theorem 1 to systems with n
degrees of freedom and at least two time s
ales; moreover the proof is de(cid:12)nitely simpli(cid:12)ed and quite
ompa
t. Inthisarti
lewehave
onsidered
ompletelyaniso
hronoussystemsonlyto(cid:12)xanexample;
generalizingto partially(or totally, thus re
overingthe results of [BB1℄) iso
hronoussystems is
om-
pletely trivial. Indeed Theorem 1 and hen
e Corollary1.5
an be provedfor very generalsystems, as
we will show in a forth
oming paper.
Moreover we have generalized the
lass of perturbing fun
tions and the \ pendulum"(the literature
onsiders only trigonometri
polynomials and the standard pendulum); the latter generalizationsare
quite te
hni
al but nevertheless non trivial and interesting, we think, as the te
hniques we propose
areeasilygeneralizableandgivea
learpi
ture ofthe limits of provingArnolddi(cid:11)usion viaMelnikov
dominan
e.
2 Perturbative
onstru
tion of the homo
lini
traje
tories
One
an use perturbation theory to (cid:12)nd the (analyti
for (cid:22)(cid:20)(cid:22)0) traje
tories on the S/U manifolds
9
of Hamiltonian (1.3)
X
k k
z(';!;t)= ((cid:22)) z (';!;t):
k
Namely we insert the expansion in (cid:22) in the Hamilton equations of system (1.3):
I_j =(cid:0)((cid:22))
os jf(q); _j =ajIj;
1 2 Xn df (2.1)
p_ =
2 sinq(1(cid:0)(1(cid:0)
)
osq)(cid:0)((cid:22)) sin idq(q); q_=p;
i=1
(cid:6) (cid:6)
and (cid:12)nd initial data I(!;';(cid:22);0 ) (and
onsequently p(!;';(cid:22);0 )) su
h that the solution of (2.1)
tends exponentially to a quasi-periodi
fun
tion of frequen
y !. Inserting in the Hamilton equations
the
onvergentpower series representation:
P1 k k P1 k k
I(t;';(cid:22))= k=0((cid:22)) I (t;'); (t;';(cid:22))= k=0((cid:22)) (t;');
P1 k k 0 P1 k k
p(t;';(cid:22))= k=0((cid:22)) p (t;'); q(t;';(cid:22))=q (t)+ k=1((cid:22)) 0(t;')
8A
losed subset of the phase spa
e is
alled minimal(with respe
t to a Hamiltonian (cid:13)ow (cid:30)th) if it is non-empty,
invariant for (cid:8)th and
ontains a dense orbit. In our
ase the minimal sets will be unstable tori T(I) with !(I)
Diophantine.
9Noti
ethattheapexkonthefun
tionsI; representstheorderintheexpansionin(cid:22)NOTanexponent. Toavoid
onfusion,whenweneedtoexponentiate wealwayssettheargumentinparentheses.
6 2 Perturbative
onstru
tion of the homo
lini
traje
tories
10
we obtain, for k >0, the hierar
hy of linear non-homogeneousequations :
I_jk =Fjk(f ihgi=0;:::;n); _jk =ajIjk; forj =1;:::;n;
h<k
p_k =
12(cid:16)
os(q0(t))(cid:0)(1(cid:0)
2)
os(2q0(t))(cid:17) 0k+F0k(cid:0)f ihgi=h0<;::k:;n(cid:1); _0k =pk; (2.2)
k 1 dk
where the fun
tions Fi are de(cid:12)ned as follows. Set: [(cid:1)℄k = k!d(cid:22)k((cid:1))j(cid:22)=0; we have:
h kX(cid:0)1 i h kX(cid:0)1 i
Fjk(t)=(cid:0) jf1( ((cid:22))h ~h(t)) (cid:0)Æj0 0f0( ((cid:22))h 0h(t)) ; j =0;:::;n
k(cid:0)1 k
h=1 h=1
where ~h(t) is the ve
tor 0h(t);:::; nh(t),
f1( ~)=Xn sin if( 0); f0( 0)=
12(cid:0)(
os 0(cid:0)1)+ 1(cid:0)2
2 sin2 0(cid:1);
i=1
(cid:12)nally Æji denotes the Krone
kerdelta. For k =0 we obtain the unperturbed homo
lini
traje
tory:
0 (cid:0)1 0 0
z (t)=(A !;p (t);'+!t;q (t)));
0 0
(q (t);p (t)) is the lower bran
h of the pendulum separatrix starting at q = (cid:25) written in Equation
(1.4).
For k >0 we have a linear non-homogeneous ODE that we
an solve by variation of
onstants. The
fundamental solution of the linearized pendulum equation is given by:
(cid:12) (cid:12)
W(t)=(cid:12)(cid:12)(cid:12)ww_00 xx_0000(cid:12)(cid:12)(cid:12) ; w0 = 21(cid:27)(t)x10 where (cid:27)(t)= sign(t)
0
2
osh(t) 1 (cid:27)(t)x00 (cid:16) (cid:0) 2(cid:1) (cid:0) 2(cid:1)2 (cid:17)
x0 =
2+sinh(t)2 ; x0 = 2
4 2 (cid:0)3+4
t+sinh(2t)+4 (cid:0)1+
tanh(t) : (2.3)
It is easily seen (see [G1℄ or [C℄) that one
an
hoose an appropriate \primitive" in the right hand
side of the (cid:12)rst
olumn of equations (2.2) so that the solutions are exponentially quasi-periodi
.
t
2.1 Whisker
al
ulus, the \primitive" =
Let us (cid:12)rst de(cid:12)ne the fun
tion spa
es on whi
h we work, all the de(cid:12)nitions and statements of this
Subse
tion and of the following one are proposed and explained in detail in [G1℄, we are simply
reformulating them to suit our needs.
De(cid:12)nition 2.1. (i) H is the ve
tor spa
e (on C) generated by monomials of the form:
j
ajtj h i('+!t)(cid:1)(cid:23) n
m = (cid:27)(t) x e whereh2Z; (cid:23) 2Z ; j 2N;
j!
(cid:0)jtj
x=e ; a=0;1; (cid:27)(t)= sign(t): (2.4)
(ii) Given two positive
onstants b and d, H(b;d) is the subset of fun
tions f(t) analyti
on the real
axis in t6=0 that admit, separately for t>0 and t<0, a (unique) representation:
Xk jtjj (cid:27)(t)
f(t)= Mj (x;'+!t); (2.5)
j!
j=0
(cid:27)(t) (cid:27)(t)
with Mj (x;') trigonometri
polynomials in ' and the fun
tion Mk not identi
ally zero .
10whenitisnotstri
tlyne
essarywewillomitthepre(cid:12)xedinitialdataoftheangles'= 1(0);(cid:1)(cid:1)(cid:1); n(0); 0(0)=(cid:25)
t
2.1 Whisker
al
ulus, the \primitive" = 7
(cid:27)(t)
The Fourier
oeÆ
ients Mj(cid:23) (x) are all holomor-
phi
in the x-plane in a region eb
(cid:0)b
f0<jxj<e g[fjargxj<dg d
and have possible polar singularities at x=0. k is
alled the t degree of f.
In Figure1 we haverepresenteda possible domain
of analyti
ity for the M(cid:23)j Noti
e that H is
on-
tained in all the spa
es H(b;d); Figure 1:
moreover if jtj > b, f(t)
an be represented as an absolutely
onvergent series of monomials of
the type m, separately for t > b and t < (cid:0)b. One
an easily
he
k that the fun
tional that a
ts on
monomials m of the form (2.4) as:
( a+1 h i( +!t)(cid:1)(cid:23)Pj jtjj(cid:0)p
t (cid:0)(cid:27) x e p=0 (j(cid:0)p)!(h(cid:0)i(cid:27)!(cid:1)(cid:23))p+1 if jhj+j(cid:23)j6=0
= (m)= (cid:27)a+1jtjj+1 (2.6)
(cid:0) ifjhj+j(cid:23)j=0
(j+1)!
is a primitive of m.
t
We
an extend = , with jtj > b, to a primitive on fun
tions f 2 H(b;d) by expanding f in the
monomials m ( we obtain absolutely
onvergentseries) and applying (2.6). Then if jtj(cid:20)b we set
Zt
t 2(cid:27)(t)b
= (cid:17)= + ; (2.7)
2(cid:27)(t)b
obviously the
hoi
e of 2b is arbitrary and this is still the same primitive of f.
t
In H(b;d) we
an extend = to
omplex values of t su
h that t2C(b;d) where:
C(b;d):=ft2C :j Im tj(cid:20)d; j Re tj(cid:20)bg[ft2C :j Im tj(cid:20)2(cid:25); j Re tj>bg;
is the domain in Figure 1 in the t variables.
t
An equivalent (and quite useful) de(cid:12)nition of = is
I Zt
t du (cid:0)(cid:27)((cid:28))u(cid:28)
= f = e f((cid:28))d(cid:28); (2.8)
2i(cid:25)u
(cid:27)(t)1+is
where(cid:27)(t)= sign(Ret), t=t1+is, with t1;s2R and the integralisperformed on the line Im(cid:28) =s;
(cid:12)nally the integrals in u have to be
onsidered to be the analyti
ontinuation on u from u positive
and large.
This de(cid:12)nition is
learly
ompatible with the formal de(cid:12)nition given above and one easily sees that
t
H(b;d) is
losed under the appli
ation of = .
De(cid:12)nition 2.2. H0(b;d) is the subspa
e of H(b;d) of fun
tions that
an be extended to analyti
fun
tions in C(b;d).
Noti
e that f is in H0(b;d) if it is in H(b;d) and f(t) is analyti
at t=0.
Remark 2.3. If f 2 H0(b;d) then generally =f 2= H0(b;d) and has a dis
ontinuity in t = 0. For
instan
e if f 2L1 is positive, then:
Z1
0(cid:0) 0+
=(f):=(= (cid:0)= )f = f 6=0:
(cid:0)1
8 2 Perturbative
onstru
tion of the homo
lini
traje
tories
0(cid:0) 0+
We
an
onstru
t operators whi
h preserve H0(b;d); let === (cid:0)= and
(cid:26) (cid:26)
t t
t = if t(cid:21)0 t = if t(cid:20)0
=+ = t =(cid:0) = t
= (cid:0)= if t<0; = += if t>0:
The operator
1 X t t 1
=(cid:26) == (cid:0) (cid:27)(t)=
2 2
(cid:26)=(cid:6)1
preserves the analyti
ity.
Now let us
ite two important properties of H0(b;d), proved in [G1℄.
Lemma 2.4. In H0(b;d) we have the following shift of
ontour formulas:
8f 2H0(b;d) and for all d>s2R
(i) =f((cid:28))==f((cid:28) +is);
I Zt
X t+is dR X (cid:0)R(cid:27)((cid:28))((cid:28)+is)
(ii) =(cid:26) f((cid:28))= e f((cid:28) +is)d(cid:28):
2i(cid:25)R
(cid:26)=(cid:6)1 (cid:26)=(cid:6)1(cid:26)1
2.2 The re
ursive equations
1 0
One
an easily verify that f ( 0(t);q0(t)) and f (q0(t)) are in H0(a;d) (and bounded at in(cid:12)nity) for
some \optimal" values a;d
orresponding respe
tively to the maximal distan
e from the imaginary
axis and the minimal distan
e form the real axis of the poles of su
h fun
tions. One
an prove by
indu
tion, see [G1℄ or [C℄ for the details, that the solutions of equations (2.2) tend to quasi-periodi
fun
tions provided that the initial data are
hosen to be:
k (cid:6) X k 0(cid:6) k (cid:6) X k 0(cid:6) 0 k
Ij(';!;0 )= (cid:22) = Fj ; p(';!;0 )= (cid:22) = x0F0:
k k
k
Moreover one
an prove that Fj (';!;t) has no
onstant
omponent. Consequently it is
onvenient
t
to express the traje
toriesin terms of the \primitives" = in the form (a0 =1):
k k k t k 0 1k 1 0k
((cid:22)) j(';t)=((cid:22)) ajQjFj +xjGj +xjGj
0 1 i
where xj =1, xj =jtj for j 6=0 while the x0 are de(cid:12)ned in equation 2.3,
t 1 t t h(cid:0) 0 1 1 0 (cid:1) i ik k1 i k
Qj[f℄= (=++=(cid:0)) xj(t)(cid:27)((cid:28))xj((cid:28))(cid:0)xj(t)(cid:27)(t)xj((cid:28)) f((cid:28)) ; Gj =((cid:22)) aj=xjFj:
2 2
For the proofs of these assertionssee [G1℄ or [C℄.
Noti
e that by our de(cid:12)nitions:
X
(cid:0) + (cid:0)1 0k (cid:0)1 0 0 (cid:0) +
Ij(';0 )(cid:0)Ij(';0 )=2aj Gj (cid:17)2aj Gj ; 2G0 =p(';0 )(cid:0)p(';0 )
k
We de(cid:12)ne the formal power series
X
lk l
Gj (')(cid:17)Gj(');j =0;:::;n; l=0;1
k
0
Noti
e that by the KAM theorem the Gj are
onvergentseries.
9
Remark 2.5. (i) We will often use formal power series and in parti
ular formal power series iden-
tities, namely identities whi
h hold only at ea
h order k in the series expansion in (cid:22); we will mark
su
h identities with the symbol A(cid:24)B.
In se
tion 4.5 we will prove that the formal power series we use are \ asymptoti
". Asa de(cid:12)nition
P
n
of asymptoti
power series we will assume that a formal power series (cid:22) an(") is asymptoti
if for
(cid:0)q (cid:0)Qn
all q >0 there exists Q>0 su
h that, for all n(cid:20)" , an(")(cid:20)" .This implies that we
an
ontrol
(cid:0)q Q
the (cid:12)rst " terms provided that (cid:22)<" .
(ii) It should be stressed that we do not need to prove
onvergen
e for all the asymptoti
power series
involved in a given identity to obtain information on those series whi
h are known to be
onvergent
(by the KAM theorem).
The following Proposition
ontains some important properties of the operators Qj all proved in
[G1℄.
Proposition 2.6 (Chier
hia). (i)The operators Qj are \symmetri
" on H(a;d):
=(fQjg)==(gQjf):
t
(ii) H0(a;d) is
losed under the appli
ation of Qj.
(iii) The operators Qj preserve parities and if f 2H0(a;d) is odd then =f =0
(iv)IfF;G2H(a;d)aresu
hthattheproje
tion onpolynomials, (cid:25)PF(cid:1)G, hasno
onstant
omponent,
then:
0(cid:27) (cid:27) (cid:27) 0(cid:27)
= G((cid:28))(cid:28)F((cid:28))=F(0 )G(0 )(cid:0)= F((cid:28))(cid:28)G((cid:28))
Proposition 2.6(iii) immediately implies the following (again proved in [G1℄)
ik
Corollary2.7. Forallk 2N,j =0;:::;ni=0;1thefun
tionGj (')iszerofor'=0. Inparti
ular
the splitting ve
tor is zero for '=0 and the system has an homo
lini
point.
i1
Proof. We pro
eed by indu
tion; by Proposition 2.6(iii) Gj (' = 0) = 0 as it is the integral of an
1
odd analyti
fun
tion. Consequently j('=0;t) is both odd and in H0(a;d). Now we suppose that
ih h
Gj ('=0)=0 and j('=0;t) is odd and in H0(a;d) for all h<k and j =0;:::;n. The fun
tion
P
k Æ h h
Fj isanoddanalyti
fun
tionoftheangles i ( jf )
omputedat = h<k((cid:22)) j('=0;t)whi
h
ik k
is again odd and in H0(a;d). We
an apply Proposition 2.6(iii) so Gj (' = 0)=0 and j(' =0;t)
is both odd and in H0(a;d).
3 Proofs of the Theorems
We de(cid:12)ne the formal power series:
a a a a
(cid:1)i;j ='iGj ; forj =1;:::;n; Æi ='iG0; fora=0;1:
Noti
e that su
h series are known a priori to be
onvergentonly for a=0.
Lemma 3.1. The stable and unstable manifolds are on the same energy surfa
e so that:
Xn
0 + (cid:0) 0 + (cid:0)
Gj(')(Ij(';0 )+Ij(';0 ))=(cid:0)G0(')(p(';0 )+p(';0 )) (3.1)
j=1
this relation implies that at the homo
lini
point '=0:
I~('=0;0+)(cid:1)0 =(cid:0)Æ0p('=0;0+):
Proof. The equation 3.1 are simply the energy
onservation at time t=0:
+ + 2 + (cid:0) (cid:0) 2 (cid:0)
(I(';0 );AI(';0 ))+p (';0 )=(I(';0 );AI(';0 ))+p (';0 );
thepotentialpartoftheHamiltonian
an
elsastheperturbationdependsonlyontheangles. Finally
0
we di(cid:11)erentiate in ' and
ompute at the homo
lini
point where Gj =0 by Corollary2.7.
10 3 Proofs of the Theorems
3.1 The formal linear equation
i
In the re
ursive
onstru
tion of j, and
onsequently of Gj, we have distinguished three \blo
ks" :
0 0k 1 1k k t k
(0) xjGj ; (1) xjGj ; (2) ((cid:22)) ajQj(Fj ); (3.2)
ih k 0k
astheGj
anbebroughtoutoftheintegralwe
ansaythat j andGj (j =1;:::;n)arepolynomials
rh
in the Gl with l = 0;:::;n, h = 1;:::;k(cid:0)1, r = 0;1. This
an be seen as a formal power series
identity:
X Xn
0 0 [r℄ r [r℄ r
Gj(')(cid:24)Jj(')+ ( Njl (')Gl(')+nj G0)+ quadrati
terms+(cid:1)(cid:1)(cid:1) [r℄=jr(cid:0)1j
r=0;1 l=1
Following [GGM1℄ we di(cid:11)erentiate this relation in the parameter ' and evaluate it on the homo-
i 0
lini
point where Gj (cid:24)0, this leads to a linear formal identity for (cid:1) :
0 0 1 0 0 1 1 0 0 1
(cid:1) (cid:24)D +N (cid:1) +N (cid:1) +n Æ +n Æ (3.3)
0 0
where Dij ='jJij'=0.
i i
Noti
e that we do not have an expli
it expression for the matri
es N and n although we have a
re
ursive algorithm for the
oeÆ
ients of the series expansion; we will use trees to (cid:12)nd su
h expli
it
0
expressions. We
an noti
e however that D is the holomorphi
part of the splitting matrix, namely
it is obtained by using only the holomorphi
blo
k (2) of (3.2) in the
onstru
tion of the homo
lini
traje
tory.
We insert the energy
onservation relations in equation (3.3):
(cid:16) (cid:17)
1(cid:0)N1+ 1 n1I~('=0;0+) (cid:1)0 (cid:24)D0+N0(cid:1)1+n0Æ1; (3.4)
p('=0;0+)
The tree representation of the traje
tories leads to the following Propositions all proved in the
next Se
tions:
Proposition 3.2. The following formal power series relations hold:
0 0 0
0
(i)D (cid:24)N ; (ii)n (cid:24) D !;
2
0 0 0
(iii) D0 is the Hessian of a fun
tion S at the homo
lini
point: Dij ='i'jS (')j'=0.
This Proposition generalizes to Hamiltonian 1.3 similar results of [GGM1℄.
Relations (i) and (ii) inserted in (3.4) dire
tly imply that:
(1(cid:0)N1+ 1 n1I~('=0;0+))(cid:1)0 (cid:24)D0(1+(cid:1)1+ 1!Æ1): (3.5)
p('=0;0+) 2
Proposition 3.3. One
an use the tree representation to (cid:12)nd appropriate bounds on the order k
k
terms of the series expansion of the formal power series of Equation (3.5). If we denote by M the
order k term of the (cid:22) expansion of a formal power series M, we have:
1k k + k + 1k 1k
1 (cid:0)1 k
max(jN j;jp ('=0;0 )j;jIj('=0;0 )j;jÆ j;j(cid:1) j)(cid:20)(k!) (C" ) :
0k
Moreover the Fourier
oeÆ
ients of the fun
tion S :
X
S0k(')= ei'(cid:1)(cid:23)S^0k((cid:23));
(cid:23)2Zn:j(cid:23)j(cid:20)k
respe
t the inequalities
(cid:26) p+7
jS^0k((cid:23))j(cid:20) (k!)
1(C"(cid:0) 2 )ke(cid:0)j!(cid:1)(cid:23)jd ;
(k!)
1Ck"(cid:0)ke(cid:0)j!(cid:1)(cid:23)j
where C;
<d are appropriate order one
onstants,
1 =4(cid:28)+4 ((cid:28) is the Diophantine exponent of !)
0
(cid:12)nally p is the degree of the pole nearest to the real axis of f(q (t)).
