Global Ins abili y in Hamil onian Sys ems
Thesis p esen ed o ob ain he Ph. D. in Applied Ma hema ics by
Uni e si a Poli `ecnica de Ca alunya
Rod igo Gon¸cal es Schae e
Supe iso : Amadeu Delshams
May 29, 2018
Con en s
1 In oduc ion 4
2 The i s case o 2 + 1/2 deg ees o eedom 11
2.1 TheSys em ................................... 11
2.2 The inne and he ou e dynamics . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Inne map................................ 13
2.2.2 Sca e ing map: Melniko po en ial and c es s . . . . . . . . . . . . 14
2.3 A nolddi usion................................. 28
2.3.1 A geome ical p oposi ion: The le el cu es o L∗(I, θ) ....... 29
2.3.2 Resul s abou global ins abili y . . . . . . . . . . . . . . . . . . . . 33
2.4 The imeo di usion .............................. 37
2.4.1 Accu acy o he sca e ing map . . . . . . . . . . . . . . . . . . . . 38
2.4.2 Es ima e o he ime o di usion . . . . . . . . . . . . . . . . . . . 40
3 Second case o 2+1/2 deg ees o eedom 46
3.1 Inne dynamics ................................. 46
3.2 Sca e ingmap ................................. 48
3.2.1 C es s and NHIM lines . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.2 Cons uc ion o sca e ing maps . . . . . . . . . . . . . . . . . . . . 57
3.3 A noldDi usion ................................ 64
3.3.1 P oo o Theo em1........................... 67
3.4 Piecewise smoo h global sca e ing maps . . . . . . . . . . . . . . . . . . . 68
4 A case o 3+1/2 deg ees o eedom 71
4.1 Unpe u bedcase................................ 72
4.2 Inne dynamics ................................. 72
4.3 Sca e ingmap ................................. 73
4.3.1 De ini ion o sca e ing map . . . . . . . . . . . . . . . . . . . . . . 73
4.3.2 C es s and NHIM lines . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.3 Symme y o he sca e ing map . . . . . . . . . . . . . . . . . . . . 78
4.4 Highways .................................... 83
1
5 Some open ques ions 87
5.1 Highways in piecewise smoo h global sca e ing maps . . . . . . . . . . . . 87
5.2 Abou he case wi h 3 + 1/2 deg ees o eedom . . . . . . . . . . . . . . 87
5.3 Abou Shadowing lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 Rela ion be ween he o mulas o he sca e ing and sepa a ix maps . . . . 88
5.5 Abou he amoun o di usion ajec o ies . . . . . . . . . . . . . . . . . . 89
5.6 Andmo eandmo e............................... 89
2
Acknowledgmen s
Ag aeixo especialmen al meu u o , Amadeu Delshams, pe ha e accep a eballa
amb mi, pe la solida i a , la paci`encia i la dedicaci´o.
Ag aeixo amb´e a en Jos´e Tom´as L´aza o pe la simpa ia in ini a i la som iu e cada eg-
ada que ens obem al depa amen . A l’E a Mi anda pe l’in e `es i la se a disponibili a .
Als p o esso s A u o Viei o i Ca les Sim´o pels comen a is desp ´es dels semina is.
Me gus a ´ıa ag adece a Albe o, mi compa˜ne o de despacho po es os 4 a˜nos, siemp e
una buena compa˜n´ıa en los ca ´es y cong esos.
Que o ag adece aos meus amigos de Cabo F io que se man i e am semp e p esen es
apesa da dis ˆancia, Gab iel, Paulo, Lucas, Ma cus, Rhenan, Guilhe me e Jona has. Aos
amigos do “Regen Mendie a” que ize am poss´ı el a sob e i ˆencia no p imei o ano de
dou o ado, Ma cos, Jo˜ao e Danilo. Aos colegas de Ce danyola Mu ilo, Leona do, e Jackson.
A Juliana pela companhia no pe ´ıodo que es e e em Ba celona. E cla o, ao O ´a io e ao
Glads on, se ei e e namen e g a o pela amizade de ocˆes... ocˆes sabem o qu˜ao di ´ıcil oi.
Gos a ia de ag adece amb´em a S e anella Boa o, po e me dado a p imei a opo -
unidade pa a que esse dou o ado se o nasse ealidade. Seu apoio e sua amizade.
Gos a ia amb´em de ag adece a minha am´ılia pelo apoio e amo que me ansmi em a
cada momen o, meus pais Selma e Luiz, minhas i m˜as Juliana e Ana Ca olina e aos meus
sob inhos Caio e Camila.
I ag aeixo mol especialmen a la N´u ia i a la se a am´ılia, que ja conside o com me a.
Sense u, es d’aix`o se ia possible. Enca a enim mol es coses pe conque i jun s.
Thanks o he e e ees o ead his hesis. I was suppo ed by he PhD g an CNPq-
Conselho Nacional de Desen ol imen o Cien ´ı ico e Tecnol´ogico.
3
Chap e 1
In oduc ion
This hesis conce ns abou global ins abili y in nea ly-in eg able Hamil onian sys ems,
also called “A nold di usion”. In [A n64], V.I. A nold p oposed an example o a nea ly-
in eg able Hamil onian wi h 2 + 1/2 deg ees o eedom
H(q, p, ϕ, I, ) = 1
2p2+I2+ε(cos q−1) (1 + µ(sin ϕ+ cos )) ,
and asse ed ha gi en any δ, K > 0, o any 0 < µ ε0, he e exis s a ajec o y o
his Hamil onian sys em such ha
I(0) < δ and I(T)> K o some ime T > 0.
No ice ha his a global ins abili y esul o he a iable I, since
˙
I=−∂H
∂ϕ =−εµ(cos q−1) cos ϕ
is ze o o ε= 0, so I emains cons an , whe eas Ican ha e a d i o ini e size o any
ε > 0 small enough.
A nold’s Hamil onian can be w i en as a nea ly-in eg able Hamil onian wi h 3 deg ees
o eedom
H∗(q, p, ϕ, I, s, A) = 1
2p2+I2+A+ε(cos q−1) (1 + µ(sin ϕ+ cos s)) ,
which o ε= 0 is an in eg able Hamil onian h(p, I, A) = 1
2(p2+I2) + A. Since hsa is ies
he (A nold) isoene ge ic non-degene acy
D2h Dh
Dh>0=−16= 0,
by he KAM heo em p o en by A nold in [A n63], he 5D phase space o His illed, up o
a se o ela i e measu e O(√ε) , wi h 3D-in a ian o i Tωwi h Diophan ine equencies
ω= (ω1, ω2,1):
|k1ω1+k2ω2+k0| ≥ γ/|k|τ o any 0 6= (k1, k2, k0)∈Z,
4

whe e γ=O(√ε), and τ≥2.
Since he 3D KAM in a ian o i do no sepa a e he 5D phase space, he e can exis
i egula o bi s ‘ a eling’ be ween o i. A nold conjec u ed in he KAM heo em in 1963
ha his was he gene al case.
In he i s pa o his hesis we conside an a p io i uns able Hamil onian wi h 2+1/2
deg ees o eedom
Hε(p, q, I, ϕ, s) = ±p2
2+ cos q−1+I2
2+εh(q, ϕ, s) (1.1)
consis ing o a pendulum and a o o plus a ime pe iodic pe u ba ion h(q, ϕ, s).
A p io i uns able Hamil onian sys ems like he abo e one we e in oduced by [Loc92,
CG94]. They consis on a o o in he a iables (I, ϕ) as an in eg able Hamil onian in
ac ion-angle a iables, a pendulum in he a iables (p, q) which ca ies ou a sepa a ix
associa ed o a saddle poin , plus a small pe u ba ion o size ε. Fo ε= 0, Hamil onian
(1.1) is in eg able and, in pa icula , he ac ion Iis cons an . We wan o desc ibe he
global ins abili y in he a iable I o |ε|non-ze o bu o he wise a bi a y small.
Fo simplici y, we e e o global ins abili y in his pape simply as A nold di usion.
Ne e heless, i is wo h ema king ha o iginally he e m A nold di usion was coined
o a p io i s able Hamil onian sys ems, which a e pe u ba ions o in eg able Hamil onian
sys ems w i en in ac ion-angle a iable. See [Be 10] o a ca e ul exposi ion o a p io i
uns able and a p io i s able Hamil onian sys ems. Fo ins ance, eplacing V(q) by εV (q),
ou Hamil onian (1.1) becomes a p io i s able. In ha case, A nold di usion would con-
sis ing on inding ajec o ies wi h la ge de ia ions (p(T), I(T)) −(p(0), I(0)). This would
be a much mo e di icul p oblem ha he one conside ed he e, because one has o con on
o exponen ially small spli ing o in a ian mani olds wi h espec o he pa ame e εas
well as o he passage h ough double esonances in he ac ion a iables p, I. In pa icula ,
exponen ial la ge es ima es o he ime o di usion wi h espec o εdue o Nekho oshe
[Nek77, LM05, BM11] would apply.
The main cha ac e is ic o an a p io i uns able Hamil onian sys em wi h 2+1/2 deg ees
o eedom is ha he e exis s a 3D No mally Hype bolic In a ian Mani old (NHIM) which
is a la ge in a ian objec wi h 4D uns able and s able in a ian mani olds.
Inside his NHIM he e exis s an inne dynamics gi en by a Hamil onian sys em wi h
1+1/2 deg ees o eedom. This Hamil onian possesses 2D in a ian o i which p e en
global ins abili y inside he 3D NHIM.
Fo ε= 0, he s able and uns able in a ian mani old coincide along a huge sepa a ix
illed wi h homoclinic o bi s o he NHIM.
Fo small |ε| 6= 0, he uns able and s able mani olds o he NHIM in gene al do no
coincide, bu o he wise in e sec ans e sely along 3D homoclinic in a ian mani olds.
Th ough each poin on each 3D homoclinic mani old, he e exis s a homoclinic o bi which
begins in a poin o he NHIM and inishes on ano he poin o he NHIM, no necessa ily
he same one. This assignmen be ween an ini ial and he inal poin on he NHIM is called
he sca e ing map. In p ac ice, one mus selec an adequa e domain o any sca e ing
map.
5
Unde he ac ion o a sca e ing map, he a iable Ican inc ease (o dec ease). The
geome ic mechanism o global ins abili y consis s on looking o ajec o ies o he sca e -
ing map wi h a la ge change on he a iable I. S anda d shadowing a gumen s p o ide he
exis ence o nea by ajec o ies o Hamil onian (1.1) wi h a la ge change on he a iable I.
We a e going o assume ha he pe u ba ion h(q, ϕ, s) depends on wo ha monics in
he a iables (ϕ, s):
h(q, ϕ, s) = (q)g(ϕ, s),
(q) = cos q, g(ϕ, s) = a1cos(k1ϕ+l1s) + a2cos(k2ϕ+l2s),(1.2)
wi h k1, k2, l1, l2∈Z.
One o he main goals o his hesis is o p o e ha o any non- i ial pe u ba ion
a1a26= 0 depending on any wo independen ha monics k1k2
l1l26= 0, he e is global ins a-
bili y o he ac ion I o any ε > 0 small enough.
Ou i s esul is ha he global ins abili y happens o any a bi a y pe u ba ion
(1.2).
Theo em 1. Assume ha a1a26= 0 and k1l2−k2l16= 0 in Hamil onian (1.1)-(1.2). Then,
o any I∗>0, he e exis s ε∗=ε∗(I∗, a1, a2)>0such ha o any ε,0< ε < ε∗, he e
exis s a ajec o y (p( ), q( ), I( ), ϕ( )) such ha o some T > 0
I(0) ≤ −I∗< I∗≤I(T).
Rema k 2. Fo a ough es ima e o ε∗∼exp(−πI∗/2) a leas o |a1/a2|<0.625, k1=
l2= 1 and l1=k2= 0, and T=T(ε∗, I∗, a1, a2)∼(Ts(I∗, a1, a2)/ε) log(C(I∗, a1, a2)/ε) o
he di usion ime, see 2.4. Analogous es ima es could be ob ained o all he o he alues
o he pa ame e s.
The p oo is based on he geome ical me hod in oduced in [DLS06] and elies on he
conc e e compu a ion o se e al sca e ing maps. A sca e ing map is a map o ans e se
homoclinic o bi s o a NHIM. Fo Hamil onian (1.1), he NHIM u ns ou o be simply
˜
Λε=˜
Λ = (0,0, I, ϕ, s):(I, ϕ, s)∈R×T2.(1.3)
In he unpe u bed case, i.e., ε= 0, o any I∗>0 he NHIM ˜
Λ possesses a 4D sepa a ix,
ha is o say, coinciden s able and uns able in a ian mani olds
W0˜
Λ = (p0(τ), q0(τ), I, ϕ, s) : τ∈R, I ∈[−I∗, I∗],(ϕ, s)∈T2,
whe e (p0, q0) a e he sepa a ices o he saddle equilib ium poin o he pendulum
(p0( ), q0( )) = ±2
cosh ,4 a c an e± .
In he pe u bed case, i.e., o small ε > 0, Wu(˜
Λε) and Ws(˜
Λε) do no coincide ( his
is he so-called spli ing o sepa a ices), and e e y local ans e sal in e sec ion be ween
6
hem gi es ise o a (local) sca e ing map which is simply he co espondence be ween
a pas asymp o ic mo ion in he NHIM o he co esponding u u e asymp o ic mo ion
ollowing a homoclinic o bi . Since he NHIM has also an inne dynamics, an adequa e
combina ion o hese wo dynamics on he NHIM, he inne one and he ou e one p o ided
by he sca e ing map, gene a es he A nold di usion as long as he ou e dynamics does
no p ese e he in a ian objec s o he inne dynamics.
Necessi y o he assump ions
I he de e minan ∆ := k1l2−k2l1o some coe icien a1,a2 anishes, o ins ance,
i he e is only one ha monic in g, he e is no global ins abili y o he ac ion I. Indeed,
looking a he equa ions associa ed o Hamil onian (1.1)
˙q=±p˙p= [±1 + ε(a1cos(k1ϕ+l1s) + a2cos(k2ϕ+l2s))] sin q
˙ϕ=I˙
I=εcos q(k1a1sin(k1ϕ+l1s) + k2a2sin(k2ϕ+l2s)) (1.4)
˙s= 1
his is clea o k1=k2= 0, since in his case Iis a cons an o mo ion. I k1o k26= 0,
say k16= 0, he change o a iables
¯ϕ=k1ϕ+l1s, ¯ϕ−¯s=k2ϕ+l2s, ¯
I=k1I+l1,
whe e =k2/k1can be assumed o sa is y 0 ≤ ≤1 wi hou loss o gene ali y, cas s
sys em (1.4) in o
˙q=±p˙p= [±1 + ε(a1cos ¯ϕ+a2cos( ¯ϕ−¯s))] sin q
˙
¯ϕ=¯
I˙
¯
I=εk2
1cos q(a1sin ¯ϕ+ a2sin( ¯ϕ−¯s))
˙
¯s= ∆/k1
which is a Hamil onian sys em wi h he Hamil onian gi en by
¯
Hε(p, q, ¯
I, ¯ϕ, ¯s) = ±p2
2+ cos q−1+¯
I2
2
+εk2
1cos q(a1cos ¯ϕ+a2cos( ¯ϕ−¯s)) .
(1.5)
I ∆ = 0 Hamil onian (1.5) is au onomous wi h 2 deg ees o eedom, and he e o e a
global d i o he ac ion Iis no possible. Only d i s o size √εa e possible due o KAM
heo em. Analogously one easily checks ha o a1a2= 0 Hamil onian (1.1) is in eg able
o au onomous.
Reduc ion o he ha monic ypes
Unde he hypo hesis (k1l2−k2l1)a1a26= 0 o Theo em 1, he case k2= 0 o Theo em 1
is p o ed in Chap e 2. Indeed, k2= 0 implies := k2/k1= 0 and i u ns ou om (1.5)
7
ha Hamil onian (1.1) is equi alen o he one wi h k1= 1, k2= 0, l1= 0, l2= 1:
Hε(p, q, I, ϕ, ) = ±p2
2+ cos q−1+I2
2+εcos q(a1cos ϕ+a2cos s),(1.6)
which is jus he Hamil onian s udied in Chap e 2. In Chap e 3 we p o e Theo em 1 o
k1k26= 0 o equi alen ly o ∈(0,1]. Fo he sake o cla i y we will explain in ull de ail
and p o e Theo em 1 along Sec ion 3.3 jus o = 1, which by (1.5) is equi alen o he
case k1= 1, k2= 1, l1= 0, l2=−1:
Hε(p, q, I, ϕ, ) = ±p2
2+ cos q−1+I2
2+εcos q(a1cos ϕ+a2cos(ϕ−s)) .(1.7)
To inish he p oo o Theo em 1, in Sec ion 3.3 we will ske ch he modi ica ions needed
o he case ∈(0,1).
Sca e ing map ypes
By he de ini ion gi en a Sec ion 2.2.2, a sca e ing map is in p inciple only locally
de ined, ha is, o a small ball o alues o he a iables (I, ϕ, s) o (I, θ =ϕ−Is), since i
depends on a non-degene a e c i ical poin τ∗=τ∗(I, ϕ, s) o a eal unc ion (2.6), depend-
ing smoo hly on he a iables (I, ϕ, s), al eady in oduced in [DLS06]. In he s udy ca ied
ou in Sec ion 3.2, i will be desc ibed whe he , in e ms o he pa ame e µ:= a1/a2
and he a iable I, a local sca e ing map can o canno be smoo hly de ined o all he
alues o he angles (ϕ, s) o θ=ϕ−Is, becoming hus a global o ex ended sca e ing
map. This desc ip ion will depend essen ially on a geome ical cha ac e iza ion o he unc-
ion τ∗(I, ϕ, s) in e ms o he in e sec ion o c es s and NHIM lines, ollowing [DH11]. Any
degene a ion o he c i ical poin τ∗=τ∗(I, ϕ, s) may gi e ise o mo e non-degene a e c i -
ical poin s and a bi u ca ion o mul iple local sca e ing maps o o a non global sca e ing
map. Di e en c i ical poin s τ∗=τ∗(I, ϕ, s) gi e ise o di e en local sca e ing maps,
and pu ing oge he di e en local sca e ing maps, one can some imes ob ain piecewise
smoo h global sca e ing maps, which a e e y use ul o design pa hs o ins abili y o he
ac ion I, and a e simply called di usion pa hs.
Fo ins ance, in Chap e 2 de o ed o he Hamil onian (1.6), i will be p o en ha
o 0 < µ =a1/a2<0.625, he e exis wo di e en global sca e ing maps. Among
he di e en kinds o associa ed o bi s o hese sca e ing maps, he e will appea wo o
hem called highways, whe e he d i o he ac ion Iwas e y as and simple. As will be
desc ibed in Sec ion 3.2, such highways do no appea o Hamil onian (1.7). Ne e heless,
as will be p o en in Sec ion 3.4, he e exis piecewise smoo h global sca e ing maps, and
he possible di usion along he discon inui y se s opens he possibili y o applying he
heo y o piecewise smoo h dynamical sys ems [Fil88].
Abou he model chosen and ela ed wo k
Hamil onian (1.1) is a s anda d example o an a p io i uns able Hamil onian sys-
em [CG94] o med by a pendulum, a o o and a pe u ba ion. I is usual in he li e a u e
8
has a non degene a e c i ical poin τ∗=τ∗(I, ϕ, s), whe e
L(I, ϕ, s) = Z+∞
−∞
( (q0(σ))g(ϕ+Iσ, s +σ; 0) − (0)g(ϕ+Iσ, s +σ; 0)) dσ.
Then, o 0<|ε|small enough, he e exis s a unique ans e sal homoclinic poin ˜z o ˜
Λε,
which is ε-close o he poin ˜z∗(I, ϕ, s) = (p0(τ∗), q0(τ∗), I, ϕ, s)∈W0(˜
Λ):
˜z= ˜z(I, ϕ, s) = (p0(τ∗) + O(ε), q0(τ∗) + O(ε), I, ϕ, s)∈Wu(˜
Λε) Ws(˜
Λε).(2.7)
The unc ion Lis called he Melniko po en ial o Hamil onian (1.1). In ou case, om
(2.2) and (2.3)
L(I, ϕ, s) = A00 +A10(I) cos ϕ+A01 cos s, (2.8)
whe e
A00 = 4 a00, A10(I) = 2π I a10
sinh(π I
2)and A01 =2π a01
sinh(π
2).(2.9)
Fig. 2.3: The Melniko po en ial, µ=a10/a01 = 0.6 and I= 1.
We now look o he c i ical poin s o (2.6) which indeed a e he solu ions o
∂L
∂τ (I, ϕ −Iτ, s −τ) = 0.
Equi alen ly, τ∗=τ∗(I, ϕ, s) sa is ies
I A10(I) sin(ϕ−I τ∗) + A10 sin(s−τ∗) = 0.(2.10)
F om a geome ical iew-poin , o any (I, ϕ, s)∈[−I∗, I∗]×T2, inding τ∗=τ∗(I, ϕ, s)
sa is ying (2.10) is equi alen o looking o he ex ema o Lon he NHIM line
R(I, ϕ, s) = {(I, ϕ −Iτ, s −τ), τ ∈R},(2.11)
15

which co esponds o he unpe u bed ajec o y o Hamil onian H0 h ough (I, ϕ, s) along
he unpe u bed NHIM.
Thus we can de ine he sca e ing map as in [DH11]. Le Wbe an open subse o
[−I∗, I∗]×T2such ha he map
(I, ϕ, s)∈W7→ τ∗(I, ϕ, s),
whe e τ∗(I, ϕ, s) is a c i ical poin o (2.6) o , equi alen ly, a solu ion o (2.10), is well de-
ined and C2. The e o e, he e exis s a unique ˜zsa is ying (2.7). Le Γ = {˜z(I, ϕ, s;ε),(I, ϕ, s)∈
W}. Fo any ˜z∈Γ he e exis unique ˜x+,−= ˜x+,−(I, ϕ, s;ε)∈˜
Λεsuch ha ˜z∈
Ws
ε(˜x−)∩Wu
ε(˜x+). Le
H+,−=[{˜x+,−(I, ϕ, s;ε),(I, ϕ, s)∈W}.
We de ine he sca e ing map associa ed o Γ as he map
S:H−−→ H+
˜x−7−→ S(˜x−) = ˜x+.
By he geome ic p ope ies o he sca e ing map (i is an exac symplec ic map
[DLS08]) we ha e, see [DH09] and [DH11], ha he sca e ing map has he explici o m
S(I, ϕ, s) = I+ε∂L∗
∂ϕ (I, ϕ, s) + O(ε2), ϕ −ε∂L∗
∂I (I, ϕ, s) + O(ε2), s,(2.12)
whe e
L∗(I, ϕ, s) = L(I, ϕ −I τ∗(I, ϕ, s), s −τ∗(I, ϕ, s)).(2.13)
The new a iable θ=ϕ−Is
No ice ha i τ∗(I, ϕ, s) is a c i ical poin o (2.6), τ∗(I, ϕ, s)−σis a c i ical poin
o
τ7−→ L(I, ϕ −I(τ+σ), s −(τ+σ)) = L(I, ϕ −Iσ −Iτ, s −σ−τ).(2.14)
Since τ∗(I, ϕ −Iσ, s −σ) is a c i ical poin o he igh hand side o (2.14), by he
uniqueness in Wwe can conclude ha
τ∗(I, ϕ −Iσ, s −σ) = τ∗(I, ϕ, s)−σ. (2.15)
Thus, by (2.13),
L∗(I, ϕ −Iσ, s −σ) = L(I, ϕ −Iσ −I(τ∗−σ), s −σ−τ∗)
=L(I, ϕ −Iτ∗, s −τ∗) = L∗(I, ϕ, s),
16
and, in pa icula o σ=s,
L∗(I, ϕ −Is, 0) = L∗(I, ϕ, s).
In oducing he new a iable
θ=ϕ−Is,
we de ine he Reduced Poinca ´e unc ion
L∗(I, θ) := L∗(I, ϕ −Is, 0) = L∗(I, ϕ, s).(2.16)
We can w i e he sca e ing map on he a iables (I, θ). F om (I0, ϕ0, s0) = S(I, ϕ, s),
we ha e ha
θ0=ϕ0−I0s0=ϕ−ε∂L∗
∂I (I, ϕ, s)−I+ε∂L∗
∂ϕ (I, ϕ, s)s+O(ε2)
=θ−ε∂L∗
∂I (I, ϕ, s) + ∂L∗
∂ϕ (I, ϕ, s)s+O(ε2).
Since ∂L∗
∂I (I, ϕ, s) = ∂L∗
∂I (I, θ)−s∂L∗
∂θ (I, θ) and ∂L∗
∂ϕ =∂L∗
∂θ (I, θ),
we conclude ha
θ0=θ−ε∂L∗
∂I (I, θ)+O(ε2) and I0=I+ε∂L∗
∂θ (I, θ)+O(ε2).
Then, in he a iables (I, θ), he sca e ing map akes he simple o m
S(I, θ) = I+ε∂L∗
∂θ (I, θ) + O(ε2), θ −ε∂L∗
∂I (I, θ) + O(ε2),(2.17)
so up o O(ε2) e ms, S(I, θ) is he −ε imes low o he au onomous Hamil onian L∗(I, θ).
In pa icula , he i e a es unde he sca e ing map ollow he le el cu es o L∗up o
O(ε2).
Rema k 6. We no ice ha he a iable θis pe iodic in he a iable ϕand quasi-pe iodic
in he a iable s. Fixing s, hen θbecomes pe iodic.
Rema k 7. No e ha i o some alues o (I, θ) we ha e ha ∇L∗(I, θ) = O(ε), hen
ε∂L∗/∂θ(I, θ) = O(ε2) and ε∂L∗/∂I(I, θ) = O(ε2). In his case, he le el cu es o L∗(I, θ)
do no p o ide he dominan pa o he sca e ing map S. The e o e, we will be able o
desc ibe p ope ly he sca e ing map h ough he le el cu es o he Reduced Poinca ´e
unc ion on he se o (I, θ) such ha k∇L∗(I, θ)k  ε.
17
Rema k 8. Using Eq.(2.15) and se ing s=σ, we ha e ha τ∗(I, ϕ−Is, 0) = τ∗(I, ϕ, s)−
s. So we can de ine
τ∗(I, θ) := τ∗(I, ϕ, s)−s(2.18)
and om (2.13) and (2.16) we can w i e L∗as
L∗(I, θ) = L(I, θ −Iτ∗(I, θ),−τ∗(I, θ)).(2.19)
Rema k 9. In he a iables (I, θ), he a iable sdoes no appea a all in he exp ession
(2.17) o he sca e ing map, a leas up o O(ε2). Howe e , sdoes appea in he exp ession
(2.12) in he o iginal a iables (I, ϕ), so we ha e in (2.12) a amily o sca e ing maps
pa ame e ized by he a iable s. Playing wi h he pa ame e s, we can ha e sca e ing
maps wi h di e en p ope ies. See Lemma 14 o an applica ion o his phenomenon.
The c es s
Fo he compu a ion o he sca e ing maps, we use an impo an geome ical objec
in oduced in [DH11], he c es s.
De ini ion 10. Fixed I, we de ine by c es s C(I) he cu es on {(I, ϕ, s),(ϕ, s)∈T2},
sa is ying
I∂L
∂ϕ(I, ϕ, s) + ∂L
∂s (I, ϕ, s)=0.
In ou case
I A10(I) sin ϕ+A01 sin s= 0.(2.20)
No e ha a poin (I, ϕ, s) belongs o a c es C(I) i i is a minimum o maximum, o
mo e gene ally, a c i ical poin o Lalong a NHIM line (2.11), ha is, τ∗(I, ϕ, s) = 0 in
(2.10), see Fig. 2.4.
Fig. 2.4: Le el cu es o L o µ=a10/a01 = 0.5 and I= 1.2. C es s (dashed) in blue and g een and he
NHIM lines in black.
18
Rema k 11. No e ha any c i ical poin o L(I, ·,·) belongs o he c es C(I). In gene al
we ha e wo cu es sa is ying Eq.(2.20), he maximum c es CM(I), and he minimum
c es Cm(I). The maximum c es con ains he poin (I, ϕ = 0, s = 0), and he minimum
c es he poin (I, ϕ =π, s =π). Fo a10 >0, a01 >0, he Melniko unc ion L(I, ·,·)
gi en in (2.8) has a maximum poin a he poin (I, ϕ, s)=(I, 0,0), and a minimum a
(I, π, π), and he unc ion (2.6) has a maximum on CM(I), and a minimum on Cm(I). Fo
o he combina ions o signs o a10, a01, he loca ion o maxima and minima changes, bu
o simplici y, we ha e p ese ed he name o maximum and minimum c es .
We now p oceed o s udy he c es s. By (2.9) we can ew i e Eq. (2.20) as
µα(I) sin ϕ+ sin s= 0,(2.21)
whe e
α(I) = IA10(I)
µA01
=sinh(π
2)I2
sinh(π I
2)and µ=a10
a01
.(2.22)
No e ha i |µα(I)|<1 we can w i e sas a unc ion o ϕ o any alue o ϕ. On he
o he hand, i |µα(I)|>1 we can w i e ϕas a unc ion o s. So, we ha e wo di e en
kinds o c es s:
•Fo |α(I)|<1/|µ|, he wo c es s a e ho izon al, see Fig. 2.5(a), wi h
CM,m(I) = {(I, ϕ, ξM,m(I, ϕ)) : ϕ∈T},
ξM(I, ϕ) = −a csin(µα(I) sin ϕ) mod 2π(2.23)
ξm(I, ϕ) = a csin(µα(I) sin ϕ) + πmod 2π.
(a) Ho izon al c es s: µ=a10/a01 =
0.6 and I= 1.2.
(b) Ve ical c es s: µ=a10/a01 =
1.2 and I= 1.
Fig. 2.5: Types o c es s.
•Fo |α(I)|>1/|µ|, he wo c es s a e e ical, see Fig. 2.5(b), wi h
CM,m(I) = {(I, ηM,m(I, s), s) : s∈T},
ηM(I, s) = −a csin(sin s/ (µα(I))) mod 2π(2.24)
ηm(I, s) = a csin(sin s/ (µα(I))) + πmod 2π.
19
Rema k 12. The case |α(I)|= 1/|µ|is singula , since bo h c es s a e piecewise NHIM
lines and hey ouch each o he a he poin s (ϕ, s)=(π/2,3π/2) ,(3π/2, π/2). See Fig. 2.6.
Fig. 2.6: Singula case: C es s o I= 1 and µ= 1.
We can desc ibe he ela ion be ween he c es s C(I) and he NHIM lines R(I, ϕ, s)
h ough he ollowing P oposi ion:
P oposi ion 13. Conside he c es C(I)de ined by (2.21) and he NHIM line R(I, ϕ, s)
de ined in (2.11).
a) Fo |µ|<0.625 he c es s a e ho izon al and he in e sec ions be ween any c es and
any NHIM line is ans e sal.
b) Fo 0.625 ≤ |µ| ≤ 0.97 he wo c es s C(I)a e s ill ho izon al, bu o some alues o
I he e exis wo NHIM lines R(I, ϕ, s)which a e quad a ically angen o he c es s.
c) Fo |µ|>0.97, he same p ope ies as s a ed in b) hold, excep ha o |µα(I)|>1,
he c es s C(I)a e e ical.
P oo . The “ho izon ali y” o a) and b) and he “ e icali y” o c) a e due he uppe bound
o |µ|. Since |α(I)|<1/0.97 (see Fig.2.7), o |µ| ≤ 0.97, he c es s a e ho izon al, ha is,
hey can be exp essed by equa ions (2.23).
The condi ion o ans e sali y is p o ed in [DH11]. Essen ially, he p oo is o ob-
se e ha |Iα(I)|<1.6 and ha he e exis s a ϕsuch ha ∂ξ(I, ϕ)/∂ϕ = 1/I i , only i ,
|Iα(I)|<1/|µ|(we will p o e i in a sligh ly di e en con ex , see he p oo o P oposi-
ion 20.)
Abou he amoun o NHIM lines angen s o C(I), he p oo is gi en in subsec ion
2.2.2.
In Figs. 2.5(a) and 2.5(b) we ha e displayed a segmen o he he NHIM line R(I, ϕ, s),
|τ|< π, and we see ha i in e sec s each c es CM(I) and Cm(I) ans e sally, gi ing ise
o wo alues τ∗
Mand τ∗
m, he e o e o wo di e en sca e ing maps. We deno e by τ∗
M he
τwi h minimum absolu e alue such ha gi en (I, ϕ, s), (I, ϕ −Iτ, s −τ)∈CM(I) and τ∗
m
is de ined analogously when (I, ϕ −Iτ, s −τ)∈Cm(I) (see [DH11]).
20

Fig. 2.7: G aph o |α(I)|
Sca e ing maps and c es s
No e ha τ∗
mand τ∗
Ma e associa ed o di e en homoclinic poin s o he NHIM ˜
Λ, and
consequen ly, o di e en homoclinic connec ions. F om his we build di e en sca e ing
maps. The mos na u al way is o associa e one sca e ing map o each c es . And we will
do his on he a iables (I, ϕ, s) and (I, θ), whe e θ=ϕ−Is.
Be o e, we make some conside a ions abou he NHIM lines de ined in (2.11). No e ha
θ:= ϕ−Is = (ϕ−Iτ)−I(s−τ),
ha is, θis cons an on each NHIM line R(I, ϕ, s), so we will also in oduce ano he
no a ion o a NHIM line R(I, ϕ, s), namely
Rθ(I) := {(I, ϕ, s) : ϕ−Is =θ}.
Since (ϕ, s)∈T2,R(I, ϕ, s) is a closed line i I∈Q, whe eas i is a dense line on T2i
I /∈Q. In his case, R(I, ϕ, s) in e sec s he c es s C(I) along an in ini e numbe o poin s.
Recall (see Rema k 6) ha θis quasi-pe iodic in he a iable s∈T. To a oid mon-
od omy wi h espec o his a iable, we a e going o conside om now on, in his Chap e ,
sas a eal a iable in an in e al o leng h 2π,−π/2< s ≤3π/2. Unde his es ic ion,
he NHIM line R(I, ϕ, s) de ined in (2.11) becomes a NHIM segmen
R(I, ϕ, s) = {(I, ϕ −Iτ, s −τ) ; −π/2< s −τ≤3π/2},(2.25)
as well as Rθ(I), which can be w i en as
Rθ(I) = {(I, ϕ, s) : ϕ−Is =θ, (ϕ, s)∈T×(−π/2,3π/2]}.(2.26)
F om now on, when we e e o R(I, ϕ, s) and Rθ(I), hey will be hese line segmen s.
No ice ha θ∈T.
We begin o conside he p ima y sca e ing map SMassocia ed o he maximum c es
CM, ha is, we look only a he in e sec ions be ween he segmen R(I, ϕ, s) gi en in (2.25)
and CM(I), pa ame e ized by τ∗
M(I, ϕ, s) = τ∗
M(I, θ) + s(see (2.18)):
CM(I)∩R(I, ϕ, s) = {(I, ϕ −Iτ∗
M(I, ϕ, s), ξM(I, ϕ −Iτ∗
M(I, ϕ, s)))}(2.27)
={(I, ϕ −Iτ∗
M(I, ϕ, s), s −τ∗
M(I, ϕ, s))}(2.28)
21
Equa ion (2.27) mo i a es us o in oduce a new a iable ψ=ϕ−Iτ∗
M(I, ϕ, s) ha will
be use ul in many con ex s.
The a iable ψ: a a iable on he c es .
Le C(I) be a c es such ha i can be pa ame e ized by ξ(I, ϕ) as in (2.23). Since
τ∗(I, ϕ, s) is he alue o τsuch ha R(I, ϕ, s), gi en in (2.25), in e sec s C(I), we de ine
ψ:= ϕ−Iτ∗(I, ϕ, s).(2.29)
By (2.18) we can also w i e ψin e ms o he a iable θ:
ψ=ϕ−I(τ∗(I, θ) + s) = θ−Iτ∗(I, θ).(2.30)
By (2.27) and (2.28),
s−τ∗(I, ϕ, s) = ξ(I, ϕ −Iτ∗(I, ϕ, s)) = ξ(I, ψ).(2.31)
In pa icula , o s= 0, ξ(I, ψ) = −τ∗(I, ϕ, 0) = −τ∗(I, θ) again by (2.18) and om (2.30)
we ha e he exp ession o θin e ms o ψ:
θ=ψ−Iξ(I, ψ).(2.32)
All he ela ions be ween he a iables (ϕ, s), θand ψa e w i en in Table 2.1 and a e
displayed in Fig. 2.8. By he de ini ions o L∗(I, ϕ, s) in (2.13), and L∗(I, θ) in (2.16) and
(2.19), we ha e ha
L∗(I, θ) = L∗(I, ϕ, s) = L(I, ψ, ξ(I, ψ)),(2.33)
So we can de ine he educed Poinca ´e unc ion in e ms o (I, ψ) simply as he es ic ion
o he Melniko po en ial L(I, ϕ, s) on he c es C(I) = {(I, ψ, ξ(I, ψ), ψ ∈T)}, i.e.,
L∗(I, ψ) := L(I, ψ, ξ(I, ψ)),(2.34)
which in ou case akes he simple and compu able o m
L∗(I, ψ) = A00 +A10(I) cos ψ+A01 cos ξ(I, ψ),(2.35)
o a ho izon al c es (3.21).
The e o e, as (I, ψ, ξ(I, ψ)) a e poin s on he c es , he domain o L∗(I, ·,·) is a subse
o C(I). So, i he e exis di e en subse s whe e L∗(I, ·,·) can be well de ined, we can
build di e en sca e ing maps associa ed o C(I).
Deno e L∗
i(I, θ) = L(I, ϕ −Iτ∗
i(I, ϕ, s), s −τ∗
i(I, ϕ, s)), i= m,M, and L∗
i(I, ψ) =
L(I, ψ, ξi(I, ψ)) om (2.33) and (2.34). We s a e he ollowing lemma
Lemma 14. a) The Poinca ´e Reduced unc ions L∗
M(I, ψ)and L∗
M(I, θ)a e e en unc-
ions in he a iable I, ha is, L∗
M(I, ψ) = L∗
M(−I, ψ)and L∗
M(I, θ) = L∗
M(−I, θ),
and consequen ly SM(I, θ)is symme ic in his a iable I. The same happens o
Sm(I, θ), ha is, o he sca e ing map associa ed o Cm(I).
22
0
π/
2
π
3
π/
2
-
π/
2
ξ
(
I,ψ
)
0
π/
2
π

(
ϕ
−
Iτ,s
−
τ
)

−
τ
∗
(
I,ϕ,s
)
−
Iτ
∗
(
I,ϕ,s
)
−
τ
∗
(
I,θ
)
−
Iτ
∗
(
I,θ
)
θ
ψ
(
ϕ,s
)
Fig. 2.8: The h ee a iables on he plane (ϕ, s):
ϕ, θ and ψ.
θ=ψ−Iξ(I, ψ)ψ=θ−Iτ∗(I, θ)
θ=ϕ−Is ϕ =θ+Is
ψ=ϕ−Iτ∗(I, ϕ, s)ϕ=ψ+I(s−ξ(I, ψ))
Table 2.1: Rela ion be ween a iables.
b) The sca e ing map o a alue o µand s=π, associa ed o he in e sec ion be ween
Rθ(I)and Cm(I)has he same geome ical p ope ies as he sca e ing map o −µ
and s= 0, associa ed o he in e sec ion be ween Rθ(I)and CM(I), i.e.,
Sµ,m(I, ϕ, π) = S−µ,M(I, ϕ, 0) = S−µ,M(I, θ)
P oo . a) This is an immedia e consequence o he ac ha unc ion A10(I) is e en and
ξM(I, ϕ) is odd in he a iable I, see (2.9) and (2.23).
b) Fi s , we look o τ∗
msuch ha he NHIM segmen Rθ(I) in e sec s he c es Cm(I).
I we ix s=π, we ha e by (2.13) and (2.8):
L∗
µ,m(I, ϕ, π) = A00 +A10(I) cos(ϕ−Iτ∗
m(I, ϕ, π)) + A01 cos(π−τ∗
m(I, ϕ, π)).(2.36)
Besides, we ha e by (2.10)
IA10(I) sin(ϕ−Iτ∗
m) + A01 sin(π−τ∗
m)=0,
which, in oducing µ(2.22), is equi alen o
µα(I) sin(ϕ−Iτ∗
m) + sin(π−τ∗
m)=0,(2.37)
o
−µα(I) sin(ϕ−Iτ∗
m) + sin(−τ∗
m)=0.(2.38)
By (2.31) and (2.23) we ha e ha π−τ∗
m=ξm(I, ϕ −Iτ∗
m) o π/2≤ξm≤3π/2 and
he e o e −π/2≤ −τ∗
m≤π/2.
By looking a (2.37) and (2.38), τ∗
m(I, ϕ, π) o µis solu ion o he same equa ion
as τ∗
M(I, ϕ, 0) o −µ, and lies in he same in e al −π/2≤ −τ∗
M≤π/2. The e o e
τ∗
m(I, ϕ, π) o µis equal o τ∗
M(I, ϕ, 0) o −µ. F om (2.36), L∗
µ,m(I, ϕ, π) sa is ies
L∗
µ,m(I, ϕ, π) = A00 +A10(I) cos(ϕ−τ∗
M(I, ϕ, 0)) + (−A01) cos(−τ∗
M(I, ϕ, 0))
=L∗
−µ,M(I, ϕ, 0).
23
Since L∗
µ,m(·,·, π) and L∗
−µ,M(·,·,0) coincide, hei de i a i es oo and his implies
ha Sµ,m(I, ϕ, π) = S−µ,M(I, ϕ, 0) = S−µ,M(I, θ).
The impo ance o he pa b) o his lemma is ha , conce ning di usion, he s udy o
a posi i e µusing SM(I, θ) is equi alen o he s udy o −µusing Sm(I, ϕ, π), i.e., i we
ensu e he di usion o a posi i e µ, we can ensu e i o a nega i e one (jus changing he
sca e ing map). Besides, since SM(I, θ) symme ic in he a iable I( om he i s pa o
he lemma), om now on we will conside always I≥0, µ > 0 and SM.
Now we a e going o desc ibe he in luence o he in e sec ions be ween he c es s and
he NHIM segmen s wi h espec o he pa ame e µdesc ibed in P oposi ion 13 on he
sca e ing map associa ed o such c es s.
Single sca e ing map: µ < 0.625
As in [DH11], assuming µ < 1/1.6 = 0.625, he c es s a e ho izon al and he e is no
angency be ween Rθ(I) and CM(I), so ha τ∗
M(I, θ) is well de ined and by (2.19) and
(3.11) he educed Poinca ´e unc ion akes he o m
L∗
M(I, θ) = A00 +A10(I) cos(θ−Iτ∗
M(I, θ)) + A01 cos(−τ∗
M(I, θ)),(2.39)
and he e o e SM(I, θ) akes he o m (2.17).
Example To illus a e his cons uc ion, we ix µ= 0.6. In his case he c es s a e
ho izon al o all I, and we display CM(I) pa ame ized by ξM(see (3.21)) in Fig.2.9 o
I= 1.2. We can see how Rθ(I) in e sec s ans e sally CM(I), as well as he phase space
o sca e ing map SMgene a ed by his in e sec ion gi en by he le el cu es o L∗
M(I, θ).
Rema k 15. Recall om Rema k 9 ha sdoes no appea in he exp ession (2.17) o
S(I, θ) and is a pa ame e in he exp ession (2.12) o S(I, ϕ, s). Compu a ionally, one
di e ence is ha in exp ession (2.12), once ixed a alue o s, one h ows om any “ini ial
poin ” (ϕ, s) he NHIM segmen R(I, ϕ, s) un il i ouches he c es C(I) a e a ime
τ∗(I, ϕ, s), ob aining a alue o L∗(I, ϕ, s) gi en by (2.13), while in exp ession (2.17), sis
ixed equal o 0 o , equi alen ly, he ini ial poin o h ow he NHIM segmen Rθ(I) is o
he o m (θ, 0) (see Fig. 2.8).
Mul iple sca e ing maps: 0.625 ≤µ≤0.97
As said be o e, o µ < 1/1.6 = 0.625 and any alue o I, he wo c es s CM(I) and
Cm(I) a e ho izon al, and he NHIM segmen Rθ(I) in e sec s ans e sely each o hem,
gi ing ise o a unique sca e ing map SMand Smassocia ed o each c es . We will now
explo e la ge alues o µ o de ec angencies be ween C(I) and Rθ(I), ha is, when he e
exis s (ϕ, I) such ha
∂ξ
∂ϕ(I, ϕ) = 1/I,
24
•Case 1 |µ|<0.625, ha is, 1/|µ|>1.6. Then, by (2.47) and (2.48),
α(I)≤1.03 <1
µand β(I)≤1.6<1
|µ|,
o all I, ha is, o I > 0, B= [0,+∞).
•Case 2 No e ha o 0.625 ≤ |µ|<0.97
α(I)≤1.03 = 1/0.97 <1
|µ|≤1.6 = β(Iβ),
and by (2.47), A= [0,+∞). Bu , now β(Ib)≥1/|µ|. Then he e exis wo alues
I∈Asuch ha β(I)=1/|µ|. De ine
I+= min {I:β(I) = 1/|µ|} and I++ = max {I:β(I)=1/|µ|}.(2.49)
By he cha ac e iza ion (2.46) o he se Bwe ha e B= [0, I+)∪(I++,+∞).
Fo 0.97 ≤ |µ| ≤ 1, he e exis Ia< I¯asuch ha α(Ij) = 1/|µ|,j∈ {a, ¯a}and
A= [0, Ia)∪(I¯a,+∞). Analogously, he e exis Ib< I¯
bsuch ha β(Ij) = 1/|µ|,
j∈ {b,¯
b}. As Ib≤Iaand I¯a< I¯
b, we ha e B= [0, Ib)∪(I¯
b,+∞), see Fig. 2.12.
Bu his he equi alen o B= [0, I+)∪(I++,+∞), whe e I+and I++ a e gi en by
(2.49).
•Case 3 This case is simila o he Case 2 o 0.97 ≤ |µ| ≤ 1. Bu now, as |µ| ≥ 1, we
ha e Ia≤Ib. So, in his case we ha e B= [0, Ia)∪(I¯
b,∞), o B= [0, I+)∪(I++,+∞),
whe e I+= min {I:α(I) = 1/|µ|} and I++ = max {I:β= 1/|µ|}.
Finally, we see ha L∗
M(I, θ) = A00 +A01 is composed by wo cu es in ec angles
(θ, I)∈((0, π)∪(π, 2π)) ×B. This is equi alen o p o e ha he de i a i e o his cu e
wi h espec o he a iable θis di e en om 0 o all Iin B. Fo any I∈B, we compu e
he exp ession o ∂L∗
M/∂θ(I, θ) which using (2.10) and he change o a iables (2.45) akes
he o m ∂L∗
M
∂θ (I, θ) = −A10(I) sin(ψ),(2.50)
and ne e anishes o ψ∈(0, π)∪(π, 2π), o equi alen ly, o θ∈(0, π)∪(π, 2π). Then
L∗
M(I, θ) = A00 +A01 is composed by wo e ical cu es on B.
As we ha e seen in Lemma 54, L∗(−I, θ) = L∗(I, θ). Then, he le el cu e L∗
M(I, θ) =
A00 +A01 is also de ined o I < 0, which concludes he p oo .
Rema k 21. Using he exp essions abo e o I+and I++ one can check ha
I+∼π
2|µ|sinh(π/2) and I++ ∼2
πlog(|2 sinh(π/2)µ|),as |µ| → +∞.
31

De ini ion 22. We call highways he wo cu es Hl⊂(0, π)×Tand H ⊂(π, 2π)×Tsuch
ha L∗(I, θ) = A00 +A01. By P oposi ion 20, hey exis a leas o I∈(−∞,−I++)∪
(−I+, I+)∪(I++,+∞) o |µ| ≥ 0.615 and o any alue I o |µ|<0.625. I a10 >0,
by (2.50), ∂L∗/∂θ is posi i e ( espec i ely nega i e) along he highway H ( esp. Hl). I
a10 <0, change Hl o H .
P oposi ion 23. Conside he Hamil onian
Hε(p, q, I, ϕ, s) = ±p2
2+ cos q−1+I2
2+εcos q(a1cos ϕ+a2cos s),
a1a26= 0.The highways ake he o m
θh(I) = 


a ccos A2(1− (I))
A1(I)+Ia ccos( (I)), I ≤0;
a ccos A2(1− (I))
A1(I)−Ia ccos( (I)), I > 0;
and
θH(I) = 

−a ccos A2(1− (I))
A1(I)−Ia ccos( (I)), I ≤0;
−a ccos A2(1− (I))
A1(I)+Ia ccos( (I)), I > 0;
whe e θh∈(0, π)and θH∈(π, 2π).
P oo . F om (2.20), (2.33) and he de ini ion o he highways, we ha e he ollowing wo
equa ions
A1(I) cos(θ−Iτ∗) + A2cos(−τ∗) = A2(2.51)
IA1(I) sin(θ−Iτ∗) + A2sin(−τ∗) = 0.
Mul iplying by I he i s equa ion we ob ain
IA1(I) cos(θ−Iτ∗) + IA2(cos(−τ∗)−1) = 0
IA1(I) sin(θ−Iτ∗) + A2sin(−τ∗) = 0.
o equi alen ly
IA1(I) cos(θ−Iτ∗) = −IA2(cos(−τ∗)−1)
IA1(I) sin(θ−Iτ∗) = −A2sin(−τ∗).
We sum hese wo equa ions squa ed and we ob ain
I2A2
1(I)=[IA2(cos(−τ∗)−1)]2+A2
2sin2(−τ∗).
A e some a i hme ical manipula ions we ob ain he ollowing equa ion o second de-
g ee in cos(−τ∗)
(I2−1)A2
2cos2(−τ∗)−2I2A2
2cos(−τ∗) + A2
2(I2+ 1) −I2A2
1(I)=0.
32
Sol ing his equa ion we ha e
cos(−τ∗) = 2I2A2
2±p4I4A4
2−4(I2−1)A2
2[A2
2(I2+ 1) −I2A2
1(I)]
2(I2−1)A2
2
.
A e mo e a i hme ical manipula ion and conside ing ha −1≤cos(−τ∗)≤1 we ha e
cos(−τ∗) = I2A2−pA2
2+ (I2−1)I2A2
1(I)
(I2−1)A2
.
In o de o simpli y he no a ion we de ine
(I) := I2A2−pA2
2+ (I2−1)I2A2
1(I)
(I2−1)A2
And he e o e,
⇒ −τ∗(I, θ) = ±a ccos( (I)).
Remembe ha we ha e wo highways. This explains why we ha e ound wo di e en
alues o he unc ion τ∗. Then we can ew i e he i s equa ion o (2.51) as
A1(I) cos(θ±Ia ccos( (I))) + A2 (I) = A2.
This immedia ely implies
θ=±a ccos A2(1 − (I))
A1(I)∓Ia ccos( (I)).
F om he ou possibili ies, by compa ing wi h nume ical esul s we ob ain
θh(I) = 


a ccos A2(1− (I))
A1(I)+Ia ccos( (I)), I ≤0;
a ccos A2(1− (I))
A1(I)−Ia ccos( (I)), I > 0;
and
θH(I) = 

−a ccos A2(1− (I))
A1(I)−Ia ccos( (I)), I ≤0;
−a ccos A2(1− (I))
A1(I)+Ia ccos( (I)), I > 0; .
2.3.2 Resul s abou global ins abili y
Now we a e going o p o e wo esul s abou exis ence o he di usion phenomenon
in ou model. The i s one is a di ec applica ion o he geome ical P oposi ion 20 jus
p o ed and desc ibes he di usion ha akes place close o he highways. The second is a
mo e gene al ype o di usion, alid also o he alues o he ac ion Iwhe e he e a e no
highways.
33
Fig. 2.13: Highways in black o µ= 0.6.
Di usion close o highways
Theo em 24. Assume ha a10 a01 6= 0 in he Hamil onian (2.1)+(2.3). Then, o any
I∗ he e exis s ε∗=ε∗(I∗)>0such ha o 0< ε < ε∗, he e exis s a ajec o y
(p( ), q( ), I( ), ϕ( )) such ha o some T > 0
I(0) ≤ −I∗;I(T)≥I∗,
whe e he admissible alues o I∗=I∗(µ)sa is y
•Fo |µ|<0.625,I∗is a bi a y I∗∈(0,+∞).
•Fo 0.625 ≤ |µ| ≤ 1,I∗∈(0, I+), whe e I+= min{I > 0 : I3sinh(π/2)/sinh(πI/2) =
1/|µ|}.
•Fo |µ| ≥ 1,I∗∈(0, I+), whe e I+={I > 0 : I2sinh(π/2)/sinh(πI/2) = 1/|µ|}.
P oo . Recall ha he educed Poinca ´e unc ion, gi en in (2.39), is
L∗
M(I, θ) = A00 +A10(I) cos(θ−Iτ∗
M(I, θ)) + A01 cos(−τ∗
M(I, θ)).
Du ing his p oo , we deno e τ∗
M(I, θ) simply by τ∗
M. Fo εsmall enough, he sca e ing
map SM(I, θ) akes he o m (2.17) o L∗=L∗
M, so ha o bi s unde he sca e ing map
a e con ained in he le el cu es o he educed Poinca ´e unc ion L∗
M, up o e o o O(ε2).
P oposi ion 20 ensu es he exis ence o he highways as wo e ical le el cu es L∗
M(I, θ) =
A00 +A01 o Iin
•(−∞,+∞) o |µ|<0.625.
•(−I+, I+),whe e
–I+= min{I > 0 : I3sinh(π/2)/sinh(πI/2) = 1/|µ|} o 0.625 ≤ |µ| ≤ 1;
–I+= min{I > 0 : I2sinh(π/2)/sinh(πI/2) = 1/|µ|} o |µ| ≥ 1.
34
Take a10 >0. Then gi en I∗>0 (wi h he es ic ion I∗< I+i |µ|>0.625), ∂L∗
M>0
along he highway H . No e ha (I0, θ0) := (0,3π/2) ∈H . Taking any (Ii, θi)∈H ,Ii>0,
i s image unde he sca e ing map (e
Ii+1,e
θi+1) = SM(Ii, θi) sa is ies e
Ii+1−Ii=O(ε)>0 and
is O(ε2)-close o H . Using he inne map on ˜
Λ, we ind (Ii+1, θi+1) = φ i+1 (e
Ii+1,e
θi+1)∈H
wi h Ii+1 −Ii=O(ε)>0. Con inuing ecu si ely in his way, we ge a pseudo-o bi
{(Ii, θi), i = 0, . . . , N} ⊂ H wi h IN≥I∗ o med by applying successi ely he sca e ing
map and he inne map. Using he symme y o H , in oducing Ii=−Ii o i < 0,
we ha e he pseudo-o bi {(Ii, θi),|i| ≤ N} ⊂ H . Using s anda d shadowing esul s
in [FM00, FM03] based on he exis ence o ans e se he e oclinic o bi s be ween non-
esonan o i (changing sligh ly Ii o ob ain an i a ional equency o he inne map, i
necessa y) o newe esul s like he co olla y 3.5 o [GLS14] whe e he ecu ence p ope y
o he inne dynamics is also used, he e exis s a ajec o y o he sys em such ha o
some T,I0≤ −I∗and I(T)≥I∗. I a10 <0, changing H o Hlall he p e ious easoning
applies.
Fig. 2.14: The di usion ajec o y in SM o µ= 0.6.
The gene al di usion
Now we p esen a heo em ha ensu es he di usion o all alues o he pa ame e
a10, a01 (as long as a10a01 6= 0) and o any alue o I∗. Besides, we p o e i using he
geome ical p ope ies o he sca e ing map ha we ha e explo ed up o now.
Theo em 25. Assume ha a10 a01 6= 0 in he Hamil onian (2.1)+(2.3). Then, o any
I∗>0, he e exis s ε∗=ε∗(I∗)>0such ha o any ε,0< ε < ε∗, he e exis s a ajec o y
(p( ), q( ), I( ), ϕ( )) such ha o some T > 0
I(0) ≤ −I∗< I∗≤I(T).
P oo . Ou p oo consis s on showing he exis ence o adequa e o bi s unde se e al sca e -
ing maps, whose o bi s will be gi en app oxima ely by he le el cu es o he co esponding
educed Poinca ´e unc ions, in such a way he alue o Iwill be inc easing. La e on, we
will combine hem wi h o bi s unde he inne map o p oduce adequa e pseudo-o bi s o
shadowing.
35
We begin wi h he simples case. Assume |µ|<0.625. In his case he highways, by
P oposi ion 20, a e de ined o any alue o I∈Rand Theo em 24 ensu es he di usion
phenomenon.
We now assume 0.625 ≤ |µ| ≤ 0.97. In his case o some alue o I he e may exis
angencies be ween he c es s CM(I) and he NHIM lines Rθ(I). Again by P oposi ion
20, in his case he highways a e de ined o all I∈(−∞,−I++)∪(−I+, I+)∪(I++,+∞)
whe e 0 < I+≤I++. The case I∗∈(0, I+) is con ained in he esul o Theo em 24. So,
we a e going o conside I∗∈[I+,+∞).
As be o e, we ha e one SM-o bi con ained in one highway whe e Iis inc easing. We
ha e o s udy he egion o Iwhe e he highways a e no de ined.
Ou s a egy is p o ing he exis ence o a sca e ing map in he side o θwhe e he Iis
inc easing, ha is, o θ∈(0, π) o θ∈(π, 2π) ( his depends on sign(a10)) whe e ∂L∗
M/∂θ
is posi i e. Then, we will use he inne map (o ano he sca e ing map S0) o changing
o pseudo-o bi (le el cu e) o L∗
M. In his way, we con inue he g ow h o I.
Fo any I∈(−I++,−I+)∪(I+, I++), he e exis angencies be ween CM(I) and Rθ(I),
i.e., he e exis s ψsuch ha ∂ξM/∂ψ = 1/I, and he e o e he e exis h ee di e en
sca e ing maps.
Conside he case wi h µ > 0. As we ha e seen in Subsec ion 2.2.2, ψ∈T7→ θ∈Tgi en
in (2.41) is no longe a change o a iables, bu we ha e h ee bijec ions θi:Di(I)→T,
i∈ {A,B,C}(see (2.42)). And o each bijec ion we ha e a sca e ing map associa ed o
i . Among hese h ee sca e ing maps, we will chose only one o he di usion. Conside
i s he case a10 >0 ( ecall ha he highway H goes om −I+ owa d I+). We chose
o ins ance, he sca e ing map associa ed o he educed Poinca ´e unc ion L∗
M,A(I, θ) =
L∗
M(I, θA(ψ)), ψ∈DA(I) since
∂L∗
M
∂θ (I, θA(ψ)) = −A10(I) sin(ψ)>0 o ψ∈DA(I)∩(π, 2π)
and he e o e he i e a es unde he sca e ing map SM.A(I, θ) (2.17) associa ed o L∗
M,A(I, θ)
inc ease he alues o I o θ∈(π, 2π). No ice ha by de ini ion o DA(I) o ψ∈
DA(I)∩(π, 2π) = (ψ2,2π) wi h ψ2∈(π, 3π/2) (see Subsec ion 2.2.2) he e a e no angen-
cies be ween he c es and he NHIM segmen .
We can now p oceed in he ollowing way. We i s cons uc a pseudo-o bi {(Ii, θi) :
i= 0, . . . , N1} ⊂ H wi h I0= 0 and IN=I+, as in he p oo o Theo em 24. No e
ha all hese poin s lie in he same le el cu e o L∗
M, ha is, L∗
M(Ii, θi) = A00 +A01,i=
0, . . . , N1. Applying he inne dynamics, we ge (IN1+1, θN1+1) = φ N1(IN1, θN1) wi h θN1+1 ∈
(θA(ψ2(IN1)),2π) and hen we cons uc a pseudo-o bi {(Ii, θi) : i=N1+1, ..., N1+M1} ⊂
L∗
M,A(IN1+1, θN1+1) = lN1+1 wi h θi∈(θN1+1,2π), 2π−θN1+M1=O(ε2). Applying he
inne dynamics, we ge (IN1+M1+1, θN1+M1+1) = φ N1+M1(IN1+M1, θN1+M1) wi h θN1+M1+1 ∈
θA(ψ2(IN1+M1),2π)). Recu si ely, we cons uc pseudo-o bi {(Ii, θi) : i= N1+ 1, ..., N2}
such ha IN2≥I++.We inally ollow he highway om I++ o I∗cons uc ing a pseudo-
o bi {Ii, θi) : i= N2, ..., IN3} ⊂ H wi h IN3=I∗.
Using he symme y p ope ies (see Lemma 54) in oducing Ii=−Ii o i < 0 we
ha e a pseudo-o bi {(Ii, θi) : |i| ≤ N3}wi h I−N3=−I∗,IN3=I∗. Using now he same
36

shadowing echniques as in he p oo o Theo em 25, he e exis s a di usion ajec o y. I
a10 <0, changing H o Hlall he p e ious easoning applies.
Rema k 26. Fo he p oo o his heo em we ha e chosen a simple pseudo-o bi , jus
choosing he sca e ing map SM,A when i was no unique. O cou se, he e is a lo o
eedom in choosing pseudo-o bi s, and we do no claim ha he one chosen he e is he
bes one conce ning minimal ime o di usion.
(a) SM o µ= 1.5 (b) SMcombined wi h inne map (in
ed)
Fig. 2.15: Fo µ= 1.5, highways a e no p ese ed. Inne map and sca e ing map can be adequa ely
combined
Rema k 27. A ough es ima e o ε∗=ε∗(I∗)o Theo em 25 . The sca e ing map
S(I, θ) (2.17) is he −ε ime map o he Hamil onian L∗(I, θ) gi en in (2.39), up o o de
O(ε2). The e o e, as al eady no iced in Rema k 7, i |∂L∗/∂θ(I, θ)| ≤ εo |∂L∗/∂I(I, θ)| ≤
ε, he le el cu es o L∗(I, θ) a e no use ul enough o desc ibe he o bi s o S. I is easy
o check ha ∇L∗(I, θ) only anishes o I= 0, θ= 0, π mod 2πand ha k∇L∗(I, θ)k.
8π|a10I|e−π|I|/2 o |I| → +∞. Thus, in gene al one has o a oid small neighbo hoods o
(I, θ) = (0,0),(0, π) and ake ca e in egions whe e |I|is e y la ge. In pa icula , he
highways Hl, H a e a om (I, θ) = (0,0),(0, π) and on hem k∇L∗(I, θ)k ≥ A10(I)(1 −
O(β(I)µ)) &4π|a10I|e−π|I|/2 o la ge |I|, om which we ge an uppe bound o ε∗(I∗),
which is exponen ially small in |I∗| o la ge |I∗|:
ε∗(I∗)<4π|a10||I∗|exp(−π|I∗|/2).
Fo smalle alues o I∗, one can compu e nume ically he le el cu es o k∇L∗(I, θ)k=ε
and ob ain ε∗> ε∗(I∗) such ha k∇L∗(I, θ)k=ε∗implies |I|>|I∗|. See Table 2.2 o
some alues o I∗, and µ= 0.9.
2.4 The ime o di usion
In his sec ion we will p o ide an es ima e o he di usion ime. Fo simplici y, we a e
going o es ima e he ime o a di usion using a highway (see De ini ion 22) as a guide,
37
I∗1 2 3 4
ε∗(I∗) 1.4 0.75 0.25 0.07
Table 2.2: Es ima es o ε∗ o µ= 0.9
ha is, we a e going o cons uc a pseudo-o bi close a he highway. This implies o
i e a e he sca e ing map using as ini ial poin a poin on a highway. As we ha e seen
be o e, see Subsec ion 2.2.2, one i e a e o SM(I, θ) is app oxima ed by −ε ime map o he
Hamil onian L∗
M(I, θ) up o O(ε2). Howe e , i we i e a e he sca e ing map a numbe n
o imes, i gene a es a p opaga ed e o wi h espec o he le el cu e o L∗
M(I, θ).
So, i s we s udy he e o gene a ed by ni e a es o he sca e ing map. La e , we will
es ima e he ime o di usion along he highway combining he sca e ing and he inne
maps.
2.4.1 Accu acy o he sca e ing map
Equa ion (2.17) o he sca e ing map Sis good enough up o an e o o O(ε2) o
unde s anding one i e a e o S. Bu i we conside Sn, ha is, n-i e a es o S, some
p oblems appea . These p oblems a e ela ed wi h he lack o p ecision o he equa ion
(2.17):
•Equa ion (2.17) o he sca e ing map has a ela i e e o o o de O(ε) and an
absolu e e o O(ε2). The e o e, o n-i e a es, when nis la ge, he e o is p opaga ed
in a such way ha i canno be disca ded.
•Highways a e uns able, i.e., he nea by le el cu es o L∗mo e away om highways
(see ins ance Fig.2.9.b).
Now, ou goal is o show how we can con ol hese e o s along a egion Uin he phase
space (I, θ) close o a highway. Basically, he con ol is o choose a good momen and
in e al o apply he inne map o come back o he highway and o main ain he e o s
small enough.
The p opaga ed e o
A e i e a ing n imes o mula (2.17) o he sca e ing map, one ge s o (In, θn) =
Sn(I0, θ0):
In=I0+ε
n−1
X
j=0
∂L∗
∂θ (Ij, θj) + O(nε2),and also θn=θ0−ε
n−1
X
j=0
∂L∗
∂I (Ij, θj) + O(nε2).
(2.52)
F om now on, in his sec ion, we will use he ollowing no a ion:
38
• S(I, θ) is he sca e ing map, see (2.17).
•ST(I, θ) = (I+ε ∂L∗/∂θ(I, θ), θ −ε ∂L∗/∂I(I, θ)) is he unca ed sca e ing map.
•S0, (I, θ)=(I( ), θ( )) is he solu ion o he Hamil onian sys em
˙
I( ) = ∂L∗
∂θ (I( ), θ( )) ˙
θ( ) = −∂L∗
∂I (I( ), θ( )),(2.53)
wi h ini ial condi ion (I(0), θ(0)) = (I, θ).
Le (Ih, θh) be a poin in he highway. The e o be ween he sca e ing map and he
le el cu e o he educed Poinca ´e unc ion a e n-i e a es is gi en by
kSn(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih, θh)k,(2.54)
whe e ∆Iand ∆θa e small. No e ha we can ew i e (2.54) as
k(Sn(Ih+ ∆I, θh+ ∆θ)−Sn
T(Ih+ ∆I, θh+ ∆θ))
+(Sn
T(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih+ ∆I, θh+ ∆θ))
+S0,nε(Ih+ ∆I, θh+ ∆θ)) −S0,nε(Ih, θh))k.
We now p oceed o s udy each sub ac ion.
•We begin wi h Sn(Ih+ ∆I, θh+ ∆θ)−Sn
T(Ih+ ∆I, θh+ ∆θ). F om (2.52), we can
eadily ob ain by induc ion ha
Sn(Ih+ ∆I, θh+ ∆θ)−Sn
T(Ih+ ∆I, θh+ ∆θ) = O(nε2).(2.55)
•Now we conside Sn
T(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih+ ∆I, θh+ ∆θ). By he de ini ion
o STwe ha e ha Sn
Tis he n-s ep o he Eule me hod wi h s ep size εin each
coo dina e o sol ing he sys em (2.53). I is no di icul o check he s anda d
bound (see, o ins ance, [SB02])
kSn
T(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih+ ∆I, θh+ ∆θ)k ≤ Lε
2[(1 + εK)n−1] ,(2.56)
whe e K:= max(I,θ)∈U
JH(I, θ) (J∇L∗(I, θ))T
, L = max(I,θ)∈Uk∇L∗(I, θ)kand
H(I, θ) is he Hessian ma ix o L∗(I, θ).
•Now we look o he las sub ac ion S0,nε(Ih+ ∆I, θh+ ∆θ)) −S0,nε(Ih, θh). Apply-
ing G ¨
onwall’s inequali y on he a ia ional equa ion associa ed o he Hamil onian
ec o ield −∇L∗(I, θ), one ge s
kS0,εn(Ih+ ∆I, θh+ ∆θ)) −S0,εn(Ih, θh)k≤k(∆I, ∆θ)keKεn.(2.57)
39
We can now conclude om (2.55), (2.56) and (2.57), ha he p opaga ed e o is
kSn(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih, θh)k≤O(nε2) + Lε
2[(1 + εK)n−1] + k(∆I, ∆θ)keKεn
To a oid la ge p opaga ed e o s, one has o choose nsuch ha nε 1. Fo ins ance,
aking
n=ε−c,(2.58)
wi h 0 < c < 1 (which implies nε 1) and k(∆I, ∆θ)k=εa,a > 0, one ge s
kSn(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih, θh)k=O(ε2−c, εa).(2.59)
2.4.2 Es ima e o he ime o di usion
In his sec ion ou goal is o es ima e he ime o di usion along he highway. We ha e
h ee di e en ypes o es ima es associa ed o he ime o di usion.
•The o al numbe o i e a es Nso he sca e ing map. This is he numbe o i e a es
ha sca e ing map spends o co e a piece o a le el cu e o he educed Poinca ´e
unc ion L∗.
•The ime unde he low along he homoclinic in a ian mani olds o e
Λ. This is he
ime spen by each applica ion o he sca e ing map ollowing he conc e e homoclinic
o bi o e
Λ up o a dis ance δo e
Λ. This ime is deno ed by Th=Th(δ).
•The ime unde he inne map. This ime appea s i we use he inne map be ween
i e a es o he sca e ing map (i is some imes called e godiza ion ime) and we
deno ed i by Ti.
Fo each i e a e o he sca e ing map we ha e o conside he ime Th. Besides, we ha e
seen in he p e ious subsec ion ha o con ol he p opaga ed e o , we i e a e successi ely
he sca e ing map jus a numbe n=ε−co imes, 0 < c 1. F om now on we deno e
his numbe nby Nss. So, a e Nss i e a es o he sca e ing maps we apply he inne
dynamics du ing some ime Ti o come back o a dis ance εa o he highway. The e o e,
he o al ime spen unde he inne map is bNs/NsscTi. We es ima e ha he di usion
ime along he highway is hus
Td=NsTh+bNs/NsscTi.(2.60)
Theo em 28. The ime o di usion Tdclose o a highway o Hamil onian (2.1)+(2.3)
be ween −I∗ o I∗, o any 0< I∗< I+, wi h I+gi en in P oposi ion 20, sa is ies he
ollowing asymp o ic exp ession
Td=Ts
ε2 log C
ε+O(εb), o ε→0,whe e 0<b<1,
40
Conside he au onomous ex ended Hamil onian
K(I, A, ϕ, s) = I2
2+A+ε(a1cos ϕ+a2cos(ϕ−s)) ,(3.4)
wi h associa ed di e en ial equa ions
˙ϕ=I˙
I=ε(a1sin ϕ+a2sin(ϕ−s))
˙s=1 ˙
A=−εa2sin(ϕ−s).
This sys em is equi alen o he sys em ep esen ed by (3.2)+(3.3). We wish o elimina e
he dependence on he angle a iables. Conside a change o a iables ε-close o he iden i y
(ϕ, s, I, A) = g(φ, σ, J, B)=(φ, σ, J, B) + O(ε)
such ha i is he one- ime low o a Hamil onian εG, i.e., g=g =1, whe e g is solu ion
o dg
d =J2∇εG ◦g ,whe e J2is he symplec ic ma ix 0 1
−1 0.
Composing Kwi h gand expanding in a Taylo se ies a ound = 0, one ob ains
K◦g=K+K, εG+1
2K, εG, εG+. . . ,
whe e {·} is he Poisson b acke . Using he expansion (3.4) o K, he equa ion abo e can
be w i en as
K◦g=J2
2+B+εa1cos φ+a2cos(φ−σ) + J2
2+B, G
+ε2
2J2
2+B, G, G+O(ε3).
(3.5)
We wan o ind Gsuch ha a1cos φ+a2cos(φ−σ) + nJ2
2+B, Go= 0, o equi alen ly,
J∂G
∂φ +∂G
∂σ =a1cos φ+a2cos(φ−σ).
Gi en a < b < 1, conside any unc ion Ψ ∈C∞(R) sa is ying Ψ(x) = 1 o x∈[−a, a]
and Ψ(x) = 0 o |x| ≥ band in oduce
G(J, B, φ, σ) := a1
J(1 −Ψ(J)) sin φ+a2
J−1(1 −Ψ(J−1)) sin(φ−σ),
Subs i u ing he abo e unc ion G(J, B, φ, σ) in (3.5) we ha e
K◦g=J2
2+B+O(ε2),(3.6)
47

o J, J −1/∈[−b, b]. Fo J∈[−a, a],
K◦g=J2
2+B+εa1cos φ+O(ε2).(3.7)
Finally, o J−1∈[−a, a],
K◦g=J2
2+B+εa2cos(φ−σ) + O(ε2).(3.8)
F om (3.7) and (3.8), one sees ha on J= 0 and J= 1 he e a e esonances o i s o de
in εwi h a pendulum-like beha io .
Coming back o he o iginal a iables, h ee kinds o in a ian o i a e ob ained. Fo
he i s o de esonance I= 0, he e is a posi i e asuch ha he in a ian o i a e gi en
by F0(I, ϕ, s) = cons an wi h
F0(I, ϕ, s) = I2
2+εa1cos ϕ+O(ε2).(3.9)
o I∈[−a, a].
Analogously, o he i s o de esonance I= 1, wi h
F1(I, ϕ, s) = (I−1)2
2+εa2cos(ϕ−s) + O(ε2),
o I−1∈[−a, a].
Rema k 32. As commen ed in [DLS06], he e exis s a seconda y esonance in I= 1/2,
bu he size o he gap in i s esonan egion is much smalle han he size o gaps in
esonan egions associa ed o I= 0 and I= 1.
Rema k 33. Fo Hamil onian (1.5) wi h 6= 1, he esonances ake place in I= 0 and
I= 1/ .
F om (3.6), on he non- esonan egion he in a ian o i has equa ions Fn (I) =
cons an wi h
Fn (I) = I2
2+O(ε2).
An illus a ion o he inne dynamics is displayed in Figu e 3.1.
3.2 Sca e ing map
We a e going o explo e he p ope ies o he sca e ing maps o Hamil onian (3.1). The
no ion o sca e ing map on a NHIM was in oduced in [DLS00]. Le Wbe an open se o
[−I∗, I∗]×T2such ha he in a ian mani olds o he NHIM ˜
Λ in oduced in (1.3) in e sec
48
Fig. 3.1: Plane ϕ×Io inne dynamics o µ= 0.75 and ε= 0.01.
ans e sally along a homoclinic mani old Γ = {˜z(I, ϕ, s;ε),(I, ϕ, s)∈W}so ha o any
˜z∈Γ he e exis unique ˜x+,−= ˜x+,−(I, ϕ, s;ε)∈˜
Λ such ha ˜z∈Ws
ε(x−)∩Wu
ε(˜x+). Le
H+,−=[{˜x+,−(I, ϕ, s;ε) : (I, ϕ, s)∈W}.
The sca e ing map associa ed o Γ is he map
S:H−−→ H+
˜x−7−→ S(˜x−) = ˜x+.
Fo he cha ac e iza ion o he sca e ing maps, i is equi ed o selec he homoclinic
mani old Γ and his is done using he Poinca ´e-Melniko heo y. F om [DH11, DLS06], we
ha e he ollowing p oposi ion (compa e wi h Chap e 2, P op. 5)
P oposi ion 34. Gi en (I, ϕ, s)∈[−I∗, I∗]×T2, assume ha he eal unc ion
τ∈R7−→ L(I, ϕ −I τ, s −τ)∈R(3.10)
has a non degene a e c i ical poin τ∗=τ∗(I, ϕ, s), whe e
L(I, ϕ, s) := Z+∞
−∞
( (q0(σ)) − (0)) g(ϕ+Iσ, s +σ; 0)dσ.
Then, o 0< ε small enough, he e exis s a unique ans e sal homoclinic poin ˜z o ˜
Λε
o Hamil onian (1.1), which is ε-close o he poin
˜z∗(I, ϕ, s) = (p0(τ∗), q0(τ∗), I, ϕ, s)∈W0(˜
Λ) :
˜z= ˜z(I, ϕ, s) = (p0(τ∗) + O(ε), q0(τ∗) + O(ε), I, ϕ, s)∈Wu(˜
Λε) Ws(˜
Λε).
The unc ion Lis called he Melniko po en ial o Hamil onian (1.1). Fo he conc e e
Hamil onian (3.1) i akes he o m
L(I, ϕ, s) = A1(I) cos ϕ+A2(I) cos(ϕ−s),(3.11)
49
whe e
A1(I) = 2πIa1
sinh(πI/2) and A2(I) = 2π(I−1)a2
sinh(π(I−1)/2).
The homoclinic mani old Γ is cha ac e ized by he unc ion τ∗(I, ϕ, s). Once a τ∗(I, ϕ, s) is
chosen, which unde he condi ions o P oposi ion 34, is locally smoo hly well de ined, by
he geome ic p ope ies o he sca e ing map, see [DH09, DH11, DLS08], he sca e ing
map has he explici local o m
S(I, ϕ, s) = I+ε∂L∗
∂ϕ (I, ϕ, s) + O(ε2), ϕ −ε∂L∗
∂I (I, ϕ, s) + O(ε2), s,
whe e
L∗(I, ϕ, s) = L(I, ϕ −Iτ∗(I, ϕ, s), s −τ∗(I, ϕ, s)).(3.12)
No ice ha he a iable sis ixed unde he sca e ing map. As a consequence, see
[DH11], in oducing he a iable
θ=ϕ−Is
and de ining he educed Poinca ´e unc ion
L∗(I, θ) := L∗(I, ϕ −Is, 0) = L∗(I, ϕ, s),(3.13)
in he a iables (I, θ), he sca e ing map has he simple o m
S(I, θ) = I+ε∂L∗
∂θ (I, θ) + O(ε2), θ −ε∂L∗
∂I (I, θ) + O(ε2),
so up o O(ε2) e ms, S(I, θ) is he ε imes low o he au onomous Hamil onian −L∗(I, θ).
In pa icula , he i e a es unde he sca e ing map ollow he le el cu es o L∗up o
O(ε2).
3.2.1 C es s and NHIM lines
We ha e seen ha he unc ion τ∗plays a cen al ole in ou s udy. The e o e, we a e
in e es ed in inding he c i ical poin s τ∗=τ∗(I, ϕ, s) o unc ion (3.10). Fo ou conc e e
case (3.11), τ∗is a solu ion o
IA1(I) sin(ϕ−Iτ∗)+(I−1)A2(I) sin(ϕ−s−(I−1)τ∗)=0.(3.14)
This equa ion can be iewed om wo equi alen ly geome ical iewpoin s. The i s one
is ha o ind τ∗=τ∗(I, ϕ, s) sa is ying (3.14) o any (I, ϕ, s)∈[−I∗, I∗]×T2is he same
as o look o he ex ema o Lon he NHIM line
R(I, ϕ, s) = {(I, ϕ −Iτ, s −τ) : τ∈R}.(3.15)
Rema k 35. Since (ϕ, s)∈T2,R(I, ϕ, s) is a closed line i I∈Qand i is a dense line on
{I}×T2i I /∈Q.
50
The o he iewpoin is ha , ixing (I, ϕ, s), a solu ion τ∗o (3.14) is equi alen o
inding in e sec ions be ween a NHIM line (3.15) and a cu e de ined by
IA1(I) sin ϕ+ (I−1)A2(I) sin(ϕ−s)=0.
These cu es a e called c es s, and in a gene al way can be de ined as ollows.
De ini ion 36. [DH11] We de ine by C es s C(I) he cu es on (I, ϕ, s), (ϕ, s)∈T2, such
ha ∂L
∂τ (I, ϕ −Iτ, s −τ)|τ=0 = 0,(3.16)
o equi alen ly,
I∂L
∂ϕ(I, ϕ, s) + ∂L
∂s (I, ϕ, s)=0.
As in ou case L(I, ϕ −Iτ, s −τ) = A1(I) cos(ϕ−Iτ) + A2(I) cos(ϕ−s−(I−1)τ),
equa ion (3.16) akes he o m (3.16). In oducing
σ=ϕ−s, (3.17)
equa ion (3.16) can be ew i en as
µα(I) sin ϕ+ sin σ= 0,(3.18)
o I6= 1, whe e
µ=a1
a2
and α(I) = I2sinh(π
2(I−1))
(I−1)2sinh(πI
2).(3.19)
F om now on, when we e e o c es s C(I) we mean he se o poin s (I, ϕ, σ) sa is ying
equa ion (3.18). See an illus a ion in Fig. 3.3.
Rema k 37. In Chap e 2 he c es s we e desc ibed on he plane (ϕ, s), whe eas now such
cu es lie on he plane (ϕ, σ). Besides, di e en ly om he cases s udied in [DH11] and in
Chap e 2, he unc ion α(I) in oduced in (3.19) is no de ined o all I. Mo e p ecisely,
i is no de ined o I= 1. Fo his alue o I, equa ion (3.18) is no adequa e, and one
has o use (3.16) o check ha o I= 1 he c es s a e jus wo e ical s aigh lines on
he plane (ϕ, σ) gi en by ϕ= 0 and ϕ=π.
Rema k 38. Fo Hamil onian (1.5) and ∈(0,1), α (I) is no de ined o I= 1/ and is
gi en by
α (I) = I2sinh π
2( I −1)
( I −1)2sinh πI
2.(3.20)
51
We a e in e es ed in unde s anding he beha io o hese c es s because, as we ha e seen
in[DH11] and Chap e 2, hei in e sec ion wi h he NHIM lines de e mine he exis ence
and beha io o sca e ing maps.
F om (3.18), when |α(I)|<1/|µ|,σcan be w i en as a unc ion o ϕ o all ϕ∈Ton
he c es C(I). On he o he hand, i |α(I)|>1/|µ|,ϕcan be w i en as a unc ion o σ o
all σ∈T.These wo condi ions gi e us wo kinds o c es s: ho izon al o |α(I)|<1/|µ|
and e ical o |α(I)|>1/|µ|. These names a e due o hei o ms on he plane (ϕ, σ).
We conside he same cha ac e iza ion used in Chap e 2:
•Fo |α(I)|<1/|µ|, he e a e wo ho izon al c es s σ=ξM,m(I, ϕ)
CM,m(I) = {(I, ϕ, ξM,m(I, ϕ)) : ϕ∈T},
ξM(I, ϕ) = −a csin(µα(I) sin ϕ) mod 2π(3.21)
ξm(I, ϕ) = a csin(µα(I) sin ϕ) + πmod 2π.
•Fo |α(I)|>1/|µ|, he e a e wo e ical c es s ϕ=ηM,m(I, σ)
CM,m(I) = {(I, ηM,m(I, σ), σ) : σ∈T},
ηM(I, σ) = −a csin(sin σ/ (µα(I))) mod 2π
ηm(I, σ) = a csin(sin σ/ (µα(I))) + πmod 2π.
Rema k 39. |α(I)|= 1/|µ|is a singula o bi u ca ion case. In his case, he c es s a e
s aigh lines and a e no di e en iable in ϕ=π/2 and ϕ= 3π/2. See Fig. 2.6.
Rema k 40. The c es con aining he poin (ϕ, σ) = (0,0) will be deno ed by CM(I) and
he c es con aining he poin (ϕ, σ) = (π, π) by Cm(I).
No e ha he unc ion |α(I)|is no bounded, indeed
lim
I→1|α(I)|= +∞.
This implies ha o any µ he e exis s a neighbo hood Uo I= 1 such ha o all I∈U
he c es s a e e ical. On he o he hand, since α(0) = 0 he e exis s a neighbo hood V
o I= 0 such ha o all I∈V he c es s a e ho izon al. We no ice he e a ema kable
di e ence wi h he Hamil onians s udied in [DH11] and Chap e 2, whe e, o |µ| ≤ 0.97,
all he c es s a e ho izon al o all I.
Now ake a look a he p ope ies o he unc ion α(I) in oduced in (3.19) o desc ibe
unde which condi ions in µ he c es s a e ho izon al o e ical. Fi s o all, obse e ha
o I6= 1, α(I) is smoo h and α0(I)6= 0,and o I= 1 α(I) is no bounded, indeed i has
a e ical asymp o e
lim
I→1−α(I) = −∞ and lim
I→1+α(I) = +∞.
52

Gi en a µ6= 0, since α(0) = 0, he e exis s a unique Ic∈(0,1) such ha |α(I)|= 1/|µ|.
So, he c es s a e ho izon al o I∈[0, Ic) and e ical o I∈(Ic,1).
O he s impo an limi s a e
lim
I→−∞ α(I) = exp(π/2) and lim
I→+∞α(I) = exp(−π/2).
The i s limi implies ha |α(I)|<exp(π/2) o I∈(−∞,0). Thus, i exp(π/2) ≤1/|µ|
he c es s a e ho izon al o I∈(−∞,0). O he wise, i 1/|µ|<exp(π/2), he e exis s a
unique Il∈(−∞,0) such ha |α(I)|= 1/|µ|and he c es s a e e ical o I∈(−∞, Il)
and ho izon al o I∈(Il,0).
The second limi implies ha |α(I)|>exp(−π/2) o I∈(1,+∞). Then, i exp(−π/2) ≥
1/|µ|, he c es s a e e ical o I∈[1,+∞). i exp(−π/2) <1/|µ|, he e exis s a
unique I ∈(1,+∞), such ha he c es s a e e ical o any Iin [1, I ) and ho izon-
al o I∈(I ,+∞).
Summa izing, o 1/|µ| ≥ exp(π/2), c es s a e ho izon al o I∈(−∞, Ic)∪(I ,+∞)
and e ical o I∈(Ic, I ). Fo exp(−π/2) <1/|µ|<exp(π/2), c es s a e ho izon al o
I∈(Il, Ic)∪(I ,+∞) and e ical o I∈(−∞, Il)∪(Ic, I ). Finally, i 1/|µ|<exp(−π/2),
c es s a e ho izon al o I∈(Il, Ic) and e ical o I∈(−∞, Il)∪(Ic,+∞).
Rema k 41. Fo ∈(0,1), α (I) (3.20) is no bounded on a neighbo hood o he esonance
I= 1/ , i.e., limI→1/ −α (I) = −∞ and limI→1/ +α (I)=+∞. The same beha io akes
place o = 1 and close o I= 1. On he o he hand, o I→ ±∞,α (I) has he same
beha io as in he case o = 0, limI→±∞ α (I) = 0. This implies ha o any alue o
µ, o Iclose enough o I= 1/ he c es s a e e ical, and o |I|la ge enough he c es s
a e ho izon al.
Example To illus a e his discussion, we p esen a conc e e example. Taking µ= 0.5,
we ha e exp(−π/2) <1/µ = 2 <exp(π/2). In his case we ha e Il≈ −1.807, Ic≈0.701
and I ≈1.367. The c es s a e ho izon al in (−1.807,0.701) ∪(1,367,+∞) and e ical in
(−∞,−1.807) ∪(0.701,1.367). We emphasize ha his scena io is e y di e en om he
case in Chap e 2. The e, o µ= 0.5 he c es s a e ho izon al o all I.
Now, we a e going o ocus on he ans e sali y o he in e sec ion be ween NHIM lines
R(I, ϕ, s) and c es s C(I). On he plane (ϕ, σ) he NHIM lines can be w i en as
RI(ϕ, σ) = {(ϕ−Iτ, σ −(I−1)τ), τ ∈R},(3.22)
so ha i s slope is (I−1)/I in such plane. The e o e, he e exis s an in e sec ion be ween
NHIM lines and c es s ha is no ans e sal i , and only i , he e exis s a angen ec o
o C(I) a a poin ha is pa allel o (I, I −1), o , using he pa ame e iza ions,
∂ξ
∂ϕ(I, ϕ) = I−1
Io ∂η
∂σ(I, σ) = I
I−1.
53
Conside ing a ho izon al pa ame e iza ion o C(I), he angency condi ion is equi alen
o ±α(I)µcos ϕ
p1−µ2α2(I) sin2ϕ=I−1
I.
The e o e, he e exis s a ϕsa is ying he abo e condi ion i , and only i ,
|β(I)| ≥ 1
|µ|,whe e β(I) = Iα(I)
I−1
and ϕ akes he o m
ϕ=±a c an sβ(I)2−(1/µ)2
(1/µ)2−α(I)2!.
In an analogous way, o a e ical pa ame e iza ion η(I, σ), he e a e angencies i , and
only i ,
|β(I)| ≤ 1
|µ|wi h σ=±a c an 
I−1
Is(1/µ)2−β(I)2
α(I)2−(1/µ)2!.
Rema k 42. Obse e ha in bo h cases, ho izon al and e ical c es s, he e a e angencies
i , and only i , |α(I)|− 1
|µ||β(I)|− 1
|µ|<0.
The unc ion |β(I)|is smoo h in R {1}and d|β(I)|/dI = 0 only o I= 0. Besides,
we ha e (see Figs. 3.2(a) and 3.2(b))
lim
I→1|β(I)|= +∞,lim
I→−∞ |β(I)|= exp(π/2) and lim
I→+∞|β(I)|= exp(−π/2).
The e o e, he e a e h ee possibili ies:
• o 1/|µ| ≥ exp(π/2), he e exis I0∈(1/2,1) and I+∈(1,+∞) such ha I0and
I+a e solu ions o |β(I)| − 1/|µ|= 0. Besides, |β(I)|<1/|µ| o I∈(−∞, I0)∪
(I+,+∞) and |β(I)|>1/|µ| o I∈(I0,1) ∪(1, I+).
• o exp(−π/2) <1/|µ|<exp(π/2), he e exis I−∈(−∞,0), I0∈(0,1) and
I+∈(1,+∞) such ha I−,I0and I+a e solu ions o |β(I)|−1/|µ|= 0. Besides,
|β(I)|<1/|µ| o I∈(I−, I0)∪(I+,+∞) and |β(I)|>1/|µ| o I∈(−∞, I−)∪
(I0,1) ∪(1, I+).
•Fo 1/|µ| ≤ exp(−π/2), he e exis I−∈(−∞,0) and I0∈(0,1/2) such ha I−and
I0a e solu ions o |β(I)|− 1/|µ|= 0. Besides, |β(I)|<1/|µ| o I∈(I−, I0) and
|β(I)|>1/|µ| o I∈(−∞, I−)∪(I0,1) ∪(1,∞).
54
Pu ing oge he his desc ip ion o |β(I)|wi h he s udy abou e ical and ho izon al
c es s and adding ha
|β(I)|<|α(I)| ∀I∈(−∞,0) ∪(0,1/2);
|β(I)|>|α(I)| ∀I∈(1/2,1) ∪(1,+∞);
|β(0)|=|α(0)|= 0 |β(1/2)|=|α(1/2)|= 1
we can s a e he p oposi ion below.
P oposi ion 43. Conside he wo c es s C(I)de ined by (3.18) and he NHIM line
RI(ϕ, σ)de ined in (3.15) o Hamil onian (3.1).
•Fo |µ| ≤ exp(−π/2), he e exis Ib< Ia< IA< IBsuch ha
– o I < Ibo IB< I,C(I)a e ho izon al and in e sec ans e sally any
RI(ϕ, σ);
– o Ib≤I < Iao IA< I ≤IB, he c es s C(I)a e ho izon al, bu now, he e
exis angencies be ween C(I)and wo NHIM lines RI(ϕ, σ);
– o Ia< I < IA, he c es s C(I)a e e ical and in e sec ans e sally any
RI(ϕ, σ).
•Fo exp(−π/2) <|µ|<exp(π/2) he e exis Ib< Ia< Ic≤IC< IA< IBsuch ha
– o I < Ibo IC< I < IA,C(I)a e e ical and in e sec ans e sally any
RI(ϕ, σ);
– o Ib≤I < Ia, he c es s C(I)a e e ical and he e exis angencies be ween
C(I)and wo NHIM lines RI(ϕ, σ);
– o Ia< I < Ico IB< I,C(I)a e ho izon al and in e sec ans e sally any
RI(ϕ, σ);
– o IA≤I≤IB, he c es s C(I)a e ho izon al and he e exis angencies be ween
C(I)and wo NHIM lines RI(ϕ, σ);
– o Ic≤I≤IC, i Ic<1/2, he c es s C(I)a e e ical and he e exis an-
gencies be ween C(I)and RI(ϕ, σ). I Ic= 1/2, om he p ope ies o α(I)and
β(I) his in e al is jus one poin . I Ic>1/2, he c es s C(I)a e ho izon al
and he e exis angencies.
•Fo |µ| ≥ exp(π/2) he e exis Ib< Ia< IA< IBsuch ha
– o I < Ibo IB< I,C(I)a e e ical and in e sec ans e sally any RI(ϕ, σ);
– o Ib≤I < Iao IA< I ≤IB, he c es s C(I)a e e ical and he e exis
angencies be ween C(I)and wo NHIM lines RI(ϕ, σ);
– o Ia< I < IA, he c es s C(I)a e ho izon al and in e sec ans e sally any
RI(ϕ, σ).
Rema k 44. No e ha we a e no conside ing he singula case |α(I)|= 1/|µ|desc ibed
in Rema k 39.
55
Example Again, o illus a e his p oposi ion, we ake he case wi h µ= 0.5, see
Fig. 3.2(a). In his case, we ha e |β(I)|= 1/µ o I≈ −2.942,0.595,1.85 and
• o I∈(−∞,−2.942) ∪(0.701,1) ∪(1,1.367) ⇒|α(I)|>1/|µ| ⇒ e ical c es s
|β(I)|>1/|µ| ⇒ no angencies
• o I∈[−2.942,−1.807) ⇒|α(I)|>1/|µ| ⇒ e ical c es s
|β(I)| ≤ 1/|µ| ⇒ angencies
• o I∈(−1.807,0.595) ∪(1.85,+∞)⇒|α(I)|<1/|µ| ⇒ ho izon al c es s
|β(I)|<1/|µ| ⇒ no angencies
• o I∈[0.595,0.701) ∪(1.367,1.85] ⇒|α(I)|<1/|µ| ⇒ ho izon al c es s
|β(I)| ≥ 1/|µ| ⇒ angencies
Once mo e, we compa e wi h he Hamil onian (1.6) s udied in Chap e 2. Fo Hamil o-
nian (1.6) and µ= 0.5 he e is no angency, bu o Hamil onian (3.1) we can ind angencies
o ho izon al and e ical c es s. Indeed, o Hamil onian (1.6) and any 0 <|µ|<0.625
he e is no angency, whe eas o any µ6= 0 he e a e angencies o Hamil onian (3.1).
(a) |α(I)|and |β(I)|:µ= 0.5, Ib≈ −2.942,
Ia≈ −1.807, Ic≈0.595, IC≈0.701, IA≈
1.367 and IB≈1.85
(b) |α (I)|and |β (I)|:µ= 0.5 and = 0.5.
Fig. 3.2: |α(I)|and |β(I)|: Beha io o he c es s and angencies.
Rema k 45. Fo ∈(0,1) in Hamil onian (1.5), β (I) is de ined by β (I) = Iα (I)/( I −
1). In his case, limI→1/ |β (I)|= +∞and limI→±∞ |β (I)|= 0. In Fig. 3.2(b), a compa -
ison be ween he unc ions α (I), β (I) and he s aigh line 1/|µ| o = 1/2 is displayed.
Fo each c es , whe e i is well de ined, he e exis s, a leas , a alue τ∗such ha
(ϕ−Iτ∗, σ −(I−1)τ∗)=(ϕ−Iτ∗, ξ(I, ϕ −Iτ∗)) o (η(I, σ −(I−1)τ∗), σ −(I−1)τ∗),
which means ha RI(ϕ, σ)∩ C(I)6=∅. This in e sec ion is in insically associa ed o a
homoclinic o bi o he NHIM. To make a choice abou how o ake such τ∗is o choose in
which homoclinic mani old Γ he homoclinic poin s ˜z∗lie. E en mo e, i is o choose wha
sca e ing map we a e going o use.
56
c) Fo I > 1, one mo e ime α(I)>0 and sin ξ1(I, ϕ) sin(ϕ)<0, bu now 0 < m =
1−1/I < 1. We i s ix θ= 3π/2 and sea ch o Isuch ha
3π
2−Iτ∗(I, 3π/2) = 0
3π
2−(I−1)τ∗(I, 3π/2) = π.
We ob ain I= 3/2, so θ−Iτ∗
1(I, θ)∈(0, π) o any I≥3/2 and θ∈(π, θ+= 3π/2).
Consequen ly, sin(θ−Iτ∗
1(I, θ)) >0 and ˙
I > 0. Fo he alues o I∈(1,3/2) we
change he s a egy. We look o θ∗such ha
θ−Iτ∗(I, θ) = 0
θ−(I−1)τ∗(I, θ) = π.
We ha e θ∗=πI and θ−Iτ∗
1(I, θ∗)∈(0, π) o any I∈(1,3/2) and θ∈(π, θ∗), so
˙
I > 0. No e ha θ∗<3π/2 and we can de ine θ+:= θ∗.
Obse e ha o I= 1 he c es s a e e ical, and o I= 0, θ=θ−Iτ∗
1(I, θ), and ˙
I > 0
o θ∈(π, 3π/2).
Conside now he case o e ical c es s (|α(I)µ|>1).
a) Fo I < 0, sin η1(I, σ) sin σ=−µα(I) sin2σ≤0 and m > 1. We ix θ= 3π/2 and
look o Isuch ha
3π/2π−Iτ∗=π
3π/2−(I−1)τ∗(I, 3π/2) = 0.
We ob ain I=−1/2 and he e o e, sin(θ−(I−1)τ∗
1(I, θ)) >0 o I∈(−∞,−1/2)
and θ∈(π, 3π/2). Consequen ly, ˙
I > 0 om (3.27). Fo I∈(−1/2,0), we ha e ha
θ+= (1 −I)πsa is ies
θ−Iτ∗(I, θ+) = π
θ+−(I−1)τ∗(I, θ+) = 0.
The e o e, sin(θ−(I−1)τ∗
1)(I, θ)>0 and ˙
I > 0 o any θ∈(π, θ+).
b) Fo 0 < I < 1 sin η1(I, σ) sin σ≥0 and m < 0. θ+= (I+ 1)πsa is ies
θ−Iτ∗(I, θ+) = π
θ+−(I−1)τ∗(I, θ+)=2π.
So, sin(θ−(I−1)τ∗
1(I, θ)) >0 and ˙
I > 0 o any θ∈(π, θ+). No e ha θ+<3π/2
o I∈(0,1/2).
c) Finally, o I > 1, sin η1(I, σ) sin σ≤0. We ha e ha θ−(I−1)τ∗
1(I, θ)∈(π, 2π),
so sin(θ−(I−1)τ∗
1(I, θ)) <0 and ˙
I > 0 o any θ∈(π, 3π/2).
63

Fo I= 0 he c es s a e ho izon al. Fo I= 1, θ=θ−(I−1)τ∗
1(I, θ), so ˙
I > 0 o
θ∈(π, 2π).
Rema k 52. I a1<0, we ha e ha he e exis s a θ−such ha ˙
I > 0 o any θ∈(θ−, π).
Rema k 53. An analogous p oposi ion holds o S2(I, θ), he sca e ing map associa ed
o he g aphs o ξ2and η2o C2(I). In such case, he e is a θ+such ha ˙
I≥0 o any
θ∈(θ+,2π) whe e θ≥3π/2 o I∈(1/2,3/2).
No e ha his p oposi ion leads us o ensu e he di usion in an analogous way o he
one used o p o e Theo em 25. Nex , he di usion mechanism is s a ed and he A nold
di usion is p o en.
3.3 A nold Di usion
In his sec ion we a e going o comple e ou goal p o ing he exis ence o global ins a-
bili y o A nold di usion, ha is, Theo em 1.
We begin by p esen ing some gene al geome ical p ope ies o he sca e ing maps ha
we ha e o ake in o accoun o p o e he heo em o di usion. The i s one educes he
s udy o sca e ing maps o posi i e alues o µ. Mo e p ecisely, we ha e he lemma below
Lemma 54. The sca e ing map o a alue o µand s=π, associa ed o he in e sec ion
be ween R(I, ϕ, s)and Cm(I)(CM(I)) has he same geome ical p ope ies as he sca e ing
map o −µand s= 0, associa ed o he in e sec ion be ween Rθ(I)and CM(I)(Cm(I)),
i.e.,
Sµ
m(M)(I, ϕ, π) = S−µ
M(m)(I, ϕ, 0) = S−µ
M(m)(I, θ)
P oo . Fi s , we look o τ∗
msuch ha he NHIM segmen R(I, ϕ, s) in e sec s he c es
Cm(I). I we ix s=π, we ha e om (3.11) and (3.12):
L∗
µ,m(I, ϕ, π) =A1(I) cos(ϕ−Iτ∗
m(I, ϕ, π))
+A2(I) cos(ϕ−π−(I−1)τ∗
m(I, ϕ, π)).(3.29)
Besides, τ∗sa is ies
µα(I) sin(ϕ−Iτ∗
m) + sin(ϕ−π−(I−1)τ∗
m) = 0,
o
−µα(I) sin(ϕ−Iτ∗
m) + sin(ϕ−(I−1)τ∗
m) = 0.
We ha e ha ϕ−π−(I−1)τ∗
m(mod 2π) = ξm(I, ϕ −Iτ∗
m) wi h π/2≤ξm≤3π/2.
Then, o each τ∗
m he e exis s a K∈Zsuch ha
π
2< ϕ −π−(I−1)τ∗
m+ 2πK < 3π
2.
64
This implies
3π
2< ϕ −(I−1)τ∗
m+ 2πK and ϕ−(I−1)τ∗
m+ 2π(K−1) <π
2.
The e o e,
ϕ−(I−1)τ∗
m(mod 2π)<π
2o ϕ−(I−1)τ∗
m(mod 2π)>3π
2.
We can conclude ha ϕ−(I−1)τ∗
m(mod 2π) = ξM(I, ϕ −Iτ∗
m). The e o e τ∗
m(I, ϕ, π) o
µis equal o τ∗
M(I, ϕ, 0) o −µ. F om (3.29), L∗
µ,m(I, ϕ, π) sa is ies
L∗
µ,m(I, ϕ, π) = A1(I) cos(ϕ−τ∗
M(I, ϕ, 0)) + (−A2(I)) cos(ϕ−(I−1)τ∗
M(I, ϕ, 0))
=L∗
−µ,M(I, ϕ, 0).
Since L∗
µ,m(·,·, π) and L∗
−µ,M(·,·,0) coincide, hei de i a i es oo and his implies ha
Sµ
m(I, ϕ, π) = S−µ
M(I, ϕ, 0) = S−µ
M(I, θ).
F om now on, jus o simpli y he exposi ion, a1and a2a e conside ed posi i e. The
same s a egy used in Chap e 2, Sec ion 2.3, is applied o p o e he exis ence he di usion:
we combine he sca e ing map in an in e al o θwhe e ˙
I > 0 and he inne map o build a
di usion pseudo-o bi . Then we apply shadowing esul s o ge he exis ence o a di usion
o bi .
Since I= 0 and I= 1 a e esonance alues, he applica ion o he inne map mus be
mo e ca e ul, because in hese esonance egions, o some o bi s, he alue o Idec eases
in o de O(√ε), i. e., he o i canno be conside ed la . We s udy he ans e sali y
be ween he olia ions o in a ian se s o he inne and he sca e ing map in esonan and
non- esonan egions and i s image unde he sca e ing map S. Fo mo e de ails and a
mo e gene al case, he eade is e e ed o [DH09].
Conside he esonan egion associa ed o I= 0. In such egion, he o i can be
app oxima ed by F0(I, ϕ) gi en in (3.9). The an e sali y be ween in a ian se s o he
inne and he sca e ing map holds i he g adien ec o s o he le el cu es o F0and L∗
a e no pa allel ec o s, o equi alen ly,
F0(I, θ),L∗(I, θ)6= 0,
whe e {,}is he Poisson b acke ,
F0,L∗=∂F0
∂θ
∂L
∂I −∂F0
∂I
∂L
∂θ .
F om (3.9), he pa ial de i a i es o F0a e
∂F0
∂I =Iand ∂F0
∂θ =−εa1sin θ,
65
and since L∗(I, θ) = A1(I) cos(θ−Iτ∗(I, θ)) + A2(I) cos(θ−(I−1)τ∗(I, θ)), we ha e he
pa ial de i a i es gi en by
∂L∗
∂θ =A1(I) sin(θ−Iτ∗)
I−1,
∂L∗
∂I =A0
1(I) cos(θ−Iτ∗) + A0
2(I) cos(θ−(I−1)τ∗)
+A1(I)τ∗sin(θ−Iτ∗) + A2(I)τ∗sin(θ−(I−1)τ∗).
No e ha i |I|>O(ε), ∂F0/∂I domina es ∂F0/∂θ, so he Poisson b acke abo e can
be educed o F0,L∗≃ −∂F0
∂I
∂L
∂θ =−IA1(I) sin(θ−Iτ∗)
I−1
Expanding sin(θ−Iτ∗) in Taylo ’s se ies a ound I= 0, we ha e
sin(θ−Iτ∗) = sin θ+O(I),
which implies {F0,L∗}= 0 i , and only i , θ≈0, π, assuming ha O(I) is small enough.
Now, we conside I=O(ε) and look a he in e sec ions be ween he NHIM lines and
he g aph o ξ1. No e ha as he alue o Iis close o 0 we can assume ha he c es s a e
ho izon al. Using Taylo ’s se ies we can w i e
sin(θ−Iτ∗) = sin θ+O(I) cos(θ−Iτ∗) = cos θ+O(I)
sin(θ−(I−1)τ∗) = O(I) cos(θ−(I−1)τ∗) = −1 + O(I).
This implies
F0,L∗=−IA1(I) sin θ
I−1−εa1sin θ(A0
1(I) cos θ−A0
2(I)
+A1(I)τ∗sin θ) + O(I2, εI).
(3.30)
Taylo expanding he unc ions A1(I), A0
1(I) and A0
2(I) a ound I= 0, we ob ain
A1(I) = 4a1+O(I2), A0
1(I) = O(I) and A0
2(I) = a2π(πco h π/2−2)csch(π/2) + O(I)
Plugging hese exp essions in (3.30), we se
F0,L∗=−4a1Isin θ
I−1−εa1sin θ[a2π(πco h π/2−2)csch(π/2)
+4a1(π−θ) sin θ] + O(I2, Iε).
The e o e,
F0,L∗= 0 ⇔a1sin θ−4I
I−1−εa2ππco h π
2−2csch π
2)
+ε4(π−θ) sin θ] = 0.
66
In o he wo ds, we do no ha e ans e sali y i , and only i , θ= 0, π o sa is ies
(π−θ) sin θ=I
εa1
+π(co h π/2−2)cschπ/2)
4,
which is no an ho izon al cu e in he plane (θ, I) and is ans e sal o an in a ian o us
o he inne dynamics.
Fo he o he esonan egion I= 1, F1is e y simila . Assuming I−1 = O(ε), we
ha e
F1,L∗=a2sin θ4I−1
I−ε[πa1(2 −πco h(π/2))csch(π/2) + 4a2sin θ].
Applying he same me hodology, we ob ain an analogous esul o he o he esonan
egion F1. In sho , we conclude ha he image S(Ti) o an in a ian o us Tio he inne
map unde he sca e ing map in e sec s an e sally ano he in a ian o us Ti+1 o he
inne map.
Finally, in he non- esonan egion, we no ice ha
{Fn ,L∗}=−∂Fn
∂I
∂L∗
∂θ =−IA1(I) sin(θ−Iτ∗)
I−1,
jus he same exp ession as he one o he esonance I= 0, so he ans e sali y be ween
in a ian se s o he inne and he sca e ing map ollows.
Now, a cons uc i e p oo o Theo em 1 is p esen ed. This p oo is simila o he p oo
p esen ed in Subsec ion 2.3.2 o Chap e 2, bu now, he e is no any piece o “highway” o
as e ical lines whe e |I|is la ge. So, he inne map is applied mo e imes.
3.3.1 P oo o Theo em 1
P oo . We conside = 1 in Hamil onian (1.5). Fi s o all we ha e o choose wha
sca e ing map we use. This choice depends on he sign o µas explained in Lemma 54.
Assuming µ > 0, we ake S1(I, θ), he global sca e ing map associa ed o he g aphs o ξ1
and η1. I a1>0, by P oposi ion 51 o any I he e exis s an in e al θ∈(π, θ+) whe e
˙
I > 0. De ine H he se (ρ, θ+)×[−I∗, I∗], whe e ρ=π+δis such ha π < ρ < θ+
and he ans e sali y be ween NHIM lines and L∗
1holds. We i s cons uc a pseudo-
o bi {(Ii, θi) : i= 0, . . . , N1} ⊂ H wi h I0=−I∗and θN1as close as possible o ρ.
No e ha all hese poin s lie in he same le el cu e o L∗
1, ha is, L∗
1(I0, θ0) = L∗
1(Ii, θi),
i= 1, . . . , N1. Applying he inne dynamics, we ge (IN1+1, θN1+1) = φ N1(IN1, θN1) wi h
θN1+1 ∈(ρ, θ+) and hen we cons uc a pseudo-o bi {(Ii, θi) : i=N1+ 1, . . . , N1+M1} ⊂
L∗
1(IN1+1, θN1+1) = lN1+1 wi h θi∈(ρ, θN1+1), θ+−θN1+M1=O(ε2). Applying he inne
dynamics, we ge (IN1+M1+1, θN1+M1+1) = φ N1+M1(IN1+M1, θN1+M1) wi h θN1+M1+1 ∈(ρ, θ+).
Recu si ely, we cons uc a pseudo-o bi {(Ii, θi) : i= N1+ 1,...,N2}such ha IN2≥I∗.
In he same way, as in he p oo o Teo em 25, we can apply shadowing echniques o
[FM00, FM03, GLS14], due o he ac ha he inne dynamics is simple enough o sa is y
67
he equi ed hypo hesis o hese e e ences, o p o e he exis ence o a di usion ajec o y.
I a10 <0, changing H o Hl= (θ+, π) all he p e ious easoning applies.
Conside ing Rema k 33, Rema k 38, Rema k 41 and Rema k 45, o any ∈(0,1), an
equi alen di usion esul is eadily ob ained. And, inally, he case o = 0 is p o ed in
Theo em 25 in Chap e 2.
3.4 Piecewise smoo h global sca e ing maps
In his sec ion, he geome ic eedom o he choice o τ∗is explo ed. Un il now, only
wo di e en sca e ing maps ha e been used o build a global one, and his was enough
o ensu e di usion. Bu , wi h his app oach, inding a di usion pseudo-o bi is no always
easy enough and his pseudo-o bi can be also complica ed. This depends simply on he
“aspec ” o he sca e ing map ob ained.
We now sugges a new c i e ion o choose τ∗: o ake he minimal alue o |τ∗| o
any (θ, I). This p o ides us wi h a piecewise smoo h global sca e ing map wi h a good
p ope y: he phase space o his sca e ing map which is O(ε2)-close o he le el se s o he
educed Poinca ´e unc ion L∗(I, θ) associa ed o he chosen τ∗is simple and“cleane ” han
he phase spaces o o he sca e ing maps displayed up o now. By a cleane sca e ing
map, we mean ha we can easily iden i y and unde s and he o bi s o he sca e ing maps,
excep o a small egion which con ains he angency locus.
Besides, he zones whe e he alue o Iis inc eased o dec eased unde he sca e ing
map is well beha ed. Idec eases o θ∈(0, π) ( he ed egion on all pic u es in Fig. 3.6)
and Iinc eases o θ∈(π, 2π) ( he g een egion on all pic u es in Fig. 3.6). So i is easy o
in e ha o inding a di usion pseudo-o bi i is enough o build a combina ion be ween
he inne map and his sca e ing map es ic ed o (π, 2π), o example i an inc eased
alue o Iis wished. The same idea used in he p oo o Theo em 1.
Obse e ha he sca e ing maps we a e now conside ing a e a mix o he sca e ing
maps s udied p e iously. As an example, we illus a e he sca e ing map ob ained o
µ= 0.9. Such sca e ing map can be di ided in o h ee egions and in each egion, he
sca e ing map coincides wi h a sca e ing map s udied be o e.
In Fig. 3.7, o egions I (0 < θ < π/2), II (π/2< θ < 3π/2) and III (3π/2< θ < 2π)
he sca e ing map has he ollowing co espondence:
I Ex ended sca e ing map S0(I, θ) associa ed o he ho izon al CM(I) “unde ” σ=ϕ.
II Ex ended sca e ing map S1(I, θ) associa ed o he ho izon al Cm(I).
III Ex ended sca e ing map S2(I, θ) associa ed o he ho izon al CM(I) “o e ” σ=ϕ.
I ex ended sca e ing maps a e no conside ed and we jus use sca e ing maps associ-
a ed o ho izon al and e ical c es s, one can see ha hese sca e ing maps can be di ided
in o 6 egions, i.e., hey can be iewed as a combina ion o up o 6 sca e ing maps.
68

(a) Piecewise sca e ing map o µ=
0.3.
(b) Piecewise sca e ing map o µ= 0.5.
(c) Piecewise sca e ing map o µ=
0.9.
(d) Piecewise sca e ing map o µ=
1.5.
Fig. 3.6: Examples o piecewise smoo h global sca e ing maps. The o bi s o sca e ing maps a e
ep esen ed by he blue lines. In he ed zones he alues o Ion such o bi s dec ease, in he g een one
he alues o Iinc ease.
Ano he p ope y o hese sca e ing maps is he loss o di e en iabili y on he s aigh
lines θ=π/2 and θ= 3π/2. The ec o ield associa ed o he Hamil onian −L∗
ide ined
a ound hese discon inui y lines beha es as he ec o ields s udied in non-smoo h dynam-
ics heo y. Mo e p ecisely, we can ind egions wi h slide and uns able slide beha io [Fil88].
In a u u e wo k, we en isage o design special pseudo-o bi s along hese discon inui y lines
using such heo y. No e ha hese pseudo-o bi s would be e y simila o he “highways”
de ined in 2.3 o Chap e 2, so in p inciple, one can expec as and simple di usion along
hese discon inui y lines.
69
Fig. 3.7: A piecewise smoo h global sca e ing map di ided in o 3 egions. The e ical black lines a e
he bounda ies o he domains o smoo h sca e ing maps.
70
Chap e 4
A case o 3+1/2 deg ees o eedom
A e a s udy abou an a p io i uns able Hamil onian sys em wi h 2 + 1/2 deg ees
o eedom, a na u al ques ion is wha happens o a simila sys em wi h mo e deg ees
o eedom. In his chap e , we y o answe , a leas pa ially, his ques ion. Pa ially
because we conside a pa icula case o 3 + 1/2 deg ees o eedom.
We a e going o conside a gene aliza ion o he Hamil onian conside ed in Chap e s 2
and 3, which is gi en by he a p io i uns able Hamil onian wi h 3+1/2 deg ees o eedom
Hε(p, q, I1, I2, ϕ1, ϕ2, s) = ±p2
2+ cos q−1+h(I1, I2) + ε (q)g(ϕ1, ϕ2, s),(4.1)
whe e (q) = cos q,h(I1, I2) = Ω1I2
1/2+Ω2I2
2/2 and
g(ϕ1, ϕ2, s) = a1cos ϕ1+a2cos ϕ2+a3cos(k·ϕ−s),
wi h k= (k1, k2)∈Z2and (ϕ1, ϕ2)∈T2.
Rema k 55. In [DLS16], he au ho s deal wi h k= (1,1) as an example o hei esul s.
In his hesis, we es ic ou a en ion o he case wi h k= (0,0). The e a e wo main
easons o his es ic ion: Fi s , his sys em is a di ec gene aliza ion o Hamil onian
(2.1)+(2.3) in Chap e 2. So, o his Hamil onian, we can expec o ind a simila beha -
io o he c es s, he exis ence o global sca e ing maps and, mo eo e , he exis ence o
highways. Besides, we ha e a well-known case o compa e wi h he new esul s ob ained.
The second eason is ha i is much easie o handle i because we educed he numbe o
pa ame e s and i s inne dynamics is simpli ied.
The e o e, om now on, we a e always assume
g(ϕ1, ϕ2, s) = a1cos ϕ1+a2cos ϕ2+a3cos s. (4.2)
Fo a simple no a ion, we deno e I= (I1, I2) and ϕ= (ϕ1, ϕ2).
71
4.1 Unpe u bed case
In he unpe u bed case (ε= 0), such sys em is he Hamil onian sys em wi h Hamil o-
nian
H0(p, q, I, ϕ, s) = ±p2
2+ cos q−1+h(I),
and equa ions
˙q=p˙p= sin q
˙ϕ1=ω1˙
I1= 0
˙ϕ2=ω2˙
I2= 0
˙s= 1,
whe e ωi= ΩiIi,i= 1,2. This sys em consis s o a pendulum plus wo o o s. F om he
equa ions abo e, I1and I2a e cons an s and he low has he o m
Φ (p, q, I, ϕ)=(p( ), q( ), I, ϕ + ω),
whe e ω= (ω1, ω2). And we ha e an in a ian se (on he ex end phase space)
TI={(0,0, I, ϕ, s); ϕ, s ∈T3}.
In his case, he NHIM is
˜
Λ = {(0,0, I, ϕ, s):(I, ϕ, s)∈R2×T3},(4.3)
4.2 Inne dynamics
The inne dynamics is de i ed om he es ic ion o he Hamil onian (4.1) and i s
equa ions o ˜
Λ, gi en in (4.3), i.e.,
Kε(I, ϕ, s) = h(I) + ε(a1cos ϕ1+a2cos ϕ2+a3cos s)
and i s equa ions
˙ϕ1=ω1˙
I1=εa1sin ϕ1
˙ϕ2=ω2˙
I2=εa2sin ϕ2
˙s= 1.
No e ha he inne dynamics is in eg able, wi h i s in eg als
F1(I1, ϕ1) = Ω1I2
1
2+a1(cos ϕ1−1) and F2(I2, ϕ2) = Ω2I2
2
2+a2(cos ϕ2−1)
in in olu ion. The inne dynamics is jus he p oduc in he spaces (I1, ϕ1), (I2, ϕ2) o he
dynamics desc ibed in Fig. 2.2, so he e a e wo esonances cen e ed a I1= 0 and I2= 0.
72
F om (4.6), i is o e i y ha Aiis an e en unc ion, and he e o e,
L∗(−I, θ) = A1cos(θ1+ω1τ∗(−I, θ))+A2cos(θ2+ω2τ∗(−I, θ))+A3cos(−τ∗(−I, θ)).
(4.17)
F om (4.12) and (4.12), τ∗(I, θ) is he solu ion o
ω1A1sin(θ1−ω1τ) + ω2A2sin(θ2−ω2τ) + A3sin(−τ)=0.(4.18)
Analogously, τ∗(−I, θ) is he solu ion o
−ω1A1sin(θ1+ω1τ)−ω2A2sin(θ2+ω2τ) + A3sin(−τ) = 0.
No e ha he abo e equa ion can be w i en as
ω1A1sin(θ1−ω1(−τ))ω2A2sin(θ2−ω2(−τ)) + A3sin(−(−τ)) = 0.(4.19)
F om he local uniqueness o he solu ion o (4.18) and (4.19) we can conclude
τ∗
j(I, θ) = −τ∗
−j(−I, θ), so τ∗
0(I, θ) = −τ∗
0(−I, θ) Applying his equali y in (4.17),
we ob ain
L∗
0(−I, θ) = A1cos(θ1−ω1τ∗
0(I, θ)) + A2cos(θ2−ω2τ∗
0(I, θ)) + A3cos(τ∗
0(I, θ)).
The e o e, L∗
0(I, θ) = L∗
0(−I, θ).
Le (I+, θ+) = S0(I, θ) and (I−, θ−) = S−1
0(−I, θ), whe e S−1
0is he in e se image o
he sca e ing map. We a e going o p o e ha I+=−I−and θ+=θ−. F om (4.9)
we ha e
I+=I+ε∂L∗
0
∂θ (I, θ) + O(ε2) and θ+=θ−ε∂L∗
0
∂I (I, θ) + O(ε2).
On he o he hand, i is easy o e i y ha S−1
0(−I, θ) is
I−=−I+ (−ε)∂L∗
0
∂θ (−I, θ) + O(ε2) and θ−=θ−(−ε)∂L∗
0
∂I (−I, θ) + O(ε2).
Now, we use he ac ha L∗
0(I, θ) = L∗
0(−I, θ), and so
I−=−I+ (−ε)∂L∗
0
∂θ (I, θ) + O(ε2) = −I+
θ−=θ−(−ε)−∂L∗
0
∂I (I, θ)+O(ε2) = θ+.
b) Analogously o he abo e case, τ∗(I, 2π−θ) is he solu ion o
ω1A1sin(2π−θ1−ω1τ) + ω2A2sin(2π−θ2−ω2τ) + A3sin(−τ)=0.
79

O , equi alen ly,
ω1A1sin(θ1−ω1(−τ)) + ω2A2sin(θ2−ω2(−τ)) + A3sin(−(−τ)) = 0.
This implies τ∗(I, θ) = −τ∗(I, 2π−θ).
As he ela ed sca e ing map depends on which in e al he unc ion τ∗(I, θ) belongs,
we can w i e τ∗
j(I, θ) = −τ∗
−j(I, 2π−θ), j∈Z. The e o e, we ha e τ∗
0(I, θ) =
−τ∗
0(I, 2π−θ). Using his equali y and he 2πpe iodici y o he cosine,
L∗
0(I, 2π−θ) = A1cos(−θ1+ω1τ∗
0(I, θ)) + A2cos(−θ2+ω2τ∗
0(I, θ)) + A3cos(τ∗
0(I, θ))
=A1cos(θ1−ω1τ∗
0(I, θ)) + A2cos(θ2−ω2τ∗
0(I, θ)) + A3cos(−τ∗
0(I, θ))
=L∗
0(I, θ).
Le (I+, θ+) = S0(I, 2π−θ) and (I−, θ−) = S−1
0(I, θ), whe e S−1
0is he in e se image
o he sca e ing map. We wan o p o e I−=I+and θ+= 2π−θ−.
F om (4.9) we ha e
I+=I+ε∂L∗
0
∂θ (I, 2π−θ) + O(ε2) = I+ε−∂L∗
0
∂θ (I, θ)+O(ε2)
=I+ (−ε)∂L∗
0
∂θ (I, θ) + O(ε2) = I−.
In he same way,
θ+= (2π−θ)−ε∂L∗
0
∂I (I, 2π−θ) + O(ε2)
= 2π−θ−(−ε)∂L∗
0
∂I (I, θ) + O(ε2)= 2π−θ−.
Theo em 62 (The gene al di usion).Conside he Hamil onian (4.1)+(4.2). Assume
a1a2a36= 0 and |a1/a3|+|a2/a3|<0.625. Then, o e e y δ < 1 he e exis s ε0>0such
ha o e e y 0<|ε|< ε0, gi en I±∈ I∗ {(0,0)}, he e exis s an o bi ˜x( )and T > 0,
such ha
|I(˜x(0)) −I−| ≤ Cδ
|I(˜x(T)) −I+| ≤ Cδ
P oo . Conside i s he case ha I2−=I2+ so ha I−,I+a e joined by a ho izon al line
γ: [0, ∗]→R2such ha γ(0) = I−,γ( ∗) = I+,I1−< I1+ and γ( )6= (0,0) o ∈[0, ∗].
Gi en a posi i e δ, de ine he ini e open co e ing o he image o he cu e γ
N
[
i=0
Bδ(γ( i)),
80
whe e Bδ(γ( i)) = {p∈R2:kγ( i)−pk∞< δ}.
Le Ii∈Bδ(γ( i)) and Ii/∈Bδ(γ( i+1)). This implies Ii
1< γ1( i)< γ1( i+1). We ake
he ec o ui=γ( i+1)−Ii. We wan o ind a ec o isa is ying
i
1ui
1>0 and i
2ui
2>0.
Assume ui
2>0 (ui
1=γ1( i+1)−Ii
1>0 ). In his case we wish j>0, j={1,2}. This
implies Ii
2< γ2( i) = γ2( i+1). Since
i=˙
I(Ii, θ∗) = −A1(I1) sin(θ∗
1−ωi
1τ∗(Ii, θ∗),−A2(I2) sin(θ∗
2−ωi
2τ∗(Ii, θ∗))
and assuming a1, a2>0, j>0 i , and only i , θ∗
j−ωi
jτ∗(Ii, θ∗)∈(π, 2π).
Since he ini ial alues (Ii, θi), we wan o use he inne dynamics o displace θi o a
poin θ∗∈(π, 2π)2. The inne dynamics is e y simple and as Chap e 2 we a e going o
assume ha i is ho izon al, i.e., i is desc ibed by he equa ions
˙
Ij= 0 and ˙ϕj=ωj, j = 1,2.
And he e o e, ϕ( ) = ω +ϕ(0). So, we wish o p o e he exis ence o a ∗such ha
θ( ∗)−ωiτ∗(Ii, θ( ∗)) ∈(π, 2π)2,
whe e θ( ) = θi+ωi .
De ine ψj( ) = θi
j( )−ωi
jτ∗(Ii, θ( )). Wi hou loss o gene ali y we can assume ωi
1≥ωi
2,
we ha e
ψ2=ωi
2
ωi
1
ψ1+¯
ψ,
whe e ¯
ψ=θi
2−ωi
2θi
1/ωi
1. Fo ωi
2/ωi
1∈R Q, (ψ1, ψ2(ψ1)) is dense in T2, hen he e exis s
a ∗such ha (ψ1( ∗), ψ2( ∗)) ∈(π, 2π).
Fo ωi
2/ωi
1=p/q ∈Q,q, p ∈Z, assume wi hou loss o gene ali y q p > 0, his implies
0< p/q ≤1. Now, we look a ψ2(ψ1) = 2πpψ1/q +¯
ψas a o a ion by he angle 2πp/q o
Con he S1. So, we w i e
l(¯
ψ) = 2πp
ql+¯
ψ.
We wan o p o e ha o any ¯
ψ he e exis s a l∈Nsuch ha l(¯
ψ)∈(π, 2π], so ha he
s aigh lines (ψ1, ψ2(ψ1)) in e sec s (π, 2π)2.
Suppose by con adic ion ha l(¯
ψ)∈(0, π], l∈N. No e ha l(¯
ψ) is a q-pe iodic
unc ion. This implies ha he e exis s a l0∈N 0 such ha l0(¯
ψ) = ¯
ψ. The e o e,
i ¯
ψ∈(π, 2π] we ob ain a con adic ion. So, assume ¯
ψ∈(0, π] and conside he o bi
O=0, 1(¯
ψ), . . . , q−1(¯
ψ). Fo q6= 1, i we so he poin s o he o bi we ha e o ob ain
qequidis an poin s in S1. Impossible i l(¯
ψ)∈(0, π], l∈ {0, . . . , q −1}.
Fo q= 1, ωi
1=ωi
2and i is easy o e i y ha (ψ1, ψ2(ψ1)) does no in e sec (π, 2π]2
only o ¯
ψ=π. We i s p o e he case ha i does no happen.
81
We wan o p o e o any k∈ {0, . . . , N},Ikis δ-close o he cu e γ. We ha e
Ii+1 =Ii+ε i+O(ε2).(4.20)
So, Ii+1 is δ-close o γi he ollowing condi ions a e sa is ied
Ii+1
2< γ2( i) + δand Ii+1
1< γ1( i+1) + δ.
F om (4.20) and i we conside only he e ms o he i s o de , hese condi ions a e
equi alen o
ε < γ2( i)−Ii
2+δ
i
2
and ε < γ1( i+1)−Ii
1+δ
i
1
.
No e ha γ2( i)−Ii
2< δ and γ( i+1)−Ii
1> δ. Besides, i
1, i
2≤ k ik∞. The e o e, i is
enough o equi e
ε < γ2( i)−Ii
2+δ
k ik∞
.(4.21)
De ine εi= sup nε:ε < γ2( i)−Ii
2+δ
k ik∞o,we ob ain o any 0 <  ≤εi,Ii+1 is δ-close o γ. Fo
u2≤0, (4.21) akes he o m
ε < Ii
2−γ2( i) + δ
k ik∞
.
Now we wish o ob ain a simila esul o any i e a e o sca e ing map. Obse e ha
k ik∞<4a, o any iand a= max {a1, a2}. The e o e, he esul is hold i we conside
ε < δ
4a.
Tha is, we ake ε0= sup ε: 0 <ε< δ
4a, and hus o any ε<ε0we ob ain a pseudo-o bi
δ-close o γ.
Now we come back o he case whe e ωi
1=ωi
2and ¯
ψ=π. In his case (ψ1, ψ2(ψ1))
in e sec s jus ((0, π)×(π, 2π)) S((π, 2π)×(0, π)). Now, we conside a ini e open co e
o he image o he s aigh line γgi en by
N
[
i=0
Bδ/2(γ( i)),
whe e Bδ/2(γ( i)) = {p∈R2:kγ( i)−pk∞< δ/2}. As ui
1, ui
2>0, we ake i
1>0 and
i
2<0. The image o he sca e ing map in he a iable Iis gi en by
Ii+1
1=Ii
1+ε i
1+O(ε2) and Ii+1
2=Ii
2+ε i
2+O(ε2).(4.22)
The p oblem is when Ii+1
2< γ2( i)−δ. F om (4.22),
Ii+1
2> γ2( i)−δ⇔δ > γ2( i)−Ii
2− i
2εi
2+O(ε2).
82
As Ii∈Bδ(γ( i)), we ha e γ2( i)−Ii
2< δ/2. Besides, k ik∞<4a. The e o e,
Ii+1
2> γ2( i)−δ⇔δ/2>4aε.
O explici ly, Ii+1
2< γ2( i)−δ o any εsa is ying
ε < δ
8a.
We p o e now ha his si ua ion is no in a ian , we mean, i is no possible in ou
pu pose o ob ain ωi+1
1=ωi+1
2and θi+1
2−θi+1
1=π. We ha e
θi+1
2−θi+1
1=θi
2−ε i
2−θi
1+ε i
1+O(ε2)
=π−ε i
2− i
1+O(ε2).
Then, θi+1
2−θi+1
1= 0 i , and only i , −ε( i
2− i
1) + O(ε2)=2πK,K∈Z. Since i
1 i
2<0,
K6= 0. F om he de ini ion o i, we ha e
i
1=−A1(I1) sin(θ1−ω1τ∗(Ii, θi)) and i
2=−A2(I2) sin(θ2−ω2τ∗(Ii, θi))
F om ωi
1=ωi
2and θi
2=θi
1+π, we ob ain A2(I1) = a2A1(I1)/a1and
i
2− i
1=a2+a1
a1A1(I1) sin(θ1−ω1τ∗(I, θ)).
The e o e
ε i
2− i
1<δ
8a4(|a1|+|a2|)< δ.
So, −ε( i
2− i
1) + O(ε2) = 2πK is sa is ied only o a del a sa is ying
δ > 2π+O(ε2).
Bu his δis oo big and i is ou ou in e es .
Fo e ical lines, he same esul can be s a ed mu a is mu andis.
Fo a mo e gene al case, ha is, C1-cu e γ: [0, ∗]→R2such ha γ(0) = I−,
γ( ∗) = I+, we ake a s ai s ep cu e γs ep, a combina ion o ho izon al and e ical lines,
in a such way ha γs ep is a good enough app oxima ion o γ, whe e “good enough” we
mean, he esul is hold o γapplying he abo e esul s ( o ho izon al and e ical lines)
o γs ep.
Using he shadowing lemmas o [FM00, FM03, GLS14] we ob ain he desi ed o bi .
4.4 Highways
In analogy wi h De ini ion 22, in Chap e 2, we de ine a Highway as an in a ian se
H={(I, Θ(I))}o he Hamil onian gi en by he educed Poinca ´e unc ion L∗(I, θ) which
83
is con ained in he le el ene gy L∗(I, θ) = A3. I is he e o e a Lag angian mani old, ha
is, Θ(I) is g adien unc ion, i.e., he e exis s a unc ion F(I) such ha Θ(I) = ∇F(I).
As Θ is a g adien unc ion, i has o sa is y he ollowing condi ion
∂Θ1
∂I2
=∂Θ2
∂I1
.
This condi ion is equi alen o
∂2F
∂I2∂I1
=∂2F
∂I1∂I2
.
P oposi ion 63. Conside he Hamil onian (4.1)+(4.2). Assume a1a2a36= 0 and |a1/a3|+
|a2/a3|<0.625. Fo I1and I2close o in ini y, he unc ion F akes he asymp o ic o m
F(I) = 3π
2(I1+I2)−X
i=1,2
2aisinh(π/2)
π4Ωiπ3ω3
i+ 6π2ω2
i+ 24πωi+ 48e−πωi/2
+O(ω2
1ω2
2eπ(ω1+ω2)/2),
(4.23)
P oo . Assume a candida e o a unc ion F(I) gi en by (4.23), such ha Θ = ∇F(I). Θ(I)
has o sa is y he ene gy le el o highways in he educed Poinca ´e unc ion
A1(I1) cos(Θ1−ω1τ∗(I, Θ)) + A2(I2) cos(Θ2−ω2τ∗(I, Θ)) (4.24)
+A3(cos(−τ∗(I, Θ)) −1) = 0,
and τ∗(I, θ) has o sa is y he equa ion o he c es
ω1A1(I1) sin(Θ1−ω1τ∗(I, Θ)) + ω2A2(I2) sin(Θ2−ω2τ∗(I, Θ))
+A3sin(−τ∗(I, θ)) = 0.(4.25)
We wan o w i e hei e sion o I1and I2close o in ini y. Using (4.23) we no ice
ha Θi= Θi(I) akes he o m
Θi= 3π/2−aisinh(π/2)ω3
ie−πωi/2+O(ω2
1ω2
2eπ(ω1+ω2)/2).
This implies
cos(Θi−ωiτ∗) = −aisinh(π/2)ω3
ie−πωi/2−ωiτ∗
∞+O(ω2
1ω2
2eπ(ω1+ω2)/2)
and
sin(Θi−ωiτ∗) = −1 + O(ω6
ie−πωi),
Besides,
cos(−τ∗(I, Θ)) = 1 −τ∗2
∞
2+O(τ∗4
∞),sin(−τ∗) = −τ∗
∞+O(τ∗
∞),
whe e τ∗
∞is an asymp o ic app oxima ion o τ∗ ha we a e going o es ima e below. Fi s ,
we no ice ha he unc ions A1(I1) and A2(I2) can be app oxima ed by
Ai(Ii) = 4πaiωie−πωi/21 + e−2πωi+. . . = 4πaiωie−πωi/2+Oωie−5πωi/2.
84

F om (4.13), he unc ion τ∗(I, Θ) sa is ies
−τ∗(I, Θ) = −a csin A1(I1)ω1
A3
sin(Θ1−ω1τ∗(I, Θ)) + A2(I2)ω2
A3
sin(Θ2−ω2τ∗),
and he e o e,
τ∗
∞≈X
i=1,2
2aisinh(π/2)ω2
ie−πωi/2+Oωie−5πωi/2.
Applying hese es ima es in Eq. (4.24) we ob ain ha he le hand o Eq. (4.24) sa is ies
X
i=1,24πaiωie−πω1/2−aisinh(π/2)ω3
ie−πωi/2−ωi2a1sinh(π/2)ω2
1e−πω1/2
+2a2sinh(π/2)ω2
2e−πω2/2−A3
2 X
i=1,2
2aisinh(π/2)ω2
ie−πωi/2!2
+O(ω2
1ω2
2e−π(ω1+ω2)/2) = O(ω2
1ω2
2e−π(ω1+ω2)/2).
In he same way, applying in Eq. (4.25) he es ima es ob ained, we ha e ha he le
hand o Eq. (4.25) sa is ies
−4πa1ω2
1e−πω1/2−4πa2ω2
2e−πω2/2+A3 X
i=1,2
2aisinh(π/2)ω2
ie−πωi/2!
+O(ω2
1ω2
2e−π(ω1+ω2)/2) = O(ω2
1ω2
2e−π(ω1+ω2)/2).
The e o e, up o o de O(ω2
1ω2
2e−π(ω1+ω2)/2), he equa ion o he c es and he ene gy
le el o he educed Poinca ´e unc ion a e sa is ied.
We inish his chap e wi h an explici equa ion o he highway in a special case.
P oposi ion 64. (Highways in a e y special case) Conside he Hamil onian (4.1)+(4.2)
and a1=a2=asa is ying 2|a/a3|<0.625 and Ω1= Ω2= Ω.
Le O=(I0, θ0),...,(IN, θN)be an o bi in a highway, N∈Nsuch ha I0
1=I0
2and
θ0
1=θ0
2. Then, Ii
1=Ii
2=¯
Iiand θi
1=θi
2=¯
θi o any i∈ {0, . . . , N}and can be desc ibed
by
¯
θh(¯
I) = 


a ccos A3(1− −(¯
I))
A(¯
I)+ ¯ωa ccos( −(¯
I)),¯
I≤0;
a ccos A3(1− −(¯
I))
A(¯
I)−¯ωa ccos( −(¯
I)), I > 0;
o
¯
θH(I) = 

−a ccos A3(1− −(¯
I))
A(¯
I)−¯ωa ccos( −(¯
I)),¯
I≤0;
−a ccos A3(1− −(¯
I))
A(¯
I)+ ¯ωa ccos( −(¯
I)),¯
I > 0; ,
whe e −(¯
I) = ¯ωA3−pA2
3+ (¯ω−1)¯
I2A2(¯
I)/[A3(¯ω2−1)] and ¯ω=¯
IΩ1.
85
P oo . We ha e ha he ajec o ies o he sca e ing map a e gi en by he ε- ime low
o he Hamil onian −L∗(I, θ) up o o de O(ε2). And such low is gi en by he ollowing
di e en ial equa ions:
˙
Ii=−Ai(Ii) sin(θi−ωiτ∗(I, θ)) (4.26)
˙
θi=−Ωi
dAi
dωi
(Ii) (cos(θi−ωiτ∗) + τ∗(I, θ)Ai(Ii) sin(θi−ωiτ∗(I, θ))) ,
o i= 1,2. Assuming Ω1= Ω2=: Ω and a1=a2=: aand aking ini ial condi ions
sa is ying I(0) = I0and θ(0) = θ0whe e I0
1=I0
2and θ0
1=θ0
2, he solu ion (I( ), θ( )) o
(4.26) sa is ies θ1( ) = θ2( ) and I1( ) = I2( ). Le O=(I0, θ0),(I1, θ1),...,(IN, θN)be
an o bi o he sca e ing map in a ε- ime low o he Hamil onian −L∗(I, θ) up o o de
O(ε2), N∈N. The e o e, Il
1=Il
2and θl
1=θl
2 o any l∈ {0, . . . , N}. We simply deno e
I1=I2and θ1=θ2.
I he o bi Ois a highway, i has o sa is y wo equa ions: he equa ion o he c es s
gi en in (4.12) and
L∗(I, θ) = A3.
Bu now, as I1=I2=: ¯
I,θ1=θ2=: ¯
θ, Ω1= Ω2and a1=a2, hese equa ions can be
ew i en as
A(¯
I) cos(¯
θ−¯ωτ∗(¯
I, ¯
θ)) + A3cos(−τ∗(¯
I, ¯
θ)) = A3
¯ωA(¯ω) sin(¯
θ−¯ωτ∗(¯
I, ¯
θ)) + A3sin(−τ∗(¯
I, ¯
θ)) = 0,
whe e ¯ω:= ω1=ω2and A(¯
I)=4π¯ωa/ sinh(π¯ω/2). F om a simila app oach used in
P oposi ion 23, we ob ain he c es s a e desc ibed by
¯
θh(¯
I) = 


a ccos A3(1− −(¯
I))
A(¯
I)+ ¯ωa ccos( −(¯
I)),¯
I≤0;
a ccos A3(1− −(¯
I))
A(¯
I)−¯ωa ccos( −(¯
I)), I > 0;
and
¯
θH(I) = 

−a ccos A3(1− −(¯
I))
A(¯
I)−¯ωa ccos( −(¯
I)),¯
I≤0;
−a ccos A3(1− −(¯
I))
A(¯
I)+ ¯ωa ccos( −(¯
I)),¯
I > 0; ,
whe e −(¯
I) = ¯ωA3−pA2
3+ (¯ω−1)¯
I2A2(¯
I)/[A3(¯ω2−1)].
86
Chap e 5
Some open ques ions
5.1 Highways in piecewise smoo h global sca e ing
maps
As we showed in Sec ion 3.4, in he piecewise smoo h global sca e ing maps he e exis
wo lines o discon inui y in he ec o ield o he sca e ing map. I seems ha we can
de ine wo special o bi s using he heo y de eloped by [Fil88], such ha hese o bi s lie
on he lines o discon inui y and beha e like he highways de ined in Chap e 1.
In a u u e wo k we plan o pe o m nume ical expe imen s o e i y whe he i is
possible o ind eal o bi s o he Hamil onian beha ing like hese special o bi s in his
egion o he phase space. A e ha we wish o exploi hese o bi s o ob ain as and
simple di usion.
5.2 Abou he case wi h 3 + 1/2 deg ees o eedom
Fo he case s udied in his hesis, i.e., he Hamil onian sys em gi en by (4.1)+(4.2), in
P oposi ion 63 we ob ain an asymp o ic app oxima ion o he highways. A nex s ep is o
check ha his app oxima ion is good enough in o de o con inue globally hose highways
o ob ain a global desc ip ion.
Besides, he e we ha e p esen ed esul s o a es ic ed se o alues o a1and a2, mo e
p ecisely, o a1and a2sa is ying |a1|+|a2| ≤ 0.625. And we ha e ob ained simila esul s
o he pa o he esul s in Chap e 2. The nex s ep is o elimina e his es ic ion o e he
alues o a1and a2and o s udy he bi u ca ion o c es s, he bi u ca ion o he sca e ing
maps and he exis ence o he highways.
Finally, we expec o s udy he case o a complemen a y pe u ba ion wi h espec
o (4.2) o co e he comple e amily case, in an analogous way ha we ha e done in
Chap e 3.
87
5.3 Abou Shadowing lemmas
Taking in o accoun ou nume ical expe imen s, he geome ical mechanisms used in
ou p oo and ou es ima es o he ime, we wish o unde s and he eal ole played by
he inne map in his mechanism.
Ou main ques ion is: Is i eally necessa y o use he inne dynamics in he building
o a pseudo-o bi o gua an ee he exis ence o eal o bi o he sys em?
In ou heo ems we we e able o use he esul s o [FM00, FM03, GLS14]. In [GLS14]
hey p o ed a shadowing lemma o pseudo-o bi s buil by using a numbe o i e a es o
sca e ing map. Bu he esul appea s no o be e y p ac ical o as di usion.
In he ongoing wo k we plan o use nume ical expe imen s o e i y he exis ence o eal
o bi s close o pseudo-o bi s o a sca e ing map (o a combina ion o mul iple sca e ing
maps). Besides, in he u u e we wish o ca y ou an analy ic app oach as well.
5.4 Rela ion be ween he o mulas o he sca e ing
and sepa a ix maps
The sepa a ix map was in oduced by Zasla skii and Filonenko in [ZF68], and has
been s udied and de eloped in [T e98, T e02, Pi 06, PT07, GKZ16, DT16]. Unde ce ain
condi ions we belie e ha he o mulas ob ained in [T e02] can be imp o ed as
I∗=I+ε∂ϕL∗(I∗, ϕ, s)−∂ϕω0
λlog κ ω0
λ+O2
ϕ∗=ϕ+ν−ε∂IL∗(I∗, ϕ, s) + ∂Iω0
λlog κω0
λ+O1
h∗=H0+ε∂sL∗(I∗, ϕ, s)−∂sω0
λlog κω0
λ+O2
s∗=s+¯
∂hω0
λlog κω0
λ+O1,
whe e λ, κ and µa e unc ions o I∗,¯
is an in ege .
s+¯
+∂hω0
λlog κω0
λ< c−1
and O1=O(ε1
4)(ε7
8) log2ε,O2=O(ε1
4)(ε5
4) log2εand L∗is a educed Poinca ´e unc ion.
Since he Sca e ing map akes he explici o m
Sε(I, θ) = I+ε∂
∂θL∗(I, θ) + O(ε2), θ −ε∂
∂I L∗(I, θ) + O(ε2).
We expec o e i y analy ically and nume ically hese equa ions.
88