a Xi :2401.05795 1 [ma h.DS] 11 Jan 2024
Chao ic sca e ing o He a oms o a Cu su ace wi h
co uga ed Mo se po en ial
Flo en ino Bo ondo∗
, E nes Fon ich†
, and Pau Ma ín‡
.
Janua y 12, 2024
Abs ac
We conside a Hamil onian sys em ha models he sca e ing o helium a oms o a coppe
su ace. The in e ac ion be ween he He and he Cu a oms is desc ibed by a co uga ed
Mo se po en ial. Using co uga ion coe icien s alues in he po en ial ob ained by i ing
o expe imen al alues, we p o e ha , p o ided some coe icien o an auxilia y unc ion is
di e en om 0, he e a e egions o he phase space, co esponding o su icien ly la ge ene gy
o he inciden a om, whe e he sca e ing is chao ic. Fu he mo e, we p o e ha he sys em
has oscilla o y mo ions.
Keywo ds: chao ic sca e ing, Hamil onian sys ems, oscilla o y o bi s, exponen ially small
spli ing, inne equa ion.
Con en s
1 In oduc ion 2
1.1 The He-Cu sca e ing p oblem. Main s a emen . . . . . . . . . . . . . . . . . . . . 2
1.2 Exponen ially small spli ing o in a ian mani olds . . . . . . . . . . . . . . . . . . 4
1.3 S uc u e o he pape .................................. 4
2 Hamil onian o mula ion o he He-Cu sca e ing p oblem and spli ing o sepa a ices 5
2.1 McGehee-like coo dina es and as dynamics ...................... 5
2.2 Dynamics o H0...................................... 6
2.3 The Melniko po en ial ................................. 8
2.4 Dynamics o he ull sys em. Spli ing o he in a ian mani olds . . . . . . . . . . 9
3 Chao ic dynamics 9
3.1 Poinca é-Ca an educ ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Local coo dina es a ound q=p= 0 and a pa abolic λ-lemma . . . . . . . . . . . . 10
3.3 The local and he global maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Symbolic dynamics. P oo o Theo em 1.1 . . . . . . . . . . . . . . . . . . . . . . . 12
4 Hamil on-Jacobi equa ion 14
4.1 No a ion .......................................... 14
4.2 De i ing he Hamil on-Jacobi equa ion . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 De ini ions and echnical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4 Solu ion o he Hamil on-Jacobi equa ion (4.3) . . . . . . . . . . . . . . . . . . . . 18
∗ .b[email p o ec ed], Depa amen o de Química, Uni e sidad Au ónoma de Mad id.
† on [email protected], Depa amen de Ma emà iques i In o mà ica, Uni e si a de Ba celona (UB), and Cen e de
Rece ca Ma emà ica (CRM).
‡[email p o ec ed], Depa amen de Ma emà iques, Uni e si a Poli ècnica de Ca alunya (UPC), and Cen e de
Rece ca Ma emà ica (CRM)
1
5 Fi s ex ension o he in a ian mani old 20
5.1 F om Hamil on-Jacobi pa ame iza ion o low pa ame iza ion . . . . . . . . . . . 20
5.1.1 P elimina ies and echnical lemmas o sol e equa ion (5.6) . . . . . . . . . . 21
5.1.2 Solu ion o equa ion (5.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Ex ension o he low pa ame iza ion . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3 F om low pa ame iza ion o Hamil on-Jacobi . . . . . . . . . . . . . . . . . . . . 29
6 The inne equa ion 31
6.1 Spaces and echnical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.2 Fixed poin equa ion ................................... 33
7 App oxima ion o he mani old in he inne domain 35
7.1 Spaces and echnical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.2 The ixed poin equa ion ................................. 39
8 Di e ence be ween solu ions o he inne equa ion 41
9 Di e ence o he solu ions o he Hamil on-Jacobi equa ion 47
9.1 S aigh ening he linea ope a o . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
9.2 The exponen ially small o mula o Φ+−Φ−. . . . . . . . . . . . . . . . . . . . . 52
10 Acknowledgemen s 55
A P oo o Theo em 3.2 55
B P oo s o claims in Appendix A 59
B.1 P oo o Lemma A.1 ................................... 59
B.2 P oo o Lemma A.4 ................................... 60
B.3 P oo o Lemma A.5 ................................... 61
1 In oduc ion
1.1 The He-Cu sca e ing p oblem. Main s a emen
We conside he mo ion o a helium a om bouncing o a coppe su ace. The p oblem a ises
om expe imen al echniques, whe e he sca e ing o noble gas a oms a e collisions wi h a
su ace is used o cha ac e ize su ace s uc u es in a non-des uc i e way [Hul92]. The co uga ion
o he su ace is modeled by he e ec i e po en ial seen by he a oms as hey come nea he
su ace. Co uga ion depends on he inciden ene gy: he highe he ene gy o he He a om, he
close will ge o he su ace. Nume ical e idence o he exis ence o chao ic sca e ing was gi en
in [GBMA97,BGB+99], whe e i was ela ed o some in a ian mani olds in he sys em. The
pu pose o he p esen pape is o p o ide an ac ual igo ous p oo o such chao ic beha io , hus
es ablishing a link be ween expe imen s and heo y on i m g ounds.
We simpli y he p oblem assuming ha he He a om mo ion akes place on a plane; ac ually
he ou -o -plane sca e ing is ela i ely small o he alues o he ene gies and inciden angles
conside ed in he expe imen s. We deno e by (x, z)∈R2 he posi ion o he He a om, whe e
xand za e he ho izon al and e ical displacemen s, espec i ely. Le (px, pz)be he conjuga e
momen a. We model he in e ac ion o he He a om wi h he coppe su ace by a co uga ed Mo se
po en ial, whe e he co uga ion ep esen s he p esence o he coppe a oms in he su ace. Mo e
conc e ely, we conside he Hamil onian
HCM(x, z, px, pz) = 1
2m(p2
x+p2
z) + VM(z) + VC2πx
a, z,(1.1)
whe e VM(z) = De−αz(e−αz −2),
VC(θ, z) = De−2αz V(θ),
V(θ) = X
n≥1
( ncos(nθ) + snsin(nθ)) .
(1.2)
2
The coe icien s nand sna e de e mined expe imen ally. Thei alues a e
1= 0.06, 2= 0.008, n= 0, n ≥3, sn= 0, n ≥1,
D= 6.35 meV, a= 3.6Å, α= 1.05 Å−1. See [GBMA97]. In pa icula , 16= 0. In he p esen
pape , ou only equi emen on Vis ha i is an analy ic e en unc ion, ha is, sn= 0, o all n.
We emphasize ha we do no equi e V o be a igonome ic polynomial. E enness is no an
impo an equi emen and he esul s will hold wi h he same echniques o non-e en V, bu
he added symme y will simpli y some o he de ails, pa icula ly he nume ical compu a ions
pe o med in he p oblem ha a e ca ied ou in [BBF+23]. Since θ= 2πx/a appea s in he
equa ion h ough V, we will ake θ∈T=R/2πZ.
The pu pose o his pape is wo old. On he one hand, we wan o p o e he p esence o chaos
in some pa s o he phase space o (1.1). He e, he no ion o chaos is he one in oduced by Smale
in [Sma65] and i is based on he p esence o a Smale ho seshoe and a conjuga ion wi h he shi
on a space o sequences o symbols. I is wo h o ema k ha he se whe e chaos akes place is a
hype bolic se . We ecall he de ini ion o symbolic dynamics wi h an in ini e numbe o symbols,
as in oduced by Mose in [Mos73]. Le S={s= (...,s−1, s0, s1,...)|si∈N}be he space o
wo sided sequences o in ini e symbols, wi h he opology induced by he neighbo hood basis o
s∗= (...,s∗
−1, s∗
0, s∗
1,...),
Ij(s∗) = {s∈S|sk=s∗
k,|k|< j}, s∗∈S.
I is well known ha he shi σ:S→Sde ined by σ(s)i=si+1 is a homeomo phism. The shi
σis ansi i e and he se o i s pe iodic o bi s is dense in S. I has sensi i e dependence on ini ial
condi ions. I is one o he pa adigms o chaos.
The second goal o he pape is o p o e he exis ence o oscilla o y o bi s, ha is, solu ions
(x( ), z( )) o (1.1) wi h he p ope y ha lim sup z( ) = ∞and lim in z( )<∞, ha is, solu ions
ha go highe and highe bu always go back again o a ini e dis ance o he Cu su ace.
The claims in he p esen pape ollow he scheme p oposed by Mose in [Mos73], which was
also used in he es ic ed plana h ee body p oblem in [LS80]. In his las pape , an impo -
an echnicali y, no p esen in Mose ’s wo k, appea ed: he exponen ially small beha io o he
di e ence be ween he mani olds which gi e ise o he in e es ing dynamics. This p oblem was
la e o e come in [GMS16]. This exponen ially small beha io also appea s in he model unde
conside a ion he e and dealing wi h i will ep esen an impo an pa o ou wo k.
Ou main heo em is he ollowing.
Theo em 1.1. The e exis s a unc ion 1o he coe icien s 1, 2,... such ha i 16= 0, hen,
o any hla ge enough, he e exis s a sec ion Σ⊂ {HCM =h}o he ec o ield associa ed o HCM
and a subse I ⊂ Σsuch ha he Poinca é map Ψ : I → I is a homeomo phism and is conjuga e
o he shi σo in ini e symbols. The se Iis a hype bolic se o Ψ.
Fu he mo e, o any hla ge enough, he Hamil onian HCM in (1.1)possesses oscilla o y o bi s.
Rema k 1.2. The unc ion 1appea s in a p oblem independen o he ene gy h. This equa ion is
usually known in he li e a u e as inne equa ion. The nume ical e idence in [BBF+23] s ongly
sugges s ha o he expe imen al alues men ioned abo e, 1is non-ze o. Fu he mo e, we p o e
ha i one eplaces iby ε˜ iin he de ini ion o V, in (1.2), ha is, i we conside he e m VCas
a small pe u ba ion, hen
1=επ
4˜ 1+O(ε2).
The es o he pape is de o ed o p o e Theo em 1.1. In pa icula , i will be an immedia e
consequence o Theo em 3.5.
We would like o ema k ha he hype bolic se , as is he case o he Si niko p oblem conside ed
in [Mos73] o he es ic ed plana h ee body p oblem in [LS80,GMS16], is ela ed o he in a ian
mani olds o a ce ain pe iodic o bi a z=∞. In ou case, since he equa ions o mo ion a e
˙x=1
mpx,
˙z=1
mpz,
˙px=−2π
aDe−2αzV′2πx
a,
˙pz=−2Dα e−αz 1−e−αz 1 + V2πx
a,
(1.3)
3
we ha e ha he se
{(x, z, px, pz)∈R/aZ×R×R2|z=∞, pz= 0, px= (2mh)1/2}(1.4)
is a pe iodic o bi o (1.3) a in ini y in he ene gy le el h. The p oo o Theo em 1.1 consis s
in checking ha his pe iodic o bi possesses in a ian s able and uns able mani olds, ha hese
in a ian mani olds in e sec ans e sally i his la ge enough and hen p o e a sui able λ-lemma
ha ensu es ha his ans e sal in e sec ion gi es ise o he s anda d isola ing blocks wi h cone
condi ions. This las pa is due o he ac ha he pe iodic o bi (1.4) is nei he hype bolic no
ellip ic, bu degene a e. Howe e , al hough degene a e, i possesses s able and uns able in a ian
mani olds. The s udy o he in a ian mani olds o hese ype o degene a e objec s goes back
o [McG73]. See also [BFdlLM07], whe e he pa ame iza ion me hod is used, and he subsequen
wo ks, [BFM20a,BFM20b,BFM17,BFM20c]. In his case, when he pe iodic o bi is degene a e,
he s anda d λ-lemma does no hold. In pa icula , i is no ue ha he o wa d images o
a mani old in e sec ing ans e sally he s able mani old o he o bi accumula e o he whole
uns able mani old. Hence, a di e en a gumen is needed o con ol he passage close o he
pe iodic o bi . This was al eady known by Mose in [Mos73]. Howe e , his p oo does no di ec ly
apply o he p esen case, because he deg ee o degene acy o ou case is di e en om he one o
he Si niko p oblem. We p esen ano he p oo , based in he ideas in [GMPS22].
1.2 Exponen ially small spli ing o in a ian mani olds
One o he main di icul ies o he p esen wo k is o es ablish ha he angle o in e sec ion be ween
he s able and uns able in a ian mani olds o ce ain pe iodic o bi s is non-ze o. Indeed, since ou
goal is o deal wi h he physical p oblem, in which he co uga ion, modelled by he unc ion V
in (1.2), is ixed, he only pa ame e we will ha e o deal wi h he p oblem will be he ene gy o
he sys em. We will see ha he angle o in e sec ion is in ac exponen ially small in he ene gy
hence p ecluding he use o he s anda d Melniko heo y.
S a ing wi h he seminal pape o Lazu kin [Laz84] (see he English ansla ion in [Laz03]),
he e is a long lis o wo ks in he li e a u e conce ning he exponen ially small spli ing o sepa a-
ices. In pa icula , he me hod in oduced in [Sau01] has been essen ial in pos e io de elopmen s
o he ield. See [BFGS12] and he e e ences he ein. Ou app oach he e is simila o he he one
es ablished in [BFGS12] and [Bal06]. I is impo an o ema k ha he esul s in hese las wo
pape s do no apply o ou se ing. As a ma e o ac , i one ies o w i e Hamil onian (1.1),
which has 2 deg ees o eedom, as a 11
2deg ees o eedom one using he Poinca é-Ca an e-
duc ion, he educed Hamil onian does no sa is y some o he hypo heses in [BFGS12,Bal06].
Pa icula ly, when w i en as an in eg able Hamil onian plus a pe u ba ion, he pe u ba ion is
no polynomial and has b anching singula i ies when ex ended o he complex domain. This is
ue e en in he McGehee a iables ha a e in oduced in Sec ion 2.
Since we a e no assuming he pe u ba ion o be small, we need o s udy he inne equa ion
associa ed o he p oblem in o de o cap u e he leading e m o he exponen ially small beha io
o he spli ing. This inne equa ion was used by Gel eich in [Gel99] o p o e he spli ing o
sepa ices in he Chi iko s anda d map. See also [MSS11a,MSS11b], whe e he McMillan map
was s udied by using esu gence heo y. The s udy o he inne equa ion o he Hénon map was
pe o med in [GS01].
The exponen ially small spli ing o in a ian mani olds also appea s in o he physical p oblems,
since i is ela ed o he exis ence o some as equencies on he sys em. In pa icula , i has
been deal wi h success ully in some p oblem o celes ial mechanics [GMS16,GMPS22,GPS23]. In
hese las cases, howe e , he Melniko unc ion p edic s co ec ly he i s o de o he spli ing.
1.3 S uc u e o he pape
The s uc u e o he pape is as ollows.
In Sec ion 2we w i e he sys em in sui able McGehee coo dina es, desc ibe he geome ic
beha iou o i and claim he main heo em o he spli ing o sepa a ices.
In Sec ion 3we apply he spli ing heo em o deduce he exis ence o chao ic dynamics o he
sys em. To do so, we need o cope wi h he p oblem ha he pe iodic o bi s unde conside a ion
a e no hype bolic bu degene a e, which implies ha he s anda d lambda lemma does no apply.
4
He e we use a e sion o he pa abolic lambda lemma in [GMPS22] o ob ain he conjuga ion o
he Be noulli shi .
Sec ions 4 o 9deal wi h he ac ual spli ing o he in a ian mani olds o he pe iodic o bi a
in ini y. In Sec ions 4and 5we ob ain sui able app oxima ions o he in a ian mani olds by sol ing
an app op ia e Hamil on-Jacobi equa ion. These app oxima ions a e alid in a ce ain complex
domain. Howe e , we a e no able o ob ain enough in o ma ion o hei di e ence. To do so, we
in oduce he inne equa ion in Sec ion 6, om which we ob ain wo solu ions. In Sec ion 7, we
check ha hese solu ions a e also good app oxima ions o he in a ian mani olds in some egion
o hei complex domain o de ini ion. In Sec ion 8we ob ain an exponen ially small o mula o
he di e ence o solu ions o he inne equa ion which, in u n, we use in Sec ion 9 o ob ain he
exponen ially small o mula o he di e ence o he in a ian mani olds.
We ha e le o Appendix A he p oo o he pa abolic lambda lemma (Theo em 3.2). Ap-
pendix B.1 con ains he p oo s o some echnical lemmas used in he p oo o Theo em 3.2.
2 Hamil onian o mula ion o he He-Cu sca e ing p oblem
and spli ing o sepa a ices
2.1 McGehee-like coo dina es and as dynamics
We a e in e es ed in mo ions o which he He pa icle a i es o z=∞wi h ze o momen um. Fo
his eason, we in oduce he McGehee-like coo dina es



Aq2=e−αz,a
2πθ=x,
Bp =pz, CI =px,
whe e
A= 2, B =aα
4πC, C2= 2mD 8π
aα2
.
Wi hou loss o gene ali y, we ake C > 0. This change ans o ms he s anda d 2- o m dx ∧dpx+
dz ∧dpzin o he b-symplec ic o m (aC/2π)ω, whe e
ω=dθ ∧dI −1
qdq ∧dp (2.1)
and he Hamil onian unc ion HCM in (1.1) becomes 8D·H, whe e
H(q, p, θ, I) = 1
2(νI2+p2)−1
2q2+1
2q4+1
2q4V(θ)(2.2)
and
ν=4π
aα2
.
Fo he expe imen al alues o he He-Cu p oblem, he alue o νis
ν= 11.051879175935...
Rescaling ime, he dynamics o HCM is equi alen o he one gene a ed by Hwi h espec o he
o m ω.
To s udy he beha io o he sys em o la ge alues o I0, we in oduce he new a iable J
by I=I0+Jin (2.2) wi h I0≫1. This change p ese es he 2- o m ω. We will deno e he
Hamil onian in hese new a iables wi h he same le e , namely,
H(q, p, θ, J) = H0(q, p, J) + H1(q, p, θ, J),(2.3)
whe e
H0(q, p, J) = 1
2(ν(I0+J)2+p2)−1
2q2+1
2q4,
H1(q, p, θ, J) = 1
2q4V(θ).
(2.4)
5

Since H0does no depend on θ, i is in eg able. We ema k ha Hlooks like a pe iodically
pe u bed Du ing equa ion. Howe e , since he ec o ield gene a ed by His ob ained h ough
he non-s anda d 2- o m ω, he sys em is no equi alen o he Du ing equa ion. Indeed, he
equa ions o mo ion gene a ed by Ha e
˙q=−q∂H
∂p =−qp, ˙
θ=∂H
∂I =νI0+νJ,
˙p=−q−∂H
∂q =−q2+ 2q4+ 2q4V(θ),˙
J=−∂H
∂θ =−q4
2V′(θ),
(2.5)
which is no a Du ing oscilla o due o he p esence o he ac o qin he (q, p)componen s o he
ec o ield.
We conside , a he ene gy le el νI2
0/2o Hin (2.3), he se
ΛνI0={q=p=J= 0, θ ∈T}.
F om (2.5), i is in a ian and he dynamics on ΛνI0is gi en by θ=θ0+νI0 , ha is, i is pe iodic
wi h equency νI0/(2π), which is as when νI0is la ge. Tha is, a he ene gy le el νI2
0/2, o
la ge alues o νI0,Hcan be seen as a as pe iodic pe u ba ion o he in eg able Hamil onian
H0. We emphasize ha he pe u ba ion H1is no small since we a e assuming ha 1is ixed
and we will conside mo ions in which qwill each size o O(1). The p oo o Theo em 1.1 consis s
in p o ing ha each o hese pe iodic o bi s possesses in a ian mani olds and ha hey in e sec
ans e sally. F om his in e sec ion we will deduce he exis ence o a ho seshoe wi h in ini ely
many symbols. Howe e , since he sys em is a as pe iodic pe u ba ion o an in eg able sys em,
i is well known ha he angle o in e sec ion o he mani olds will be smalle han any powe o he
in e se o he equency which makes he ques ion o compu ing i a beyond all o de s phenomenon.
Rema k 2.1. Al hough H1is no small, i is well known ha he ac ha depends on a as
angle implies ha i s con ibu ion a e ages ou up o an exponen ially small emainde . Indeed,
we can apply a s ep o a e aging and s ill ob ain an explici exp ession o he emainde , which is
o size O((νI0)−1).
I is immedia e o check ha he change
θ= Θ, J =K+A′(Θ)Q4, q =Q, p =P+ 4A(Θ)Q4,
p ese es he 2- o m ωin (2.1). In hese new a iables, Hbecomes
e
H(Q, P, Θ, K) =1
2(ν(I0+K−A′(Θ)Q4)2+ (P+ 4A(Θ)Q4)2)−1
2Q2+1
2Q4+1
2Q4V(Θ)
=1
2(ν(I0+K)2+P2)−1
2Q2+1
2Q4+1
2V(Θ) −νI0A′(Θ)Q4
−νA′(Θ)KQ4+ 4A(Θ)PQ4+ 8A(Θ)2Q8+1
2νA′(Θ)2Q8.
Taking Asuch ha A′(Θ) = −V(θ)/(2νI0)and in oducing W(Θ) = RΘV(ϑ)dϑ, we ob ain
e
H(Q, P, Θ, K) = H0(Q, P, K) + e
H1(Q, P, Θ, K),
whe e
e
H1(Q, P, Θ, K) = 1
2νI0−νV (Θ)KQ4+ 4W(Θ)PQ4+1
4I0
V(Θ)2Q8+4
νI0
W(Θ)2Q8.
The compu a ions o he Melniko po en ial in Sec ion 2.3 below lead o a o mula o he same o de .
We will no ollow his app oach. On he con a y, we will wo k di ec ly wi h he Hamil onian
H=H0+H1in (2.3).
2.2 Dynamics o H0
In iew o Rema k 2.1, al hough H1is no small, we can see H0as a e e ence sys em o H. He e
we desc ibe i s dynamics. Fi s we obse e ha , since H0does no depend on θ,Jis a conse ed
quan i y. Hence, H0is in eg able. Using he 2- o m ωin (2.1), he equa ions o mo ion o H0a e
˙q=−qp, ˙
θ=νI0+νJ,
˙p=−q2+ 2q4,˙
J= 0.(2.6)
6
The se
ΛνI0={q=p=J= 0, θ ∈T}
is a pe iodic o bi in {H0=νI2
0/2}. F om (2.6), as we ha e al eady men ioned, i is clea ha
he es ic ion o ΛνI0 o {H0=νI2
0/2, J = 0}is no hype bolic. Howe e , i possesses in a ian
s able and uns able mani olds, ha gene a e wo homoclinic loops, gi en by
W0
νI0=(q, p, θ, J)|J= 0, θ ∈T, H0(q, p, 0) = 1
2νI2
0.
See Figu e 1. We will ocus on he loop wi h q≥0. The igu e is sligh ly misleading: o any
J, he se {q= 0, p =p0}is a pe iodic o bi . In pa icula , {q= 0}is in a ian , e en o
he ull Hamil onian H. This means ha , in ac , in he (q, p)-plane, (q, p) = (0,0) is no a
opological saddle because, in pa icula , is no an isola ed equilib ium poin because q= 0 is a
line o equilib ia. As a consequence, he s anda d λ-lemma is no longe ue and ano he s a emen
will be necessa y. The needed esul is Theo em 3.2 below. We will be in e es ed in he pa o
he phase space whe e q > 0.
Figu e 1: The p ojec ion o he in a ian mani olds W0
νI0on o he (q, p)-plane. Fo each p=p0,
{(0, p0, θ, 0) |θ∈T}is a pe iodic o bi .
The ollowing lemma, whose p oo is a s aigh o wa d compu a ion, p o ides he ime pa ame-
iza ion o he igh hand side homoclinic o bi .
Lemma 2.2. The ime pa ame iza ion o he homoclinic o bi o (0,0) o (H0)|J=0 ha passes
h ough (1,0) a = 0 is gi en by
qh( ) = 1
√1 + 2, ph( ) = −˙qh( )
qh( )=
1 + 2.
As a consequence,
Γ0(u, θ) = 



qh(u)
ph(u)
θ
0



(2.7)
pa ame izes W0
νI0. Deno ing φ0
he low gene a ed by H0, his pa ame iza ion sa is ies φ0
◦
Γ0(u, θ) = Γ0(u+ , θ +νI0 ).
7
2.3 The Melniko po en ial
As usual, we de ine he Melniko po en ial associa ed o he Hamil onian Has
L(u, θ) = −Z∞
−∞
H1(qh(u+ ), ph(u+ ), θ +νI0 , 0) d . (2.8)
Expanding Vin Fou ie se ies,
V(θ) = X
k∈Z
V[k]eikθ,
by (2.4), we can w i e
H1(q, p, θ, 0) = q4
2V(θ) = q4
2X
k∈Z
V[k]eikθ.
Since Vis eal analy ic, e en and has ze o a e age, we ha e ha
V[k]∈R, V [0] = 0, V [k]=V[−k], k ∈Z {0}
and he e exis s σ0>0and K > 0such ha , o all k∈Z,
|V[k]| ≤ Ke−|k|σ0.(2.9)
Hence,
L(u, θ) = −Z∞
−∞
H1(qh( ), ph( ), θ −νI0u+νI0 , 0) d =−X
k∈Z
eik(θ−νI0u)Z∞
−∞
q4
h( )
2V[k]eikνI0 d .
Tha is,
L(u, θ) = X
k∈Z
L[k](νI0)eik(θ−νI0u),
whe e
L[k](νI0) = −1
2V[k]Z∞
−∞
q4
h( )eikνI0 d .
P oposi ion 2.3. The Melniko po en ial Lsa is ies
L[k](νI0) = −πνI0V[k]
4e−|k|νI0|k|+1
νI0, k ∈Z.
In pa icula , since V[0] = 0 hen L[0](νI0) = 0.
P oo . Since, in iew o Lemma 2.2,q4
his a me omo phic unc ion ha has poles o o de 2a ±i,
he claim ollows om a s aigh o wa d esidue compu a ion.
Co olla y 2.4. I 16= 0 in (1.2), hen
L(u, θ) = −π
4νI0 1e−νI01 + 1
νI0cos(θ−νI0u) + O(e−νI0).
I he coe icien s iin he de ini ion o Vin (1.2) sa is y i=ε˜ iand ˜ 16= 0, he s anda d
Melniko heo y implies ha he dis ance be ween he uns able and s able in a ian mani olds
o ΛνI0is gi en by he de i a i es o εL plus an e o o size O(ε2). Howe e , since Lhas size
O(e−νI0), his means ha a he ene gy le el νI2
0/2,εL gi es he leading o de o he dis ance
only i ε < O(e−νI0), ha is, o a ixed alue o he ene gy, he in a ian mani olds o in ini y
spli only i he co uga ion is exponen ially small in he ene gy. We will see ha εand νI0can be
aken as independen pa ame e s while he exponen ially small o mula o he dis ance will emain
alid. The physical implica ions a e impo an : we will see ha , o a gi en co uga ion, i ce ain
coe icien is di e en om 0( ha depends on he coe icien s o he co uga ion and is gene ically
non-ze o), o any la ge enough ene gy, he in a ian mani olds o in ini y spli . Mo eo e , his
coe icien depends on i s o de on L[1] i εis small.
I is impo an o ema k ha i 1= 0, he Melniko unc ion is o o de e−2νI0. Howe e , in
his case, he ue beha io , gi en by o mula (2.11) in Theo em 2.5, below, can be o o de e−νI0,
i ce ain coe icien 1is di e en om 0. See [BMS23].
8
2.4 Dynamics o he ull sys em. Spli ing o he in a ian mani olds
The pe iodic o bi ΛνI0={q=p=J= 0, θ ∈T}o H0 emains when we conside he ull sys em
Hin (2.3), wi h he same equency νI0. Al hough i is no hype bolic, we will p o e ha i has
in a ian uns able and s able mani olds, which, i ce ain non-degene acy condi ion holds, no longe
coincide. Nex heo em summa izes he claim. I is he i s s ep in he p oo o Theo em 1.1.
Theo em 2.5. I νI0is la ge enough, he pe iodic o bi ΛνI0={q=p=J= 0, θ ∈T}possesses
in a ian uns able and s able mani olds, W±
νI0. Mo eo e , o any 0< u0< u1, he e exis analy ic
unc ions Φ±: [u0, u1]×T→Rsuch ha
Γ±(u, θ) = 



qh(u)
ph(u)−1∂uΦ±(u, θ)
θ
∂θΦ±(u, θ)



(2.10)
a e pa ame iza ions o pieces o W±
νI0sa is ying
(1) Γ±= Γ0+O((νI0)−1), whe e Γ0is gi en in (2.7),
(2) he e exis s 1,Λ0∈Rsuch ha , o any j, k ≥0,0≤j+k≤3,
∂j
u∂k
θ(Φ+(u, θ)−Φ−(u, θ)−Λ0)
= (−1)j(νI0)j+1e−νI02 1∂j+k
αcos α|α=(θ−νI0u)+O1
log(νI0).(2.11)
The coe icien s 1and Λ0do no depend on νI0. They only depend on he coe icien s io
he unc ion Vin H1. I one assumes ha i=ε˜ i, i.e. he co uga ion e m H1is a small
pe u ba ion o H0, hen
1=π
2˜ 1ε+O(ε2).(2.12)
Rema k 2.6. Fo he expe imen al alues o Vco esponding o he in e ac ion be ween helium
and coppe a oms, nume ical expe imen s in [BBF+23] sugges ha 16= 0. Howe e , i i=ε˜ i,
we will see ha 1is an analy ic unc ion o ε. Then, (2.12)implies ha , i ˜ 16= 0, 1can only
anish o a disc e e numbe o alues o ε.
Co olla y 2.7. I νI0is la ge enough, he in a ian mani olds W±
νI0o ΛνI0in e sec ans e sely
along wo p ima y homoclinic o bi s.
P oo . Since W±
νI0a e con ained in he le el νI2
0/2o H, he exp ession o Γ±in (2.10) implies
ha hei in e sec ions a e gi en by ∂uΦ+(u, θ)−∂uΦ−(u, θ) = 0. Then, (2.11) allows us o apply
he s anda d implici unc ion heo em o ob ain ha he p ima y homoclinic o bi s a e gi en by
θ−νI0u+O((νI0)−1) = kπ.
Theo em 2.5 p o ides an exponen ially small o mula o he di e ence o he in a ian mani olds
W±
νI0, wi h he ene gy o he sys em as pa ame e . We will ind sui able app oxima ions o he
in a ian mani olds o ΛνI0which ex end o complex alues o hei a iables. The leading e m
o hese app oxima ions, o bo h he s able and uns able mani olds, will be he unpe u bed
sepa a ix Γ0. This is done in Sec ions 4and 5. In many cases, his is enough o ob ain he
exponen ially small o mula, whose leading e m is gi en by he Melniko po en ial (2.8). He e,
howe e , in o de o cap u e he leading e m o he exponen ially small di e ence be ween he
mani olds, we need o eso o wha is o en known as he inne equa ion. This is done in Sec ion 6
and 8. In Sec ion 7 he mani olds a e compa ed wi h he solu ions o he inne equa ion and, inally,
all he in o ma ion is ga he ed in Sec ion 9, comple ing he p oo o Theo em 2.5.
3 Chao ic dynamics
He e we de ine a e u n map on a sui able sec ion such ha i possesses an in a ian hype bolic se
and when es ic ed o his se , i is opologically conjuga e o he shi o in ini e symbols. The
cons uc ion is analogous o he one o Mose in [Mos73].
9
equa ion (4.6) becomes
L(Φ1) = F(Φ1).(4.9)
We ema k ha
F(0)(u, θ) = −H1(qh(u), θ) = −1
2
1
(1 + u2)2V(θ).(4.10)
To sol e (4.9), we will ew i e i as a ixed poin equa ion by means o a sui able igh in e se Go
L, o be de ined la e on, ha is,
Φ1=G ◦F(Φ1).(4.11)
We will de o e he es o he sec ion o p o e he exis ence o a solu ion o (4.6) wi h he
bounda y condi ions as u→ −∞ in (4.4).
4.3 De ini ions and echnical lemmas
Le 0< β1< β2< π/2be ixed. Fo κ > 1and δ∈(0,1/2), we conside he complex domain
D+
κ,δ ={u∈C| |ℑu|<− an β1ℜu+ 1 −κ(νI0)−1,|ℑu|> an β2ℜu+ 1 −δ.(4.12)
We will be in e es ed in he case whe e I0and κa e big bu sa is y κ(νI0)−1< δ. Obse e ha ,
i κ(νI0)−1< δ, hen d(D+
κ,δ,±i) = Cκ(νI0)−1wi h C= cos β1. See Figu e 2.
Figu e 2: The domain D+
κ,δ de ined in (4.12).
To sol e equa ion (4.9), o , s ∈Rwe in oduce he Banach space o eal analy ic, 2π-pe iodic
in θ, unc ions
X ,s ={Ψ : D+
κ,δ ×Tσ→C|Ψ eal analy ic,kΨk ,s <∞}
and, aking in o accoun he Fou ie expansion Ψ(u, θ) = Pk∈ZΨ[k](u)eikθ, he no m
kΨk ,s =X
k∈ZkΨ[k]k ,se|k|σ,
whe e, o an analy ic unc ion :D+
κ,δ →C,
k k ,s = max 

sup
u∈D+
κ,δ,ℜu≤−u0|u (u)|,sup
u∈D+
κ,δ,ℜu>−u0|(1 + u2)s (u)|

,
whe e u0∈Ris chosen such ha u0≥max{1,(1 −δ)/ an β2}.
16

Lemma 4.2. Le , 1, 2, s, s1, s2∈R. We ha e
(1) I Ψ∈ X + 1,s+ 2wi h 1, 2≥0 hen Ψ∈ X ,s and
kΨk ,s ≤KνI0
κ 2
kΨk + 1,s+ 2,
whe e K > 0is independen o νI0and κ.
(2) I Ψ1∈ X 1,s1and Ψ2∈ X 2,s2 hen Ψ1Ψ2∈ X 1+ 2,s1+s2and
kΨ1Ψ2k 1+ 2,s1+s2≤ kΨ1k 1,s1kΨ2k 2,s2.
The p oo o he p e ious lemma is s aigh o wa d om he ideas in [Sau01].
We also in oduce
e
X ,s ={Ψ∈ X ,s |∂uΨ∈ X +1,s+1, ∂θΨ∈ X +1,s+1,TΨU ,s <∞},
whe e
TΨU ,s =kΨk ,s +k∂uΨk +1,s+1 +νI0k∂θΨk +1,s+1.
Gi en Ψ∈ X ,s, wi h > 0,s≥0, we o mally de ine
Gu(Ψ)(u, θ) = Z0
−∞
Ψ(u+ξ, θ +νI0ξ)dξ =Zu
−∞
Ψ(ξ, θ +νI0(ξ−u)) dξ. (4.13)
Clea ly, when Gu(Ψ) is well de ined, i Ψis eal analy ic, so is Gu(Ψ). The ope a o Gu o mally
sa is ies L◦Gu(Ψ) = Ψ, whe e Lis he di e en ial ope a o in oduced in (4.7).
The nex echnical lemma will be he main ool o ind he desi ed solu ions o equa ion (4.11).
I s p oo is an immedia e a ia ion o he a gumen s in [GOS10].
Lemma 4.3. Le Gube he ope a o de ined in (4.13).
(1) I Ψ∈ X ,s, wi h > 1,s≥0, hen Gu(Ψ) ∈ X −1,s and
kGu(Ψ)k −1,s ≤KkΨk ,s.
I , s > 0and hΨi= 0, hen Gu(Ψ) ∈ X ,s and
kGu(Ψ)k ,s ≤K(νI0)−1kΨk ,s.
(2) I Ψ∈ X ,s, wi h , s > 1,Gu(Ψ) ∈ X −1,s−1and
kGu(Ψ)k −1,s−1≤KkΨk ,s.
(3) I Ψ∈ X ,s, wi h ≥1,s > 0,∂uGu(Ψ), ∂θGu(Ψ) ∈ X ,s and
k∂uGu(Ψ)k ,s ≤KkΨk ,s,k∂θGu(Ψ)k ,s ≤K(νI0)−1kΨk ,s.
As a consequence, i Ψ∈ X ,s, wi h , s > 1,Gu(Ψ) ∈e
X −1,s−1and
TGu(Ψ)U −1,s−1≤KkΨk ,s.
The cons an Konly depends on , s and he cons an s in ol ed in he de ini ion o D+
κ,δ bu i is
independen o νI0and κ.
17
4.4 Solu ion o he Hamil on-Jacobi equa ion (4.3)
Le Fand Gube he ope a o s de ined by (4.8) and (4.13), espec i ely. We o mally in oduce
L+
ou (u, θ) = Gu◦F(0)(u, θ) = −Zu
−∞
H1(qh(s), θ +νI0(s−u)) ds
=−Zu
−∞
1
2(1 + s2)2V(θ−νI0(s−u)) ds.
(4.14)
Fo la e use we also in oduce
L−
ou (u, θ) = −L+
ou (−u, −θ) = Z∞
u
1
2(1 + s2)2V(θ+νI0(s−u)) ds
and
Lou (u, θ) = L+
ou (u, θ)−L−
ou (u, θ) = −Z∞
−∞
1
2(1 + s2)2V(θ+νI0(s−u)) ds.
Obse e ha Lou =L, whe e Lis he Melniko po en ial in oduced in (2.8).
P oposi ion 4.4. The unc ion L+
ou ∈ X4,2in D+
κ,δ and sa is ies
kL+
ou k4,2,k∂θL+
ou k4,2,k∂uL+
ou k5,3≤K
νI0
,
o some K > 0, independen o νI0and κ. Fu he mo e, hL+
ou i= 0.
P oo . We ecall ha
F(0)(u, θ) = −H1(qh(u), θ) = −1
2q4
h(u)V(θ) = −1
2
1
(1 + u2)2V(θ),
whe e H1was in oduced in (2.4) and he unc ion Vin (1.2). Clea ly, F(0) ∈ X4,2and hF(0)i= 0.
The bounds on kL+
ou k4,2and on k∂θL+
ou k4,2 ollow om (1) and (3) o Lemma 4.3, espec i ely.
To ob ain he bound on k∂uL+
ou k5,3we obse e ha , by in eg a ion by pa s, i k6= 0,
∂uZu
−∞
1
2
1
(1 + s2)2eikνI0(s−u)eikθ ds =−2Zu
−∞
s
(1 + s2)3eikνI0(s−u)eikθ ds.
Hence, de ining
g(u, θ) = −2u
(1 + u2)3V(θ),
we ha e ha ∂uL+
ou =Gu(g). Since hgi= 0 and g∈ X5,3, using again (1) o Lemma 4.3,
k∂uL+
ou k5,3=kGu(g)k5,3≤K(νI0)−1kgk5,3.
The las claim ollows om he ac ha hF(0)i= 0.
We use L+
ou o ew i e equa ion (4.11) as a new ixed poin equa ion, wi h be e con ol o i s
solu ion. To do so, we in oduce Φ2by Φ1=L+
ou + Φ2. Then, Φ1is a solu ion o (4.11) i and
only i Φ2sa is ies
Φ2=Gu◦e
F(Φ2),
whe e e
F(Φ2) = F(L+
ou + Φ2)−F(0).(4.15)
P oposi ion 4.5. The e exis s K∗>0such ha , i κis big enough, he ope a o Gu◦e
F:
BK∗(νI0)−2⊂e
X5,3→ BK∗(νI0)−2is well de ined and a con ac ion. Le Φ+
2be i s ixed poin .
As a consequence, since Φ+
2is he ixed poin o Gu◦e
F, hen Φ+
1=L+
ou + Φ+
2is a solu ion o
equa ion (4.9).
18
P oo . We i s claim ha e
F(0) ∈ X6,4and ke
F(0)k6,4≤K1(νI0)−2, o some K1independen o
νI0. Indeed, by he de ini ions o Fin (4.8) and e
Fin (4.15), using Lemma 4.2, P oposi ion 4.4,
he ac ha p−2
h∈ X−2,−2and
k∂θL+
ou k2,2≤ k∂θL+
ou k4,2≤K
νI0
,
we ha e ha
ke
F(0)k6,4=
1
2p2
h
(∂uL+
ou )2+ν
2(∂θL+
ou )26,4
≤
1
2p2
h
∂uL+
ou 1,1k∂uL+
ou k5,3+ν
2k∂θL+
ou k2,2k∂θL+
ou k4,2
≤K
2kp−2
hk−2,−2k∂uL+
ou k3,3
1
νI0
+νK
2k∂θL+
ou k2,2
1
νI0
≤K2
2kp−2
hk−2,−2
1
(νI0)2+νK2
2(νI0)2
≤K1
(νI0)2.
Hence, by he las claim o Lemma 4.3,TGu◦e
F(0)U5,3≤Kke
F(0)k6,4≤KK1
(νI0)2<∞.
We ake
K∗= 2KK1.
Le Ψ∈ BK∗(νI0)−2⊂e
X5,3. In iew o (4.8) and (4.15), we w i e
e
F(Ψ) = e
F(0) + e
F(Ψ) −e
F(0) = F0−F1−F2,
whe e
F0=e
F(0),
F1=1
2p2
h(∂uL+
ou +∂uΨ)2−(∂uL+
ou )2,
F2=ν
2(∂θL+
ou +∂θΨ)2−(∂θL+
ou )2.
We s a wi h F1. Using ha kp−2
hk−2,−2<∞,k∂uL+
ou k5,3≤K(νI0)−1,Ψ∈ BK∗(νI0)−2⊂
e
X5,3and Lemma 4.2, we ha e ha
kp−2
h(2∂uL+
ou +∂uΨ)k0,0≤ kp−2
hk−2,−2k2∂uL+
ou +∂uΨk2,2
≤K2νI0
κk∂uL+
ou k5,3+(νI0)2
κ2k∂uΨk6,4≤K1
κ+K∗
κ2(4.16)
o some cons an s Kindependen o νI0and κ.
Then, using again Lemma 4.2,
kF1k6,4=1
2kp−2
h(2∂uL+
ou +∂uΨ)∂uΨk6,4
≤1
2kp−2
h(2∂uL+
ou +∂uΨ)k0,0k∂uΨk6,4
≤K
21
κ+K∗
κ2K∗
(νI0)2.
Now we deal wi h F2. We obse e ha , since kL+
ou k4,2=K(νI0)−1and Ψ∈ BK∗(νI0)−2⊂e
X5,3,
by Lemma 4.2,
k2∂θL+
ou +∂θΨk0,0≤2K(νI0)2
κ2k∂θL+
ou k4,2+K(νI0)4
κ4k∂θΨk6,4≤KνI0
κ2+K∗νI0
κ4.(4.17)
Then,
kF2k6,4≤ν
2k2∂θL+
ou +∂θΨk0,0k∂θΨk6,4≤ν
2K1 + K∗
κ21
κ2(νI0)2.
19
Then, e
F(Ψ) ∈ X6,4and, by Lemma (4.3), aking κla ge enough, Gu◦e
F(Ψ) ∈ BK∗(νI0)−2⊂e
X5,3.
Now we check ha e
Fis Lipschi z wi h Lip e
F ≤ Kκ−1. Le Ψ,Ψ′∈ BK∗(νI0)−2⊂e
X5,3. Obse e
ha kp−2
h(2∂uL+
ou +∂uΨ + ∂uΨ′)k0,0and k2∂θL+
ou +∂θΨ + ∂θΨ′k0,0a e bounded as in (4.16)
and (4.17). Then,
ke
F(Ψ) −e
F(Ψ′)k6,4≤1
2kp−2
h(2∂uL+
ou +∂uΨ + ∂uΨ′)k0,0k∂uΨ−∂uΨ′k6,4
+νk2∂θL+
ou +∂θΨ + ∂θΨ′k0,0k∂θΨ−∂θΨ′k6,4
≤K1
κ+2K∗
κ2TΨ−Ψ′U5,3+νK νI0
κ2+ 2K∗νI0
κ4(νI0)−1TΨ−Ψ′U5,3
≤K
κTΨ−Ψ′U5,3,
o some cons an K2>0. Then, by he las claim o Lemma 4.3,
TGu◦e
F(Ψ) −Gu◦e
F(Ψ′)U5,3≤Kke
F(Ψ) −e
F(Ψ′)k6,4≤K
κTΨ−Ψ′U5,3.
The claim ollows aking κla ge enough. Then, we easily check ha Gu◦e
Fsends he ball BK∗(νI0)−2
in o i sel and has a unique ixed poin he e.
5 Fi s ex ension o he in a ian mani old
In Sec ion 4we ha e ound he ex ension o he local uns able in a ian mani old o he complex
domain D+
κ,δ ×Tσ(see Figu e 2) by means o he unc ion Φ+= Φ0+L+
ou + Φ+
2, whe e Φ0is
in oduced in (4.5), L+
ou in (4.14) and Φ+
2in P oposi ion 4.5. Hence, as commen ed in Rema k 4.1,
he pa ame iza ion o he s able mani old is gi en by Φ−(u, θ) = −Φ+(−u, −θ). I is de ined in
D−
κ,δ ×Tσ, whe e D−
κ,δ =−D+
κ,δ. This ex ension o Φ+is no enough o ou pu poses, because
D+
κ,δ ∩ D−
κ,δ ∩R=∅and we canno compu e he di e ence be ween he mani olds in he eals.
In his sec ion we will ex end Φ+ o a la ge domain, D+
κ,δ ∪D+
κ,ex , de ined below in (5.21) (see
Figu e 3). Once ex ended, he pa ame iza ions o he uns able and s able mani olds will be de ined
in a common domain con aining an in e al o R. We will compu e he di e ence o he mani olds
in his common domain in Sec ion 9.
We ema k ha we ha e no been able o ind he ex ension o Φ+in a single s ep because
he Hamil on-Jacobi equa ion (4.3) is no de ined a u= 0. The eason lies in he ac ha he
me hod we ha e used o ind Φ+ equi es compu ing some in eg als along s aigh lines wi h some
slope in he complex domain whe e he a iable uli es. These s aigh lines canno go h ough
0and hence he cu en me hod does no allow us o ex end Φ+beyond u= 0 di ec ly. In his
sec ion we will ind an ex ension o Φ+ o D+
κ,ex by choosing ano he ype o pa ame iza ion o
he in a ian mani old, and hen we will go back o he o iginal ype o pa ame iza ion. Howe e ,
his ex ension will no be de ined a u= 0.
5.1 F om Hamil on-Jacobi pa ame iza ion o low pa ame iza ion
Taking in o accoun (4.1) and (4.2), he solu ion Φ+= Φ0+L+
ou + Φ+
2o he Hamil on-Jacobi
equa ion (4.3) ob ained in Sec ion 4p o ides he pa ame iza ion o he uns able mani old




q
p
θ
J



= Γ+(u, θ) = 



qh(u)
1
ph(u)∂uΦ+(u, θ)
θ
∂θΦ+(u, θ)



= Γ0(u, θ) + Γ1(u, θ) + Γ2(u, θ),(5.1)
whe e
Γ0(u, θ) = 



qh(u)
1
ph(u)∂uΦ0(u)
θ
0



,Γ1(u, θ) = 



0
1
ph(u)∂uL+
ou (u, θ)
0
∂θL+
ou (u, θ)



(5.2)
and Γ2= Γ+−Γ0−Γ1. We ecall ha ph(0) = 0. Howe e , Γ0is analy ic a u= 0 because
ph(u)−1∂uΦ0(u) = ph(u).
20
Le X= (Xq, Xp, Xθ, XJ)⊤be he ec o ield co esponding o H, in (2.3), using he 2- o m
in (2.1), and le φ be i s low. We look o a change o a iables
(u, θ) = ( + 1( , ϕ), ϕ + 2( , ϕ))
such ha conjuga es Xon he uns able in a ian mani old, pa ame ized by
e
Γ+( , ϕ) = Γ+( + 1( , ϕ), ϕ + 2( , ϕ)) (5.3)
o he ec o ield (1, νI0), ha is,
φ (e
Γ+( , ϕ)) = e
Γ+( + , ϕ +νI0 )
o , equi alen ly,
L(e
Γ+) = X◦e
Γ+,(5.4)
whe e L(Γ) = ∂ Γ + νI0∂ϕΓis he ope a o de ined in (4.7). An immedia e compu a ion shows
ha (5.3) sa is ies (5.4) i and only i = ( 1, 2)is a solu ion o
[∂uΓ+◦(Id + )](1 + L( 1)) + [∂θΓ+◦(Id + )](νI0+L( 2)) = X◦Γ+◦(Id + ).(5.5)
We emphasize ha he abo e equa ion has ou componen s. Howe e , he symplec ic cha ac e o
he ec o ield Xensu es ha i wo o hem a e sa is ied, so a e he o he wo. We choose o sol e
he equa ions co esponding o he i s and hi d componen s o (5.5). Taking in o accoun (5.1),
he equali y ˙qh=−qhphand he ac ha ∂uΦ0=p2
h, we can w i e hese wo equa ions as
L( ) = A◦(Id + ),(5.6)
whe e
A=A1
A2=p−2
h(∂uL+
ou +∂uΦ+
2)
ν(∂θL+
ou +∂θΦ+
2).(5.7)
We deno e N( ) he igh hand side o (5.6). We ha e ha
N( ) = A+DA +R( ),(5.8)
whe e DA deno es he de i a i e o Aand R( ) = A◦(Id + )−A−DA . We will see in a
momen ha equa ion (5.6) canno be sol ed di ec ly as a ixed poin equa ion because he linea
e m DA in (5.8) is oo la ge. We will need o ew i e i in a be e sui ed way.
5.1.1 P elimina ies and echnical lemmas o sol e equa ion (5.6)
To sol e equa ion (5.6), we conside = ( 1, 2)∈ X ,s ×X +1,s+1, wi h he no m
k k ,s =k 1k ,s +k 2k +1,s+1
and he ope a o Guin (4.13), ac ing on each componen . Also, gi en a ma ix unc ion
M=M1,1M1,2
M2,1M2,2
wi h M1,1, M1,2∈ X ,s and M2,1, M2,2∈ X +1,s+1 we de ine
kMk ,s = max {kM1,1k ,s +kM2,1k +1,s+1, νI0(kM1,2k ,s +kM2,2k +1,s+1)}.(5.9)
I ollows immedia ely om Lemma 4.2 ha , i , s ∈Rand ˜ , ˜s≥0,
kMk ,s ≤KνI0
κ˜s
kMk +˜ ,s+˜s(5.10)
and, o any , ′, s, s′∈R,
kM k + ′,s+s′≤ kMk ,sk k ′,s′.(5.11)
21

Indeed, inequali y (5.10) ollows immedia ely om (1) o Lemma 4.2. As o (5.11), using (2) o
Lemma 4.2, we ha e ha
k(M )1k + ′,s+s′+k(M )2k + ′+1,s+s′+1
=kM1,1 1+M1,2 2k + ′,s+s′+kM2,1 1+M2,2 2k + ′+1,s+s′+1
≤ kM1,1 1k + ′,s+s′+kM1,2 2k + ′,s+s′+kM2,1 1k + ′+1,s+s′+1 +kM2,2 2k + ′+1,s+s′+1
≤(kM1,1k ,s +kM2,1k +1,s+1)k 1k ′,s′+KνI0
κ(kM1,2k ,s +kM2,2k +1,s+1)k 2k ′+1,s′+1
≤ kMk ,sk k ′,s′,
i κ≥K.
We will look o he solu ion o equa ion (5.6) in a domain sligh ly smalle han D+
κ,δ ×Tσ.
The no ms we use depend on he choice o he domains. In pa icula , hey depend on κ,δand
σ. Below, we will inc ease κ o ˜κand dec ease δ o ˜
δ. In o de o ha e he dis ance om he
poin s o he bounda y o D+
˜κ,˜
δ o D+
κ,δ cons an , when aking ˜κ > κ we will ake δ−˜
δ=˜κ−κ
νI0
cos β1
cos β2,
some imes wi hou explici men ion o i . To cla i y he exposi ion, ill he end o he sec ion,
we include σand κas subsc ip s in he no ms, bu no δ, since we will unde s and ha when
changing he domain he p e ious ule applies. In his way k k ,s,κ,σ will deno e he no m k k ,s
o de ined ei he in D+
κ,δ o in D+
κ,δ ×Tσ, depending on he se ing. We ema k ha i ˜κ > κ
(wi h ˜κ(νI0)−1<˜
δ) and 0<˜σ < σ, hen D+
˜κ,˜
δ×T˜σ⊂ D+
κ,δ ×Tσ. Analogously, we will deno e he
spaces by X ,s,κ,σ o cla i y hei dependence on he domain. The educ ion o domain will only
be done a ini e numbe o imes.
Lemma 5.1. Fo all ˜κ > κ and 0<˜σ < σ, he e exis s C > 0such ha i B∈ X ,s,κ,σ, hen, o
all m, n ∈N,∂m
u∂n
θB∈ X ,s,˜κ,˜σwi h
k∂m
uBk ,s,˜κ,˜σ≤C˜κ
κs(νI0)mm!
(˜κ−κ)m
1
(cos β1)mkBk ,s,κ,σ, m ≥0,
k∂m
u∂n
θBk ,s,˜κ,˜σ≤C˜κ
κs(νI0)mm!n!
(˜κ−κ)m(σ−˜σ)n+1
1
(cos β1)mkBk ,s,κ,σ, m ≥0, n ≥1.
When applying he p e ious lemma, we will choose ˜κ=κ+κ0, whe e κ0>0is ixed. In his
way, ˜κ/κ = 1 + κ0/κ < 2, i κis la ge enough.
P oo o Lemma 5.1.We no e ha
(∂m
u∂n
θB)[k]= (ik)n∂m
uB[k].
By Cauchy o mula,
∂m
uB[k](u) = m!
2πi Zγu
B[k](z)
(z−u)m+1 dz,
aking γu o be he ci cle o adius (˜κ−κ)(νI0)−1cos β1, we immedia ely ha e, o some C1>0,
k∂m
uB[k]k ,s,˜κ,˜σ≤C1˜κ
κsm!(νI0)m
(˜κ−κ)m
1
(cos β1)mkB[k]k ,s,κ,σ.
Hence, since kB[k]k ,s,κ,σ ≤ kBk ,s,κ,σe−|k|σ, we ha e ha
k∂m
u∂n
θB[k]k ,s,˜κ,˜σ≤C1˜κ
κsm!|k|n(νI0)m
(˜κ−κ)m
1
(cos β1)me−|k|σkBk ,s,κ,σ.
The e o e,
k∂m
u∂n
θBk ,s,˜κ,˜σ=X
k∈Zk∂m
u∂n
θB[k]k ,s,˜κ,˜σe|k|˜σ
≤C1
(cos β1)m˜κ
κsm!(νI0)m
(˜κ−κ)mkBk ,s,κ,σ X
k∈Z|k|ne−|k|(σ−˜σ)
≤C1C2
(cos β1)m˜κ
κsm!n!(νI0)m
(˜κ−κ)m(σ−˜σ)n+1 kBk ,s,κ,σ,
whe e we ha e used ha Pk≥0kne−bk ≤C2n!/bn+1, o b > 0and n≥1, o some C2.
22
Lemma 5.2. Le Cbe he cons an gi en by Lemma 5.1 and ′≥0and s′>0. Gi en K > 0and
δ > 0, o any 0<˜σ < σ, he e exis κ0>0such ha o any ˜κ > κ +δ > κ ≥κ0, i B∈ X ,s,κ,σ
and ∈ X ′,s′,˜κ,˜σ×X ′+1,s′+1,˜κ,˜σwi h k k ′,s′,˜κ,˜σ≤K/(νI0)s′+1, hen B◦(Id + )∈ X ,s,˜κ,˜σand
kB◦(Id + )k ,s,˜κ,˜σ≤2C
σ−˜σ˜κ
κs
kBk ,s,κ,σ.
P oo . Fi s o all, by (1) o Lemma 4.2, we ha e ha
k 1k0,0,˜κ,˜σ≤K(νI0)s′
˜κs′k 1k ′,s′,˜κ,˜σ≤K
˜κs′νI0
,
k 2k0,0,˜κ,˜σ≤K(νI0)s′+1
˜κs′+1 k 2k ′+1,s′+1,˜κ,˜σ≤K
˜κs′+1 .
Then, expanding Bin Taylo se ies and using Lemma 5.1, we ha e ha , w i ing α1= cos β1,
kB◦(Id + )k ,s,˜κ,˜σ≤X
j≥0
1
j!
j
X
ℓ=0 j
ℓk∂j−ℓ
u∂ℓ
θB j−ℓ
1 ℓ
2k ,s,˜κ,˜σ
≤X
j≥0
j
X
ℓ=0
1
(j−ℓ)!ℓ!k∂j−ℓ
u∂ℓ
θBk ,s,˜κ,˜σk 1kj−ℓ
0,0,˜κ,˜σk 2kℓ
0,0,˜κ,˜σ
≤C
σ−˜σ˜κ
κsX
j≥0
j
X
ℓ=0 K
˜κs′νI0j−ℓK
˜κs′+1 ℓ(νI0)j−ℓ
(α1(˜κ−κ))j−ℓ(σ−˜σ)ℓkBk ,s,κ,σ
≤C
σ−˜σ˜κ
κsX
j≥0
Kj
j
X
ℓ=0
1
(˜κs′(α1(˜κ−κ))j−ℓ
1
(˜κs′+1(σ−˜σ))ℓkBk ,s,κ,σ
≤C
σ−˜σ˜κ
κsX
j≥0K1
˜κs′α1(˜κ−κ)+1
˜κs′+1(σ−˜σ)j
kBk ,s,κ,σ
≤2C
σ−˜σ˜κ
κs
kBk ,s,κ,σ,
whe e we ha e chosen κ0such ha
K
˜κs′1
α1(˜κ−κ)+1
˜κ(σ−˜σ)≤1
2.
5.1.2 Solu ion o equa ion (5.6)
Le Gube he ope a o de ined in (4.13). We o mally de ine
0=Gu(A), F0=Gu(DA).(5.12)
Lemma 5.3. Le Abe he unc ion de ined in (5.7). Le ˜κ > κ and 0<˜σ < σ, wi h κbig. Then,
he e exis s K > 0such ha
(1) kAk3,1,κ,σ ≤K(νI0)−1,
(2) k 0k3,1,κ,σ ≤K(νI0)−2,
(3) kDAk3,1,˜κ,˜σ≤K,
(4) kF0k3,1,˜κ,˜σ≤K(νI0)−1.
P oo . We w i e A=A0+A1, whe e
A0=p−2
h∂uL+
ou
ν∂θL+
ou , A1=p−2
h∂uΦ+
2
ν∂θΦ+
2.
23
Taking in o accoun ha p−2
h∈ X−2,−2,κ,σ and he p ope ies o L+
ou in P oposi ion 4.4, we ha e
ha kA0k3,1,κ,σ ≤K(νI0)−1and hA0i= 0. Also, by he p ope ies o Φ+
2in P oposi ion 4.5, we
ha e ha kA1k4,2,κ,σ ≤K(νI0)−2. Hence, kA1k3,1,κ,σ ≤Kκ−1(νI0)−1. This p o es (1).
We bound kGu(A0)k3,1,κ,σ and kGu(A1)k3,1,κ,σ sepa a ely.
Since hA0i= 0, by (1) o Lemma 4.3,
kGu(A0)k3,1,κ,σ ≤K
νI0kA0k3,1,κ,σ ≤K
(νI0)2.
Using again ha p−2
h∈ X−2,−2, he p ope ies o Φ+
2in P oposi ion 4.5 imply ha kA1k4,2,κ,σ ≤
K(νI0)−2. Then, by (2) o Lemma 4.3,
kGu(A1)k3,1,κ,σ ≤KkA1k4,2,κ,σ ≤K
(νI0)2,
om which (2) ollows.
We w i e DA =DA0+DA1. F om he p e ious bounds o A0and A1, applying Lemma 5.1
and he de ini ion o he ma ix no m (5.9), we ha e ha , in he educed domain,
kDA0k3,1,˜κ,˜σ≤K,
kDA1k4,2,˜κ,˜σ≤K
κνI0
.
He e he cons an s depend on ˜κ−κand σ−˜σ. This p o es (3).
We inally p o e (4). Since hA0i= 0, we ha e ha hDA0i= 0. By (1) o Lemma 4.3,
kGu(DA0)k3,1,˜κ,˜σ≤K
νI0
.
Mo eo e , by (2) o Lemma 4.3,
kGu(DA1)k3,1,˜κ,˜σ≤KkDA1k4,2,˜κ,˜σ≤K
νI0
.
Since F0=Gu(DA0) + Gu(DA1), he claim ollows.
We in oduce ˜
by
= 0+ (Id + F0)˜
.
Obse e ha , by (4) o Lemma 5.3 and (1) o Lemma 4.2,
kF0k0,0,˜κ,˜σ≤KνI0
κkF0k3,1,˜κ,˜σ≤K
κ.
Hence, i κis la ge enough, Id+F0is in e ible and k(Id+F0)−1k0,0,κ,σ ≤2. Using ha , by (5.12),
L( 0) = Aand L(F0) = DA, we ew i e equa ion (5.6) as
L(˜
) = e
N(˜
),(5.13)
whe e
e
N(˜
) = (Id + F0)−1DA 0+ (Id + F0)−1DA F0˜
+ (Id + F0)−1R 0+ (Id + F0)˜

and Rwas in oduced in (5.8). Using he ope a o Gu, we ew i e equa ion (5.13) as he ixed
poin equa ion ˜
=Gu◦e
N(˜
).(5.14)
P oposi ion 5.4. Fo all ˆκ > ˜κ > κ and 0<ˆσ < ˜σ < σ, wi h κbig enough, equa ion (5.14)has
a solu ion ˜
+, de ined in D+
ˆκ,ˆ
δ×Tˆσ, and ˜
+∈ X3,1,ˆκ,ˆσ× X4,2,ˆκ,ˆσwi h k˜
+k3,1,ˆκ,ˆσ≤C(νI0)−2.
Consequen ly, e
Γ+= Γ+◦(Id+ 0+(I+F0)˜
+)sa is ies he in a iance equa ion (5.4)in D+
ˆκ,δ ×Tˆσ.
24
P oo o P oposi ion 5.4.We claim ha he e exis s K1>0such ha
kGu◦e
N(0)k3,1,ˆκ,ˆσ≤K1
(νI0)2.
Indeed, i s we no ice ha
e
N(0) = (Id+F0)−1(DA 0+R( 0)) = (Id+F0)−1DA 0+Z1
0
(1 −s)D2A◦(Id + s 0)ds ( 0)⊗2
= (Id + F0)−1DA 0+Z1
0
(1 −s) 0
1∂uDA ◦(Id + s 0) 0+ 0
2∂θDA ◦(Id + s 0) 0ds.
Nex , we no e ha
kDA 0k4,2,ˆκ,ˆσ≤ kDA 0k6,2,ˆκ,ˆσ≤ kDAk3,1,ˆκ,ˆσk 0k3,1,ˆκ,ˆσ≤K
(νI0)2.
By (3) o Lemma 5.3 and Lemma 5.1, o ˆκ > ˜κand 0<ˆσ < ˜σ, we ha e
k∂uDAk3,1,ˆκ,ˆσ≤KνI0kDAk3,1,˜κ,˜σ≤KνI0,
k∂θDAk3,1,ˆκ,ˆσ≤KkDAk3,1,˜κ,˜σ≤K. (5.15)
Mo eo e , by (1) o Lemma 4.2 and (2) o Lemma 5.3,
k 0
1∂uDA ◦(Id + s 0) 0k4,2,ˆκ,ˆσ≤KνI0
κk 0
1∂uDA ◦(Id + s 0) 0k9,3,ˆκ,ˆσ≤K
κ(νI0)2
and, since 0
2∈ X4,2,ˆκ,ˆσ,
k 0
2∂θDA ◦(Id + s 0) 0k4,2,ˆκ,ˆσ≤KνI0
κ2
k 0
2∂ DA ◦(Id + s 0) 0k10,4,ˆκ,ˆσ≤K
κ2(νI0)2.
Hence,
kGu◦e
N(0)k3,1,ˆκ,ˆσ≤Kke
N(0)k4,2,ˆκ,ˆσ≤K1
κ(νI0)2.
Le C= 2K1.
Le , ′∈ X3,1,ˆκ,ˆσ×X4,2,ˆκ,ˆσwi h k k3,1,ˆκ,ˆσ,k ′k3,1,ˆκ,ˆσ≤e
C. We ha e ha
e
N( )−e
N( ′) = (I+F0)−1(M1+M2) ( − ′),(5.16)
whe e M1=DA F0,
M2( , ′) = Z1
0Z1
0
D2A◦(Id + us( , ′)) d us( , ′)ds (I+F0)
wi h us( , ′) = 0+ (Id + F0) ′+s(I+F0)( − ′).
Then, using (1) o Lemma 4.2 and (3) and (4) o Lemma 5.3, we ha e ha
kM1k1,1,ˆκ,ˆσ≤KνI0
κkDAF 0k6,2,ˆκ,ˆσ≤KνI0
κkDAk3,1,ˆκ,ˆσkF0k3,1,ˆκ,ˆσ≤K
κ.(5.17)
Obse e ha , by (2) o Lemma 5.3 and he hypo heses on , ′,
kus( , ′)k3,1,ˆκ,ˆσ≤K(νI0)−2.
Hence, also using (5.15)
kus,1∂uDA ◦(Id + us)k1,1,ˆκ,ˆσ≤KνI0
κkus,1∂uDA ◦(Id + us)k6,2,ˆκ,ˆσ
≤KνI0
κkus,1k3,1,ˆκ,ˆσk∂uDA ◦(Id + us)k3,1,ˆκ,ˆσ≤K
κ
25
whe e
Lin(T2) = ∂ T2+∂θT2,
Fin(T2) = −ν
2(∂θL+
in)2+ 2 2(∂ L+
in)2+ν∂θL+
in∂θT2+ν
2(∂θT2)2+ 4 2∂ L+
in∂ T2+ 2 2(∂ T2)2,
whe e we ha e used ha Lin(L+
in) = −E(T0).
6.1 Spaces and echnical lemmas
To deal wi h equa ion (6.9) we ha e o in oduce some unc ion spaces and p o ide basic p ope ies
o he ope a o Lin. We conside he domain
D+
κ,in ={ ∈C| ℑ < −( an β1)ℜ −κ}∪{ ∈C| ℑ > ( an β1)ℜ +κ},(6.10)
wi h κ > 1. See Figu e 5. We no ice ha he image o D+
κ,δ ∪ D+
κ,ex by he ans o ma ion
u=i+ (νI0)−1 is con ained in D+
κ,in.
Figu e 5: The domain D+
κ,in de ined in (6.10), shaded in g ay.
To sol e equa ion (6.9), o ∈R, we in oduce he Banach space o 2π-pe iodic in θ, analy ic
unc ions
Y ={R:D+
κ,in ×Tσ→C| kRk <∞},(6.11)
whe e, aking in o accoun he Fou ie expansion o R( , θ) = Pk∈ZR[k]( )eikθ ,
kRk =X
k∈ZkR[k]k e|k|σ,(6.12)
and, o an analy ic unc ion :D+
κ,in →C,
k k = sup
∈D+
κ,in | ( )|.
Lemma 6.1. Le , 1, 2∈R.
(1) I R∈ Y +s,s≥0, hen R∈ Y and he e exis s K > 0such ha
kRk ≤K1
κskRk +s.
32

(2) I R1∈ Y 1and R2∈ Y 2, hen R1R2∈ Y 1+ 2and
kR1R2k 1+ 2≤ kR1k 1kR2k 2.
We also in oduce he Banach space
e
Y ={R∈ Y |∂ R, ∂θR∈ X +1,TRU <∞},(6.13)
wi h he no m
TRU =kRk +k∂ Rk +1 +k∂θRk +1.(6.14)
We will use he nex echnical lemma se e al imes. I is analogous o Lemma 4.3.
Lemma 6.2. Le Gu
in be he ope a o de ined in (6.7). The e exis s K > 0such ha
(1) I R∈ Y wi h > 0and hRi= 0, hen Gu
in(R)∈ Y and
kGu
in(R)k ≤KkRk .
(2) I R∈ Y wi h > 1, hen Gu
in(R)∈ Y −1and
kGu
in(R)k −1≤KkRk .
(3) I R∈ Y wi h > 0, hen ∂ Gu
in(R), ∂θGu
in(R)∈ Y and
k∂ Gu
in(R)k ≤KkRk ,k∂θGu
in(R)k ≤KkRk .
As a consequence, i R∈ Y , wi h > 1,Gu
in(R)∈e
Y −1and
TGu
in(R)U −1≤KkRk .
The cons an Konly depends on and he cons an s in ol ed in he de ini ion o D+
κ,in.
6.2 Fixed poin equa ion
We conside he equa ion
T2=Gu
in ◦Fin(T2).(6.15)
We decompose
Fin(T2) = −(F0+F1+F2+F3+F4),(6.16)
wi h
F0( , θ) = ν
2(∂θL+
in( , θ))2+ 2 2(∂ L+
in( , θ))2,
F1( , θ) = ν∂θL+
in( , θ)∂θT2( , θ),
F2( , θ) = ν
2(∂θT2( , θ))2,
F3( , θ) = 4 2∂ L+
in( , θ)∂ T2( , θ),
F4( , θ) = 2 2(∂ T2( , θ))2.
I T2is a solu ion o he ixed poin equa ion (6.15) i is also a solu ion o (6.9). In iew o (6.8),
since V[0] = 0, we ha e ha (L+
in)[0] = 0. The ollowing p oposi ion is analogous o P oposi ion 4.4.
I s p oo ollows exac ly he same lines.
P oposi ion 6.3. We ha e L+
in, ∂θL+
in ∈ Y2and ∂ L+
in ∈ Y3. Mo eo e , he e exis s K > 0,
independen o κ, such ha kL+
ink2,k∂θL+
ink2,k∂ L+
ink3≤K.
Mo i a ed by P oposi ion 6.3, we in oduce he cons an
ΘV=kL+
ink2
2+k∂θL+
ink2
2+k∂ L+
ink2
31/2.(6.17)
Rema k 6.4. I one eplaces iby ε˜ iin he de ini ion o Vin (1.2), hen ΘV=O(ε).
33
P oposi ion 6.5. The e exis K1>0and κ0such ha , i κ > κ0, he ope a o Gu
in ◦Fin :BK1Θ2
V⊂
e
Y3→ BK1Θ2
Vis well de ined and a con ac ion. Hence, equa ion (6.15)has a solu ion T+
2∈e
Y3
sa is ying
TT+
2U3≤K1Θ2
V.(6.18)
P oo . We s a by bounding F0. By P oposi ion 6.3,
kF0k4≤ν
2(∂θL+
in)24+ 2k 2(∂ L+
in)2k4≤ν
2k∂θL+
ink2
2+ 2k 2k−2k∂ L+
ink2
3< K2Θ2
V,(6.19)
o some K2>0. We ake K1= 2kGu
inkK2and K∗=K1Θ2
V. In his p oo kGu
inks ands o he
ope a o no m o Gu
in :Y4→e
Y3. Le T2∈ BK∗⊂e
Y3. We claim ha Fin(T2)∈ Y4and
kFin(T2)k4≤ kF0k4+Kνk∂θL+
ink2
κ2+KνK∗
2κ4+K4k∂ L+
ink3
κ+K2K∗
κ2K∗.(6.20)
Indeed, nex we deal wi h Fj,j= 1,...,4. By P oposi ion 6.3 and Lemma 6.1
kF1k4=kν∂θL+
in∂θT2k4=νk∂θL+
ink0k∂θT2k4≤Kνk∂θL+
ink2
κ2TT2U3≤Kνk∂θL+
ink2K∗
κ2,
since k∂θT2k4≤TT2U3≤K∗. Again by Lemma 6.1,
kF2k4=ν
2(∂θT2( , θ))24≤ν
2k∂θT2k0k∂θT2k4≤Kν
2κ4k∂θT2k4k∂θT2k4≤Kν(K∗)2
2κ4.
Now, by P oposi ion 6.3, Lemma 6.1 and he ac ha k∂ T2k4≤TT2U3≤K∗,
kF3k4=4 2∂ L+
in∂ T24≤4 2k−2k∂ L+
ink2k∂ T24≤K4k∂ L+
ink3
κK∗.
Finally,
kF4k4=2 2(∂ T2( , θ))24≤2 2∂ T2( , θ)0k∂ T2( , θ)k4
≤ k2 2k−2k∂ T2( , θ)k2k∂ T2( , θ)k4≤K2
κ2k∂ T2( , θ)k4k∂ T2( , θ)k4≤K2(K∗)2
κ2.
This p o es (6.20). Now we choose κ0>1such ha
Kνk∂θL+
ink2
κ2
0
+νK∗
2κ4
0
+4k∂ L+
ink3
κ0
+2K∗
κ2
0K∗< K2Θ2
V.
Hence, o any κ > κ0, i T2∈ BK∗, by he las claim o Lemma 6.2,
TGu
in ◦Fin(T2)U3≤ kGu
inkkFin(T2)k4≤2kGu
inkK2Θ2
V=K∗.
Nex we check ha Gu
in ◦Fin is a con ac ion on BK∗. We claim ha he e exis s K3>0such ha ,
i T2, T ′
2∈ BK∗,
kFin(T2)−Fin(T′
2)k4≤K3
κTT2−T′
2U3.(6.21)
To p o e his claim, we w i e
Fin(T2)−Fin(T′
2) = −
4
X
i=1
(Fi(T2)−Fi(T′
2)) ,
whe e F1(R) = ν∂θL+
in∂θR,
F2(R) = 1
2ν(∂θR)2,
F3(R)( , θ) = 4 2∂ L+
in( , θ)∂ R( , θ),
F4(R)( , θ) = 2 2(∂ R( , θ))2.
34
We bound each di e ence sepa a ely. Fi s , by P oposi ion 6.3 and Lemma 6.1, we ha e ha
kF1(T2)−F1(T′
2)k4≤ kν∂θL+
ink0k∂θT2−∂θT′
2k4≤Kνk∂θL+
ink2
κ2TT2−T′
2U3.
Second,
kF2(T2)−F2(T′
2)k4=ν
2k(∂θT2+∂θT′
2)(∂θT2−∂θT′
2)k4
≤ν
2k(∂θT2+∂θT′
2)k0k∂θT2−∂θT′
2k4
≤Kν
2
1
κ4k∂θT2+∂θT′
2k4k∂θT2−∂θT′
2k4
≤KK∗ν
κ4TT2−T′
2U3.
Thi d, using again P oposi ion 6.3 and Lemma 6.1,
kF3(T2)−F3(T′
2)k4≤4 2∂ L+
in0k∂ T2−∂ T′
2k4≤4∂ L+
in3
κTT2−T′
2U3.
And ou h,
kF4(T2)−F4(T′
2)k4≤ k2 2(∂ T2+∂ T′
2)k0k∂ T2−∂ T′
2k4
≤ k2 2k−2k∂ T2+∂ T′
2k2k∂ T2−∂ T′
2k4
≤K2
κ2k∂ T2+∂ T′
2k4k∂ T2−∂ T′
2k4
≤4KK∗
κ2TT2−T′
2U3.
Inequali y (6.21) ollows om combining hese ou s eps, since κ > κ0>1and κ0is big.
Finally, we check ha he ope a o Gu
in ◦Fin is a con ac ion in BK∗. Indeed, i T2, T ′
2∈ BK∗,
using he las s a emen o Lemma 6.2 and (6.21),
TGu
in ◦Fin(T2)−Gu
in ◦Fin(T′
2)U3≤ kGu
inkkFin(T2)−Fin(T′
2)k4≤ kGu
inkK3
κTT2−T′
2U3.
Hence, by he s anda d a gumen , aking κ0la ge enough, Gu
in ◦Fin sends BK∗in o i sel and has
a unique ixed poin T+
2in his ball. The bound (6.18) ollows om he de ini ion o K∗.
Co olla y 6.6. The inne equa ion (6.2)admi s a solu ion
T+=T0+L+
in +T+
2,
whe e T0is gi en by (6.3),L+
in is de ined in (6.8)and T+
2is gi en by P oposi ion 6.5.
7 App oxima ion o he mani old in he inne domain
Le T+=T0+L+
in +T+
2be he solu ion o he inne equa ion (6.2) in he domain D+
κ,in gi en by
Co olla y 6.6.
F om P oposi ion 4.5, we ha e ha he solu ion Φ+(u, θ)o he Hamil on-Jacobi equa ion (4.3)
and i s ex ension o he domain D+
κ,δ ∪D+
κ,ex B∗
ρ(whe e B∗
ρis a ixed small ball cen e ed a he
o igin), wi h anishing con ou condi ions a ℜu=−∞, is gi en by
Φ+= Φ0+L+
ou + Φ+
2,
whe e Φ0was in oduced in (4.5), L+
ou in (4.14) and Φ+
2∈e
X5,3, hence
Φ+
2(u, θ)∼(νI0)−2
(u−i)3.
The app oxima ion o Φ+by Φ0+L+
ou is no good enough i u−i∼(νI0)−1, al hough P opo-
si ion 5.8 ensu es ha he mani olds can be ex ended o D+
κ,δ ∪ D+
κ,ex B∗
ρ. In o de o ob-
ain a be e app oxima ion o Φ+, we compa e i wi h he solu ion o he inne equa ion (6.2),
T+=T0+L+
in +T+
2, gi en by Co olla y 6.6.
35
We ecall ha he unc ion
S+( , θ) = (νI0)−1Φ+(i+ (νI0)−1 , θ)
is a solu ion o he Hamil on-Jacobi equa ion (6.1) in he ans o med domain.
Ac ually, we wan o compa e S+wi h he solu ion T+o he inne equa ion. Bo h unc ions
a e al eady de e mined. We w i e
S+=T0+L+
in +T+
2+S+
2
and de i e bounds and p ope ies o S+
2 om he equa ion i sa is ies.
Using ha T+=T0+L+
in +T+
2is a solu ion o he inne equa ion (6.2), we ha e ha S+
2is a
solu ion o
Lin(S2) = b
Fin,a(S2),(7.1)
whe e Lin(S2) = ∂ S2+∂θS2was de ined in (6.6) and
b
Fin,a(S2) = −e
E+ν∂θT+∂θS2+1
2ν(∂θS2)2+ (2A1∂ T+−1)∂ S2+A1(∂ S2)2(7.2)
wi h
A1( ) = 1
2
2(2i+ (νI0)−1 )2
(i+ (νI0)−1 )2,
A2( , θ) = −1
2
(i+ (νI0)−1 )2
2(2i+ (νI0)−1 )2+1
2
1
2(2i+ (νI0)−1 )2V(θ),
e
E( , θ) = (A1( )−2 2)(∂ T+
0)2+A2( , θ) + 1
8 2+1
8 2V(θ).
(7.3)
Le β1>0be he angle in ol ed in he de ini ion o D+
κ,δ (see (4.12)). Fix β3and β4sa is ying
0< β3< β4< β1. Fo α∈(0,1) and κ > 1, we de ine he domain
D+
κ,α ={ ∈C| −(νI0)α− an β4ℜ < ℑ < min{−κ− an β1ℜu, −κ− an β3ℜu}}.(7.4)
See Figu e 6. Obse e ha d(D+
κ,α,0) = κcos β3and | | ≤ K(νI0)α o ∈ D+
κ,α.
Figu e 6: The domain D+
κ,α de ined in (7.4), shaded in g ay.
36
In o de o sol e equa ion (7.1), we conside he ollowing igh in e ses, b
Guand b
Gu
s, o Lin.
Fi s , b
Guis de e mined by he exp ession o he Fou ie coe icien s o he image b
Gu(S2):
b
Gu(S2)[k]( ) = Z
+
0
eik( − )S[k]
2( )d i k≥0,
b
Gu(S2)[k]( ) = Z
−
0
eik( − )S[k]
2( )d i k < 0,
(7.5)
whe e
−
0=(νI0)α−κ
an β1− an β4
+− an β4
an β1− an β4
+ 1(νI0)α+ an β4
an β1− an β4
κi,
+
0=−(νI0)α+κ
an β4− an β3
+ an β4
an β4− an β3−1(νI0)α− an β4
an β4− an β3
κi,
wi h α∈(0,1) and 0< β3< β4< β1< π. See Figu e 6. The mos gene al igh in e se o Lin has
he o m b
Gu
s(S2) = S2,0+b
Gu(S2),(7.6)
wi h S2,0∈ke Lin. Wi h his ope a o we conside he ixed poin equa ion
S2=b
Gu
s◦b
Fin,a(S2)(7.7)
and we choose S2,0so ha he solu ion S+
2o (7.7) is he analy ical con inua ion o S+−T0−
L+
in −T+
2 o he domain D+
κ,α. No e again ha i S+
2is a ixed poin o b
Gu
s◦b
Fin,a, i is also a
solu ion o (7.1).
Since S2,0∈ke Lin,
S2,0( , θ) = X
k∈Z
S[k]
2,0( )eikθ =X
k∈Z
g[k]eik(θ− ),
which de ines he coe icien s g[k]. I , mo eo e , S2is a solu ion o (7.7), we ha e
S2( , θ) = S2,0( , θ) + b
Gu◦b
Fin,a(S2)( , θ).
Fo k≥0, he Fou ie coe icien s sa is y
S[k]
2( ) = S[k]
2,0( ) + Z
+
0
eik( − )(b
Fin,a(S2))[k]( )d
and, aking in o accoun ha , a he poin s ( +
0, θ)o he bounda y,
S2( , θ) = S+( , θ)−T0( , θ)−L+
in( , θ)−T+
2( , θ),
e alua ing a = +
0, we ge
g[k]=S[k]
2,0( +
0)eik +
0= (S+−T0−L+
in −T+
2)[k]( +
0)eik +
0
and inally,
S[k]
2,0( ) = g[k]e−ik = (S+−T0−L+
in −T+
2)[k]( +
0)e−ik( − +
0).(7.8)
Analogously, we ob ain ha o k < 0
S[k]
2,0( ) = g[k]e−ik = (S+−T0−L+
in −T+
2)[k]( −
0)e−ik( − −
0).(7.9)
The e o e, we choose
S2,0( , θ) = X
k∈Z
S[k]
2,0( )eikθ (7.10)
wi h he coe icien s (7.8) and (7.9).
37

7.1 Spaces and echnical lemmas
To sol e equa ion (7.7), o ∈R, we in oduce he Banach space o analy ic, 2π-pe iodic in θ
unc ions
Z ={R:D+
κ,α ×Tσ→C| kRk <∞},
wi h he no m k·k de ined as ollows. Using ha R( , θ) = Pk∈ZR[k]( )eikθ,
kRk =X
k∈ZkR[k]k e|k|σ,
whe e, o an analy ic unc ion :D+
κ,α →C,
k k = sup
∈D+
κ,α | ( )|.
The ollowing lemma is e y simila o Lemma 6.1.
Lemma 7.1. Le , 1, 2∈R. The e exis s K > 0such ha he ollowing holds.
(1) I R∈ Z +swi h s≥0, hen R∈ Z and
kRk ≤K1
κskRk +s.
(2) I R1∈ Z 1and R2∈ Z 2, hen R1R2∈ Z 1+ 2and
kR1R2k 1+ 2≤ kR1k 1kR2k 2.
We also in oduce
e
Z ={R∈ Z |∂ R, ∂θR∈ Z +1,TRU <∞},
wi h he no m
TRU =kRk +k∂ Rk +1 +k∂θRk +1.
The ollowing esul is comple ely analogous o Lemma 6.2. The only di e ence is he domain
o he unc ions o he space.
Lemma 7.2. Le b
Gube he ope a o de ined in (7.5). The e exis s K > 0such ha
(1) I R∈ Z , wi h > 0, and R[0] = 0,
kb
Gu(R)k ≤KkRk .
(2) I R∈ Z wi h > 1,b
Gu(R)∈ Z −1and
kb
Gu(R)k −1≤KkRk .
(3) I R∈ Z wi h > 0, hen ∂ b
Gu(R), ∂θb
Gu(R)∈ Z and
k∂ b
Gu(R)k ≤KkRk ,k∂θb
Gu(R)k ≤KkRk ,
As a consequence, i R∈ Z , wi h > 1,b
Gu(R)∈e
Z −1and
Tb
Gu(R)U −1≤KkRk .
The cons an Konly depends on and he cons an s in ol ed in he de ini ion o D+
κ,α.
38
7.2 The ixed poin equa ion
Be o e sol ing he ixed poin equa ion (7.7) we deal wi h some p elimina y es ima es.
Lemma 7.3. Le A1and e
Ebe he unc ions in oduced in (7.3). The e exis s K > 0such ha , in
D+
κ,α, we ha e
(1) e
E ∈ Z2wi h ke
Ek2≤K(νI0)α−1.
(2) A1∈ Z−2wi h kA1k−2≤K.
(3) 2A1∂ T+−1∈ Z0wi h k2A1∂ T+−1k0≤K(νI0)α−1+Kκ−1.
(4) ∂θT+∈ Z2wi h k∂θT+k2≤K.
P oo . We ecall ha in he domain D+
κ,α,K1κ < | |< K2(νI0)α, o some K1, K2>0. We s a
by p o ing (1). We ha e ha
A1( )−2 2=1
2
2(2i+ (νI0)−1 )2
(i+ (νI0)−1 )2−2 2= 2 2(νI0)−1 (i+3
4(νI0)−1 )
(1 −i(νI0)−1 )2,(7.11)
which implies ha kA1( )−2 2k−2≤K(νI0)α−1. Recall ha 0< α < 1.
We claim ha he e exis s a cons an Ksuch ha k∂ T+k2≤K. Indeed, since T+=T0+L+
in+
T+
2, by P oposi ions 6.3 and 6.5,k∂ L+
ink3≤Kand TT+
2U3≤K. Hence, by (1) o Lemma 7.1,
k∂ L+
ink2≤Kκ−1and k∂ T+
2k2≤Kκ−2. Since ∂ T0= 1/(4 2), he claim ollows immedia ely
om he p e ious bounds. Then, by (2) o Lemma 7.1,
(A1( )−2 2)(∂ T+)22≤ kA1( )−2 2k−2k∂ T+k2
2≤K3(νI0)α−1.
By he de ini ion o A2in (7.3), we ha e ha
A2( , θ) + 1
8 2+1
8 2V(θ) = −1
2
(i+ (νI0)−1 )2
2(2i+ (νI0)−1 )2+1
8 2+1
2
1
2(2i+ (νI0)−1 )2+1
8 2V(θ)
=1
2O((νI0)−1 )
which implies A2( , θ) + 1
8 2+1
8 2V(θ)2≤K(νI0)α−1.
Hence, (1) ollows. (2) is an immedia e consequence o (7.11).
Now we deal wi h (3). We ha e ha
2A1∂ T+−1 = 2A1∂ T0−1+2A1∂ (L+
in +T+
2) = −1
4
(νI0)−1 (4i+ 3(νI0)−1 )
(i+ (νI0)−1 )2+2A1∂ (L+
in +T+
2)
which implies ha
k2A1∂ T+−1k0≤ k2A1∂ T0−1k0+k2A1∂ (L+
in +T+
2)k0
≤K(νI0)α−1+k2A1k−2k∂ (L+
in +T+
2)k2
≤K(νI0)α−1+K
κk∂ (L+
in +T+
2)k3.
(4) Follows di ec ly om P oposi ions 6.3 and 6.5.
P oposi ion 7.4. The unc ion S2,0de ined in (7.10)sa is ies S2,0∈ Z0and
kS[k]
2,0k1≤Kmax{(νI0)−2α,(νI0)−1+αlog(νI0)}e−|k|σ, k 6= 0.
Consequen ly, in D+
κ,α ×Tσ′wi h 0< σ′< σ,
kS2,0k1≤Kmax{(νI0)−2α,(νI0)−1+αlog(νI0)}.
39
P oo . We ha e ha
(S+−T0−L+
in −T+
2)( , θ) = (νI0)−1Φ0(i+ (νI0)−1 )−T0( )
+ (νI0)−1L+
ou (i+ (νI0)−1 , θ)−L+
in( , θ) + (νI0)−1Φ+
2(i+ (νI0)−1 , θ)−T+
2( , θ),
whe e Φ0was in oduced in (4.5), T0in (6.3), L+
ou in (4.14), L+
in in (6.8), Φ+
2in P oposi ion 4.5
and T+
2in P oposi ion 6.5.
Fi s , an explici compu a ion shows ha , in he domain unde conside a ion,
(νI0)−1Φ0(i+ (νI0)−1 )−T0( ) = i
4(νI0)−1log((νI0)−1 ) + (νI0)−1O((νI0)−1 ).
Nex , also an explici compu a ion shows ha
(νI0)−1(L+
ou )[k](i+ (νI0)−1 )−(L+
in)[k]( ) = −V[k]Z
−∞
4(νI0)−1i+ (νI0)−2s
8s(2 −(νI0)−1is)2eik(s− )ds
=O(νI0)−1
(1 + O((νI0)−1 ))V[k].
Mo eo e , since Φ+
2∈e
X5,3wi h kΦ+
2k5,3≤K(νI0)−2,
|(νI0)−1(Φ+
2)[k](i+ (νI0)−1 )| ≤ K| |−3e−|k|σ.
Finally, since, by P oposi ion 6.5,T+
2∈e
Y3wi h kT+
2k3≤K,
|(T+
2)[k]( )| ≤ K| |−3e−|k|σ.
Since | ±
0|=O((νI0)α), he bounds o he Fou ie coe icien s ollow. To ge he bound in he k·k1
no m we ha e o es ic he domain o D+
κ,α ×Tσ′wi h 0< σ′< σ < σ0, whe e σ0was in oduced
in (2.9).
We conside he ixed poin equa ion
S2=b
Gu
s◦b
Fin,a(S2),(7.12)
whe e b
Gu
sand b
Fin,a we e de ined in (7.6) and (7.2), espec i ely.
P oposi ion 7.5. Le α∈(0,1). Then, equa ion (7.12)has a solu ion S+
2∈e
Z1in D+
κ,α ×Tσ′
wi h kS+
2k1≤K(νI0)−2α+K(νI0)−1+αlog(νI0).
P oo . We i s check ha he e exis s K∗>0such ha BK∗⊂e
Z1sa is ies b
Gu
s◦b
Fin,a(BK∗)⊂ BK∗.
To do so, gi en S2∈ BK∗, we w i e
b
Fin,a(S2) = F0+F1+F2+F3+F4,
wi h
F0=e
E,
F1=ν∂θT+∂θS2,
F2=1
2ν(∂θS2)2,
F3= (2A1∂ T+−1)∂ S2,
F4=A1(∂ S2)2,
whe e e
E,A1and A2we e in oduced in (7.3). We claim ha Fi∈ Z2,i= 0,...,4, and he e exis s
K > 0such ha
kF0k2≤K(νI0)α−1,(7.13)
kF1k2≤Kκ−2K∗,(7.14)
kF2k2≤Kκ−2(K∗)2,(7.15)
kF3k2≤K(νI0)α−1+κ−1K∗,(7.16)
kF4k2≤K(K∗)2.(7.17)
40
Bound (7.13) is simply (1) o Lemma 7.3.
Since k∂θT+k2≤K, by (1) o Lemma 7.1, we ha e ha
kν∂θT+∂θS2k2≤νk∂θT+k1k∂θS2k1≤ν
κ2k∂θT+k2k∂θS2k2≤Kκ−2K∗,
which p o es (7.14).
Bound (7.15) ollows analogously om

1
2ν(∂θS2)22≤ν
2k∂θS2k0k∂θS2k2≤ν
2
1
κ2(K∗)2.
By (3) o Lemma 7.3, bound (7.16) ollows om
k(2A1∂ T+−1)∂ S2k2≤ k2A1∂ T+−1k0k∂ S2k2≤K(νI0)α−1+κ−1K∗.
Bound (7.17) ollows om (2) o Lemma 7.3 and
kA1(∂ S2)2k2≤ kA1k−2k∂ S2k2
2≤K(K∗)2.
In pa icula , om (7.13), P oposi ion 7.4 and he las claim o Lemma 7.2, since b
Fin,a(0) = F0,
we deduce ha
Tb
Gu
s◦b
Fin,a(0)U1=TS2,0+b
Gu◦b
Fin,a(0)U1≤TS2,0U1+Tb
Gu◦b
Fin,a(0)U1
≤Kmax{(νI0)−2α,(νI0)−1+αlog(νI0)}+KkF0k2≤K1max{(νI0)−2α,(νI0)−1+αlog(νI0)},
o some K1>0. By he las claim o Lemma 7.2, he same ype o compu a ions imply ha , i
S2, S′
2∈ BK∗,
Tb
Gu
s◦b
Fin,a(S2)−b
Gu
s◦b
Fin,a(S′
2)U1≤Kkb
Fin,a(S2)−b
Fin,a(S′
2)k2
≤Kκ−1+ (νI0)α−1+K∗TS2−S′
2U1.
Then, aking
K∗= 2Tb
Gu
s◦b
Fin,a(0)U1= 2K1max{(νI0)−2α,(νI0)−1+αlog(νI0)},
he claim ollows wi h he usual a gumen , aking κand νI0la ge enough so ha
Kκ−1+ (νI0)α−1+K∗<1/2.
8 Di e ence be ween solu ions o he inne equa ion
In his sec ion we compu e he di e ence be ween he unc ions T±=T0+L±
in +T±
2, wi h T+is
gi en in D+
κ,in ×Tσwhe e D+
κ,in is de ined in (6.10) and T0( ) = −1/(4 )is in oduced in (6.3), L+
in
is de ined in (6.8) and T+
2is gi en by P oposi ion 6.5. Mo eo e , he unc ion T−is de ined by
T−( , θ) = −T+(−¯ , −¯
θ) = T0( )−L+
in(−¯ , −¯
θ)−T+
2(−¯ , −¯
θ)(8.1)
in he domain (−D+
κ,in)×Tσ.
We also de ine
L−
in( , θ) = −L+
in(−¯ , −¯
θ)and T−
2( , θ) = −T+
2(−¯ , −¯
θ).(8.2)
The ollowing lemma is an immedia e compu a ion. I ollows om he ac ha V es ic ed o
Ris an e en unc ion, i.e. only depends on cosinus. We ecall ha E(T0)( , θ) = −V(θ)/8 2.
Lemma 8.1. We ha e ha
L−
in( , θ) = Z∞
E(T0)(s, θ +s− )ds =−X
k∈Z
V[k]
8eik(θ− )Z∞
1
s2eiks ds.
I sa is ies he analogous bounds o L+
in in P oposi ion 6.3.
41
Figu e 8: The domain Dκde ined in (9.3), shaded in g ay. Compa e wi h Figu e 4.
9.1 S aigh ening he linea ope a o
Since Φ±a e solu ions o (4.3), e
∆ou sa is ies
e
Le
∆ou = 0,(9.6)
whe e
e
Le
∆ = 1 + B
1 + A∂ue
∆ + νI0∂θe
∆
and
A=ν
2νI0∂θΦ++∂θΦ−,
B=1
2
(1 + u2)2
u2∂uΦ++∂uΦ−−1.
(9.7)
We look o a change o a iables o he o m u= +X( , θ)such ha
∆ou ( , θ) = e
∆ou ( +X( , θ), θ)
sa is ies
L∆ou = 0,
whe e L∆ = ∂u∆ + νI0∂θ∆was in oduced in (4.7). No ice ha , p o ided 7→ +X( , θ)is
in e ible, e
∆ou sa is ies (9.6) i and only i ∆sa is ies
1
1 + ∂ X 1 + B
1 + A|(u,θ)=( +X( ,θ),θ)−νI0∂θX!∂ ∆ + νI0∂θ∆ = 0.
Hence, we need o impose ha Xsa is ies
LX=B−A
1 + A|(u,θ)=( +X( ,θ),θ)
.(9.8)
To sol e his equa ion, we conside he in e se Gou ,d o he ope a o Lde ined by he Fou ie
coe icien s o he image Gou ,d(X):
Gou ,d(X)[k](u) = Zu
u+
0
eikνI0(s−u)X[k](s)ds, k > 0,
Gou ,d(X)[0](u) = Zu
u0
X[0](s)ds,
Gou ,d(X)[k](u) = Zu
u−
0
eikνI0(s−u)X[k](s)ds, k < 0,
(9.9)
48

whe e u+
0,u−
0=u+
0and u0∈Ra e opmos , bo ommos and eal le mos poin s in Dκ, espec-
i ely. See Figu e 8.
Rema k 9.1. The choice o he poin s u±
0and u0implies ha , i X(u, θ)is eal analy ic, so is
Gou ,d(X).
To sol e equa ion (9.8), o ∈R, we in oduce he Banach space o analy ic, 2π-pe iodic in θ
unc ions
Xs={X:Dκ×Tσ→C| kXks<∞},
wi h he no m
kXks=X
k∈ZkX[k]kse|k|σ,
whe e, o an analy ic unc ion :Dκ→C,
k ks= sup
u∈Dκ|(1 + u2)s (u)|.
Nex , we s a e wo echnical lemmas comple ely analogous o Lemmas 4.2 and 4.3.
Lemma 9.2. Le s1, s2∈R.
(1) I X∈ Xs1+s2and s2≥0, hen X∈ Xs1and he e exis s K > 0, independen o s1, s2such
ha
kXks1≤KνI0
κs2
kXks1+s2.
(2) I X1∈ Xs1and X2∈ Xs2, hen X1X2∈ Xs1+s2and
kX1X2ks1+s2≤ kX1ks1kX2ks2.
Lemma 9.3. Le s∈Rand Gou ,d be he ope a o de ined by (9.9).
(1) I X∈ Xswi h s≥0, hen Gou ,d(X)∈ Xsand
kGou ,d(X)ks≤KkXks.
I , u he mo e, hXi= 0,
kGou ,d(X)ks≤K(νI0)−1kXks.
(2) I X∈ Xswi h s > 1, hen Gou ,d(X)∈ Xs−1and
kGou ,d(X)ks−1≤KkXks.
(3) I X∈ Xswi h s > 0, hen ∂uGou ,d(X), ∂θGou ,d(X)∈ Xsand
k∂uGou ,d(X)ks≤KkXks,k∂θGou ,d(X)ks≤K(νI0)−1kXks.
Now, we summa ize he p ope ies o Aand Bwe need.
Lemma 9.4. Le Aand Bbe he unc ions in oduced in (9.7). The e exis s K > 0such ha
(1) A∈ X2wi h kAk2≤K(νI0)−2.
(2) B=B1+B2whe e B1∈ X1,hB1i= 0,kB1k1≤K(νI0)−1and B2∈ X2wi h kB2k2≤
K(νI0)−2. Consequen ly, B∈ X1wi h kBk1≤K(νI0)−1.
P oo . To p o e (1), we use he de ini ions in (9.1) and (9.7). F om P oposi ion 4.4 we ha e
k∂θL+
ou k2≤K(νI0)−1and, om P oposi ion 4.5 and (1) o Lemma 9.2,
k∂θΦ+
2k2≤νI0
κ2
k∂θΦ+
2k4≤νI0
κ2
(νI0)−1TΦ+
2U3≤Kκ−2(νI0)−1.
49
Since he bounds o L−
ou and Φ−
2a e he same as he ones o L+
ou and Φ+
2, espec i ely, and
∂θΦ0= 0, we ha e
kAk2=ν
2νI0∂θΦ++∂θΦ−2≤ν
νI0k∂θL+
ou k2+k∂θΦ+
2k2≤K(νI0)−2.
Now we p o e (2). Using he de ini ion o Φ0in (4.5) we can check ha B=B1+B2, whe e
B1(u, θ) = 1
2
(1 + u2)2
u2(∂uL+
ou (u, θ) + ∂uL−
ou (u, θ)),
B2(u, θ) = 1
2
(1 + u2)2
u2(∂uΦ+
2(u, θ) + ∂uΦ−
2(u, θ)).
By P oposi ion 4.4,k∂uL+
ou k3≤K(νI0)−1and since k(1+u2)2/u2k−2=O(1), by (2) o Lemma 9.2,
kB1k1≤
1
2
(1 + u2)2
u2−2k∂uL+
ou +∂uL−
ou k3≤K(νI0)−1.
Since hL+
ou i= 0 and (1 + u2)2/u2does no depend on θ,hB1i= 0.
Finally, by P oposi ion 4.5,k∂uΦ+
2k4≤TΦ+
2U3≤K(νI0)−2. Then,
kB2k2≤
1
2
(1 + u2)2
u2−2k∂uΦ+
2+∂uΦ−
2k4≤K(νI0)−2.
Using he ope a o Gou ,d, we conside he ixed poin equa ion
X=Gou ,d ◦Fou ,d(X),(9.10)
whe e
Fou ,d(X) = B−A
1 + A|(u,θ)=( +X( ,θ),θ)
.(9.11)
I is clea ha i Xsa is ies (9.10), hen Xis a solu ion o (9.8). Le
X0=Gou ,d ◦Fou ,d(0).
P oposi ion 9.5. X0∈ X1and kX0k1≤K(νI0)−2. Fu he mo e, X0is eal analy ic.
P oo . We ha e ha Fou ,d(0) = B−A
1+A=B1+B2−A(B1+1)
1+A.
By Lemma 9.4,kB1k1≤K(νI0)−1and hB1i= 0. Then, by (1) o Lemma 9.3,kGou ,d(B1)k1≤
K(νI0)−1kB1k1≤K(νI0)−2.
Also, by Lemma 9.4,kB2k2≤K(νI0)−2. Then, he claim ollows om he bound

B2−A(B1+ 1)
1 + A2≤ kB2k2k(1 + A)−1k0+kAk2kB1+ 1k0k(1 + A)−1k0
and (2) o Lemma 9.3.
The ollowing lemma is analogous o Lemma 8.3. The only di e ence is he geome y o he
domain. Ac ually Dκdepends on κ, δ, β1and β2. Ha ing ixed some κ0,δ0,β1and β2wi h β2> β1
we will w i e b
Dκ o deno e a amily o domains wi h κ > κ0and δ < δ0and wi h he same angles
β1, β2such ha i ˜κ > κ > κ0 he dis ance om b
D˜κ o he bounda y o b
Dκis (˜κ−κ)(νI0)−1cos β1.
This implies de e mining ˜
δ, depending on ˜κso ha (δ−˜
δ) cos β2= (˜κ−κ)(νI0)−1cos β1. No e
ha a change in ˜κo o de one p oduces a change in ˜
δo o de (νI0)−1.
Lemma 9.6. Le C∈ Xsin b
Dκ×Tσand X∈ X in b
D˜κ×Tσwi h s, ≥1and ˜κ > κ. We de ine
CX( , θ) = C( +X( , θ), θ),( , θ)∈b
D˜κ×Tσ.
Then, he e exis s κ0such ha o any ˜κ > κ > κ0 he e exis K0, K1>0, depending on ˜κ, κ bu
independen on νI0, such ha he ollowing holds.
50
(1) ∂j
C∈ Xs+jin b
D˜κ×Tσand k∂j
Cks+j≤K1j!Kj
0kCks,j≥0.
(2) I X∈ X in b
D˜κ×Tσ, hen CX∈Xsin b
D˜κ×Tσand
kCXks≤ kCks
K
1−K0(νI0˜κ−1) kXk
.
(3) I X, X′∈ X in b
D˜κ×Tσ, wi h kXk ,kX′k ≤K,
kCX−CX′ks+1 ≤K(νI0˜κ−1) kCkskX−X′k .
P oo . (1) is an immedia e consequence o Cauchy es ima es in he educed domain.
Using (1) o Lemma 9.2, (2) ollows om
kCX( , θ)ks=X
j≥0
1
j!∂j
C( , θ)Xj( , θ)s
≤X
j≥0
1
j!k∂j
C( , θ)kskXkj
0
≤X
j≥0
K
j!νI0
˜κ j
k∂j
C( , θ)ks+jkXkj
≤KK1X
j≥0 νI0
˜κ
K0kXk !j
kCks
=kCks
KK1
1−νI0
˜κ K0kXk
.
(3) ollows om
C( +X( , θ), θ)−C( +X′( , θ), θ)
=Z1
0
∂ C( +X′( , θ) + s(X( , θ)−X′( , θ)), θ)ds (X−X′),
(1) and (2) and he ac ha kXk0≤K(νI0/˜κ) kXk .
To sol e equa ion (9.10), we in oduce e
Xby se ing X=X0+e
X. Then, Xis a solu ion
o (9.10) i and only i e
Xsa is ies
e
X=Gou ,d ◦e
Fou ,d(e
X),(9.12)
whe e e
Fou ,d(e
X) = Fou ,d(X0+e
X)−Fou ,d(0).
P oposi ion 9.7. The e exis s κ0such ha , o any κ > κ0, equa ion (9.12)has a unique solu ion
e
X1∈ X1wi h ke
X1k1≤K(νI0)−2/κ2. As a consequence, X=X0+e
X1∈ X1is a solu ion o (9.10),
kXk1≤K(νI0)−2and i is eal analy ic.
P oo . We i s ema k ha , by Lemma 9.2 kAk0≤K(νI0)2kAk2/κ2,kAk1≤KνI0kAk2/κ, and
aking in o accoun Lemma 9.4,(B−A)/(1 + A)∈ X1wi h k(B−A)/(1 + A)k1≤ kB−Ak1k(1 +
A)−1k0≤K(νI0)−1.
Since X0∈ X1in b
Dκwi h kX0k1≤K(νI0)−2, by (3) o Lemma 9.6, aking X′= 0,s= 1,
= 1 and ˜κ > κ we ha e ha e
Fou ,d(0) ∈ X2in b
D˜κwi h
ke
Fou ,d(0)k2≤KνI0
˜κ
B−A
1 + A1kX0k1≤Kκ−1(νI0)−2.
Hence, by (2) o Lemma 9.3,kGou ,d ◦e
Fou ,d(0)k1≤ kGou ,dkke
Fou ,d(0)k2≤K(νI0)−2/˜κ, whe e
he e kGou ,dkis he no m o he ope a o Gou ,d :X2→ X1. Le K∗= 2kGou ,d ◦e
Fou ,d(0)k1. Wi h
he same a gumen , i X, X′∈ X1wi h kXk1,kX′k1≤K∗,
ke
Fou ,d(X)−e
Fou ,d(X′)k2≤KνI0
˜κ
B−A
1 + A1kX−X′k1≤K
˜κkX−X′k1in b
D˜κ,
51
o some K > 0, independen o νI0and ˜κ. Finally, using again (2) o Lemma 9.3,
kGou ,d ◦e
Fou ,d(X)−Gou ,d ◦e
Fou ,d(X′)k1
≤ kGou ,dkke
Fou ,d(X)−e
Fou ,d(X′)k2≤kGou ,dkK
˜κkX−X′k1,
which p o es ha Gou ,d ◦e
Fou ,d is a con ac ion in b
D˜κi ˜κis la ge enough. Now we can check
ha Gou ,d ◦e
Fou ,d sends BK∗⊂ X1in o i sel and he e o e Gou ,d ◦e
Fou ,d has a unique ixed
poin in ha ball. The eal analy ici y claim ollows om he de ini ions o Fou ,d and e
Fou ,d and
Rema k 9.1.
P oposi ion 9.8. Le Xbe he unc ion, in he domain ˆ
Dκ×Tσ, gi en by P oposi ion 9.7. Then,
he map
Θ( , θ) =  +X( , θ)
θ
is a well de ined change o a iables in ˆ
Dκ×Tσ. I s in e se,
Θ−1(u, θ) = u+Y(u, θ)
θ=u−X(u, θ) + b
X(u, θ)
θ,
is well de ined and eal analy ic in ˆ
D˜κ×Tσ, o some ˜κ > κ, wi h Y∈ X1and kYk1≤K(νI0)−2.
Mo eo e Y=−X+b
Xwi h b
X∈ X1and kb
Xk1≤κ−1(νI0)−3in ˆ
D˜κ×Tσ.
P oo . The unc ion Yis he solu ion o he ixed poin equa ion
Y=P(Y),
whe e
P(Y)(u, θ) = −X(u+Y(u, θ), θ).
Clea ly, P(0) = −X∈ X1, wi h kP(0)k1=kXk1≤K(νI0)−2. We de ine K∗= 2K1kXk1.
Then, by (2) o Lemma 9.6, i Y∈ BK∗⊂ X1,kP(Y)k1≤K(νI0)−2. Since, by P oposi-
ion 9.7,X∈ X1, using (3) o Lemma 9.6,P es ic ed o BK∗is Lipschi z wi h Lipschi z cons an
K˜κ−1νI0kXk1≤K˜κ−1(νI0)−1. Hence, i is a con ac ion i ˜κ−1(νI0)−1is small enough. Now,
one easily checks ha P(BK∗)⊂ BK∗. Le Y0be he unique ixed poin o Pin BK∗. Then, w i ing
Y0=−X+b
X,
kb
Xk1=kY0+Xk1=kP(Y0)−P(0)k1≤K1
κνI0kY0k1≤K1
κ(νI0)3.
9.2 The exponen ially small o mula o Φ+−Φ−
In his sec ion we inally ob ain he o mula o he di e ence e
∆ou = Φ+−Φ− o eal alues o
uand θ. Fi s , we in oduce
Υ+(u, θ) = νI0∆in(νI0(u+Y(u, θ)−i), θ),
whe e he unc ion Yis gi en by P oposi ion 9.8.Υ+is de ined in {u∈C| ℑu < 1−κ(νI0)−1+
min{ an β1ℜu, − an β1ℜu}} which con ains he domain Dκ. We also de ine
Υ−(u, θ) = Υ+(u, θ)(9.13)
and
Υ=Υ++ Υ−.(9.14)
This de ini ion implies ha Υis eal analy ic.
We ecall ha , since Lin∆in = 0 (see Theo em 8.6), we ha e
∆in( , θ) = X
k≥1
keik(θ− ),
52
whe e he coe icien s kdo no depend on νI0. Then, in oducing
Υk=νI0 ke−kνI0, k ≥1,(9.15)
we ha e
Υ+(u, θ) = X
k≥1
Υkeik(θ−νI0u−νI0Y(u,θ))
and
Υ−(u, θ) = X
k≤−1
Υ−keik(θ−νI0u−νI0Y(u,θ)),
since Yis eal analy ic.
The unc ion Υwill be he desi ed i s o de o Φ+−Φ−. In o de o p o e his ac , we
in oduce
E(u, θ) = e
∆ou (u, θ)−Υ(u, θ)
de ined in Dκ.
P oposi ion 9.9. Le N∈Nbe ixed. The e exis s K > 0and s > 0such ha , o all (u, θ)∈
Dslog(νI0)×Tσsuch ha νI0(u−i)∈ D+
slog(νI0),α ∩−D+
slog(νI0),α,0≤j+k≤N,
|∂j
u∂k
θE(u, θ)| ≤ K(νI0)j+1 max{(νI0)−2α,(νI0)−1+αlog(νI0),(νI0)−s}
log(νI0),
whe e he domain Dκwas in oduced in (9.3)and D+
κ,α in (7.4).
P oo . Since Eis eal analy ic, i is enough o bound i o ℑu≥0. We w i e
E=E1+E2+E3,
whe e
E1(u, θ) = Φ+(u, θ)−Φ−(u, θ)−νI0T+(νI0(u−i), θ)−T−(νI0(u−i), θ),
he unc ion T+was gi en by Co olla y 6.6 and T−was in oduced in (8.1),
E2(u, θ) = νI0T+(νI0(u−i), θ)−T−(νI0(u−i), θ)−Υ+(u, θ)
and
E3(u, θ) = −Υ−(u, θ).
We bound each e m sepa a ely. We s a wi h E1. We claim ha
|E1(u, θ)| ≤ KνI0
max{(νI0)−2α,(νI0)−1+αlog(νI0)}
slog(νI0).(9.16)
Indeed, o (u, θ)∈ Dslog(νI0)×Tσsuch ha νI0(u−i)∈ D+
slog(νI0),α ∩−D+
slog(νI0),α, by
P oposi ion 7.5,
|E1(u, θ)| ≤ 2νI0kS+
2k0≤KνI0
slog(νI0)kS+
2k1≤KνI0
max{(νI0)−2α,(νI0)−1+αlog(νI0)}
slog(νI0).
To bound E2we obse e ha , by Theo em 8.6 and P oposi ions 9.5 and 9.8,
E2(u, θ) = νI0∆in(νI0(u−i) + Z(νI0(u−i), θ), θ)−∆in(νI0(u−i+Y(u, θ)), θ).
Also, aking in o accoun ha kZk1≤Kand kYk1=K(νI0)−2, we ha e ha
|Z(νI0(u−i), θ)| ≤ K
νI0|u−i|,
|νI0Y(u, θ)| ≤ K
νI0|u−i|.
53

Hence, by he mean alue heo em, o (u, θ)∈ Dslog(νI0)×Tσwe ha e ha
|E2(u, θ)| ≤ KνI0
s(νI0)slog(νI0)4.
Finally, o bound E3we i s obse e ha , o (u, θ)∈ Dslog(νI0)×Tσwi h ℑu≥0, since Yis
eal analy ic, which implies ha ℑY(ℜu, ℜθ) = 0, and kYk1=O((νI0)−2),
ℑ(u+Y(u, θ)) ≥(1 −k∂uYk0)ℑu≥(1 −K(νI0)2/(slog(νI0))2kYk1)ℑu
≥(1 −K/(slog(νI0))2)ℑu≥0,
whe e we ha e used Cauchy es ima es o ela e ∂uYand Y, and we ha e sligh ly educed he
domain by conside ing a bigge alue o s. Hence, om he de ini ion o Υ−in (9.13) and (9.15),
o (u, θ)∈ Dslog(νI0)×Tσwi h ℑu > 0,
|E3(u, θ)| ≤ X
k≤−1
νI0| −k|ek(νI0−σ)ek(νI0ℑu(1+O((νI0)−1)))
≤X
k≤−1
νI0| −k|ek(νI0−σ)≤KνI0e−(νI0−σ).
The bounds o he s a emen ollow applying s anda d Cauchy es ima es and sligh ly educing he
domain.
Finally, we need he ollowing elemen a y lemma.
Lemma 9.10. Le Ψ : Dslog(νI0)×Tσ→Cbe an analy ic unc ion. We w i e Ψ(u, θ) =
Pk∈ZΨ[k](u)eikθ. Assume
(i) LΨ=0.
(ii) The e exis s M > 0such ha |Ψ[k](±(i−is log(νI0)/(νI0)))| ≤ M.
Then, Ψ(u, θ) = Pk∈ZΛkeik(θ−νI0u)and
|Λ±k| ≤ M(νI0)kse−kνI0, k ≥1.
P oo . Since LΨ = 0, he e exis s a 2π-pe iodic unc ion Λ(φ) = Pk∈ZΛkeikφ such ha Ψ(u, θ) =
Λ(θ−νI0u). Since Ψis also pe iodic wi h espec o θ, we ha e ha Ψ(u, θ) = Pk∈ZΨ[k](u)eikθ,
ha is
Ψ[k](u) = Λke−ikνI0u.
The claim ollows e alua ing he abo e equali y a u=i−is log(νI0)/(νI0), o k≥1, and a
u=−i+is log(νI0)/(νI0), o k≤ −1.
Theo em 9.11. Fix αsuch ha min{2α, 1−α}>0. Take 0< s < min{2α, 1−α}and ix
N∈Nas in P oposi ion 9.9. The e exis s Λ0∈Rsuch ha , o all (u, θ)∈(Dslog(νI0)∩R)×Tσ,
0≤j+k≤N. Le
∂j
u∂k
θ(e
∆ou (u, θ)−Λ0) = ∂j
u∂k
θΥ(u+Y(u, θ), θ) + OνI0e−νI0
log(νI0)
=νI0e−νI02 1∂j
u∂k
θ(cos(θ−νI0u)) + O1
log(νI0),
whe e 1is gi en in Theo em 8.6. Fu he mo e, i one eplaces he iin he de ini ion o he
coe icien s io Vin (1.2)by ε i, we ha e ha
1=επ 1
4+O(ε2).(9.17)
54
P oo . By P oposi ion 9.7 and Theo em 8.6, we ha e ha
e
E(u, θ) = e
∆ou (u+X(u, θ), θ)−Υ(u+X(u, θ), θ)
sa is ies Le
E= 0. Since kXk1≤K(νI0)−2,e
Esa is ies he same bounds as Egi en by P oposi ion 9.9.
W i ing e
E(u, θ) = Pk∈Ze
Ekeik(θ−νI0u), by Lemma 9.10 we ha e ha
|e
E±1| ≤ KνI0
log(νI0)e−νI0,
|e
E±k| ≤ KνI0
log(νI0)e−kνI0(1−slog(νI0))/(νI0), k ≥2.
Hence, o (u, θ)∈(Dslog(νI0)∩R)×Tσ,
|e
E(u, θ)−e
E0| ≤ KνI0
log(νI0)e−νI0.(9.18)
Since kYk1≤K(νI0)−2, we ha e ha |Y(u, θ)| ≤ K(νI0)−2 o u∈R. Then, om he
de ini ion o Υin (9.14) and using ha 1∈R, we ha e ha , o (u, θ)∈(Dslog(νI0)∩R)×Tσ,
Υ(u, θ) = νI0
X
k≥1
ke−kνI0eik(θ−νI0(u+Y(u,θ))) +X
k≤−1
−kekνI0eik(θ−νI0(u+Y(u,θ)))

=νI0e−νI02 1cos(θ−νI0u) + O((νI0)−1).
Hence, using again kYk1≤K(νI0)−2,Ealso sa is ies (9.18), o (u, θ)∈(Dslog(νI0)∩R)×Tσ, and,
om e
∆ou = Υ + E,
e
∆ou (u, θ)−Λ0=νI0e−νI02 1cos(θ−νI0u) + O((log(νI0))−1),
whe e Λ0=e
E0. The las claim ollows immedia ely om (9.15) and Theo em 8.6.
10 Acknowledgemen s
F. B has been pa ially suppo ed by he g an PID2021-122711NB-C’21, E.F. has been pa ially
suppo ed by he g an PID2021-125535NB-I00, and P.M. has been pa ially suppo ed by he
g an PID2021-123968NB-I00, unded by he Spanish S a e Resea ch Agency h ough he p og ams
MCIN/AEI/10.13039/501100011033 and “ERDF A way o making Eu ope”.
Also, E.F. and P.M. au ho s ha e been pa ially suppo ed by he Spanish S a e Resea ch
Agency, h ough he Se e o Ochoa and Ma ía de Maez u P og am o Cen e s and Uni s o Excel-
lence in R&D (CEX2020-001084-M).
A P oo o Theo em 3.2
We i s in oduce a new ime in sys em (3.3) so ha he o igin becomes a ue saddle. Since he
solu ions o (3.3) wi h ini ial condi ion (u0, 0, 0)∈Vρwi h u0, 0≥0and u0+ 0>0sa is y
u( ) + ( )≥0while hey belong o Vρ, we de ine he new ime ssuch ha d /ds = (u+ )−1.
Equa ion (3.3) becomes
u′=u(1 + O1(u, )),
′=− (1 + O1(u, )),
′= (u+ )−1,
(A.1)
whe e ′deno es d/ds. The O1(u, ) e ms depend on sand a e uni o mly bounded in e ms o
(u, )in Vρ.
Gi en w0= (u0, 0, 0)∈Vρ, we de ine
sw0= sup {s > 0|w(˜s)∈Vρ,∀˜s∈[0, s)},(A.2)
whe e wis he solu ion o (A.1) wi h ini ial condi ion w0.
Nex lemma implies (1) o Theo em 3.2. I s p oo is pos poned o Appendix B.
55
Lemma A.1. The e exis ρ∈(0,1) and C > 0, sa is ying Cρ < 7/8, such ha he solu ion
w= (u, , )o (3.3)wi h ini ial condi ion w0= (u0, 0, 0)∈Vρwi h u0, 0>0sa is ies
log ρ
u01
1+Cρ !≤sw0≤log ρ
u01
1−Cρ !.
Mo eo e , o any 0< a ≤ρand 0< δ < a/2, he “Poinca é map"
Ψ : Σ1
a,δ →Σ0
a,δ1−Ca ,
whe e he se s Σ0
a,δ and Σ1
a,δ a e de ined in (3.4), is well de ined and, i w= (u0, a, 0)∈Σ1
a,δ and
Ψ(w) = (a, 1, 1), hen
u1+Ca
0≤ 1≤u1−Ca
0,
e
C1u−(1−Ca)/2
0≤ 1− 0≤e
C2u−(1+Ca)/2
0
o some cons an s e
C1,e
C2>0depending only on ρ.
Le w= (u, , )be a solu ion o (3.3) wi h ini ial condi ion w0∈Vρ. We in oduce
τ= /u (A.3)
and we will w i e Oi=Oi(u, ).
We ha e ha 0< τ(s)<∞, o all ssuch ha w∈Vρ. I is immedia e om (A.1) ha
dτ
ds =−(2 + O1)τ. (A.4)
The a ia ional equa ions a ound a solu ion o sys em (3.3) a e


˙
U
˙
V
˙
T
=
2u+ +O2u(1 + O1)uO2
− (1 + O1)−u−2 +O2 O2
0 0 0 

U
V
T
.(A.5)
To p o e (2) o Theo em 3.2, we will s udy he beha io o he solu ions o (A.5) wi h ini ial
condi ion U=U06= 0 along solu ions o (3.3) wi h ini ial condi ion w0= (u0, 0, 0)∈Vρ∩{u, >
0} o 0=asmall bu ixed and u0a bi a ily small.
Equa ions (A.5) become, in he ime sin which he equa ions in (A.1) a e w i en, and using
τin (A.3), 
U′
V′
T′
=

2+τ+O1
1+τ
1+O1
1+τuO1
−(1+O1)τ
1+τ−1+2τ+O1
1+τ O1
0 0 0 

U
V
T
.(A.6)
P oposi ion A.2. The e exis s α∗, wi h 0< α∗<5/12 such ha o any ρ > 0small enough,
any w= (u, , ), solu ion o (A.1)wi h w|s=0 =w0= (u0, 0, 0)∈Vρand any α∗
0∈[0, α∗], he e
exis s α: [0, sw0]→R,C∞, whe e sw0was de ined in (A.2), wi h α(0) = α∗
0, such ha , in oducing
he new a iable e
V=V+αU,
equa ion (A.6)becomes


U′
e
V′
T′
=

2+τ+O1
1+τ−α1+O1
1+τ
1+O1
1+τuO1
0−1+2τ+O1
1+τ+α1+O1
1+τ O1
0 0 0 


U
e
V
T
.(A.7)
Fu he mo e, o s∈(0, sw0],
0< α(s)<55
128
τ(s)
1 + τ(s).(A.8)
56
P oo . Gi en αand e
V=V+αU, since τ > 0, he equa ion o e
Vis
e
V′=−(1 + O1)τ
1 + τ+α′+ (3 + O1)α−α21 + O1
1 + τU(A.9)
+−1 + 2τ+O1
1 + τ+α1 + O1
1 + τe
V+ ( +αu)TO1.(A.10)
The claim will ollow inding an app op ia e solu ion o
α′=ν0+ν1α+ν2α2,(A.11)
whe e
ν0=(1 + O1)τ
1 + τ, ν1(s) = −3 + O1, ν2=1 + O1
1 + τ.
Le (w, α) = ν0+ν1α+ν2α2be he igh hand side o (A.11), whe e we ha e omi ed he
dependence o νi,i= 1,2,3, on wand s. We in oduce α0and α1, he nullclines o (A.11), by
(w, α) = ν2(α−α0(τ))(α−α1(τ)),
and R, whe e
α0(τ) = −ν1
2ν2 1−1−4ν0ν2
ν2
11/2!
=3
2+O1 1 + τ−(1 + τ)2−4
9+O1τ1/2!
=3
2+O11 + τ−pR(τ).
To comple e he p oo o P oposi ion A.2, we need he ollowing wo auxilia y lemmas.
Lemma A.3. The unc ion α0has he ollowing p ope ies. Fo (u, )∈Va( ha is, 0< τ < ∞),
(1) 2√2/3 + O1≤pR(τ)/(1 + τ)<1,
(2) limτ→∞ α0(τ) = 1/3 + O1,
(3) limτ→0α0(τ)/τ = 1/3 + O1,
(4)
d
dsα0=−(1 + O1)√R−τ+ 1 + O1
√Rα0,
(5)
−(2 + O1)
√Rα0≤d
dsα0≤ −(16/9 + O1)
√Rα0
(6) limτ→0(dα0/ds)/α0=−2 + O1.
Fu he mo e,
0< α0(τ)<11
32
τ
1 + τ.(A.12)
P oo . I ems (1) o (6) a e p o en in [GK12]. The bound (A.12) ollows om a di ec compu a ion.
Nex lemma p o ides solu ions o (A.11) close o he nullcline α0. The p oo s o he nex wo
lemmas a e gi en in Appendix B.
Lemma A.4. Fo any 0< ρ < 1small enough, he ollowing is ue. Fo any solu ion w= (u, , )
o (3.3)wi h ini ial condi ion w0∈Vρ×T, i αis a solu ion o (A.11)wi h 0≤α(s0)≤
5α0(τ(s0))/4 o some 0< s0< sw0, hen 0< α(s)<5α0(τ(s))/4 o all s∈[s0, sw0].
57