Global instability in Hamiltonian systems

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