Global instability in Hamiltonian systems

Global Instability in Hamiltonian Systems Thesis presented to obtain the Ph. D. in Applied Mathematics by Universitat Polit`ecnica de Catalunya Rodrigo Gon¸calves Schaefer Supervisor: Amadeu Delshams May 29, 2018 Contents 1 Introduction 4 2 The first case for 2 + 1/2 degrees of freedom 11 2.1 TheSystem ................................... 11 2.2 The inner and the outer dynamics . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Innermap................................ 13 2.2.2 Scattering map: Melnikov potential and crests . . . . . . . . . . . . 14 2.3 Arnolddiffusion................................. 28 2.3.1 A geometrical proposition: The level curves of L∗(I, θ) ....... 29 2.3.2 Results about global instability . . . . . . . . . . . . . . . . . . . . 33 2.4 Thetimeofdiffusion .............................. 37 2.4.1 Accuracy of the scattering map . . . . . . . . . . . . . . . . . . . . 38 2.4.2 Estimate for the time of diffusion . . . . . . . . . . . . . . . . . . . 40 3 Second case for 2+1/2 degrees of freedom 46 3.1 Innerdynamics ................................. 46 3.2 Scatteringmap ................................. 48 3.2.1 Crests and NHIM lines . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.2 Construction of scattering maps . . . . . . . . . . . . . . . . . . . . 57 3.3 ArnoldDiffusion ................................ 64 3.3.1 ProofofTheorem1........................... 67 3.4 Piecewise smooth global scattering maps . . . . . . . . . . . . . . . . . . . 68 4 A case of 3+1/2 degrees of freedom 71 4.1 Unperturbedcase................................ 72 4.2 Innerdynamics ................................. 72 4.3 Scatteringmap ................................. 73 4.3.1 Definition of scattering map . . . . . . . . . . . . . . . . . . . . . . 73 4.3.2 Crests and NHIM lines . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3.3 Symmetry of the scattering map . . . . . . . . . . . . . . . . . . . . 78 4.4 Highways .................................... 83 1 5 Some open questions 87 5.1 Highways in piecewise smooth global scattering maps . . . . . . . . . . . . 87 5.2 About the case with 3 + 1/2 degrees of freedom . . . . . . . . . . . . . . 87 5.3 About Shadowing lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 Relation between the formulas of the scattering and separatrix maps . . . . 88 5.5 About the amount of diffusion trajectories . . . . . . . . . . . . . . . . . . 89 5.6 Andmoreandmore............................... 89 2 Acknowledgments Agraeixo especialment al meu tutor, Amadeu Delshams, per haver acceptat treballar amb mi, per la solidaritat, la paci`encia i la dedicaci´o. Agraeixo tamb´e a en Jos´e Tom´as L´azaro per la simpatia infinita i la somriure cada vegada que ens trobem al departament. A l’Eva Miranda per l’inter`es i la seva disponibilitat. Als professors Arturo Vieiro i Carles Sim´o pels comentaris despr´es dels seminaris. Me gustar´ıa agradecer a Alberto, mi compa˜nero de despacho por estos 4 a˜nos, siempre una buena compa˜n´ıa en los caf´es y congresos. Quero agradecer aos meus amigos de Cabo Frio que se mantiveram sempre presentes apesar da distˆancia, Gabriel, Paulo, Lucas, Marcus, Rhenan, Guilherme e Jonathas. Aos amigos do “Regent Mendieta” que fizeram poss´ıvel a sobrevivˆencia no primeiro ano de doutorado, Marcos, Jo˜ao e Danilo. Aos colegas de Cerdanyola Murilo, Leonardo, e Jackson. A Juliana pela companhia no per´ıodo que esteve em Barcelona. E claro, ao Ot´avio e ao Gladston, serei eternamente grato pela amizade de vocˆes...vocˆes sabem o qu˜ao dif´ıcil foi. Gostaria de agradecer tamb´em a Stefanella Boatto, por ter me dado a primeira oportunidade para que esse doutorado se tornasse realidade. Seu apoio e sua amizade. Gostaria tamb´em de agradecer a minha fam´ılia pelo apoio e amor que me transmitem a cada momento, meus pais Selma e Luiz, minhas irm˜as Juliana e Ana Carolina e aos meus sobrinhos Caio e Camila. I agraeixo molt especialment a la N´uria i a la seva fam´ılia, que ja considero com meva. Sense tu, res d’aix`o seria possible. Encara tenim moltes coses per conquerir junts. Thanks to the referees to read this thesis. I was supported by the PhD grant CNPqConselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico. 3 Chapter 1 Introduction This thesis concerns about global instability in nearly-integrable Hamiltonian systems, also called “Arnold diffusion”. In [Arn64], V.I. Arnold proposed an example of a nearlyintegrable Hamiltonian with 2 + 1/2 degrees of freedom H(q, p, ϕ, I, t) = 1 2p2+I2+ε(cos q−1) (1 + µ(sin ϕ+ cos t)) , and asserted that given any δ, K > 0, for any 0 < µ ε0, there exists a trajectory of this Hamiltonian system such that I(0) < δ and I(T)> K for some time T > 0. Notice that this a global instability result for the variable I, since ˙ I=−∂H ∂ϕ =−εµ(cos q−1) cos ϕ is zero for ε= 0, so Iremains constant, whereas Ican have a drift of finite size for any ε > 0 small enough. Arnold’s Hamiltonian can be written as a nearly-integrable Hamiltonian with 3 degrees of freedom H∗(q, p, ϕ, I, s, A) = 1 2p2+I2+A+ε(cos q−1) (1 + µ(sin ϕ+ cos s)) , which for ε= 0 is an integrable Hamiltonian h(p, I, A) = 1 2(p2+I2) + A. Since hsatisfies the (Arnold) isoenergetic non-degeneracy D2h Dh Dh>0=−16= 0, by the KAM theorem proven by Arnold in [Arn63], the 5D phase space of His filled, up to a set of relative measure O(√ε) , with 3D-invariant tori Tωwith Diophantine frequencies ω= (ω1, ω2,1): |k1ω1+k2ω2+k0| ≥ γ/|k|τfor any 0 6= (k1, k2, k0)∈Z, 4 where γ=O(√ε), and τ≥2. Since the 3D KAM invariant tori do not separate the 5D phase space, there can exist irregular orbits ‘traveling’ between tori. Arnold conjectured in the KAM theorem in 1963 that this was the general case. In the first part of this thesis we consider an a priori unstable Hamiltonian with 2+1/2 degrees of freedom Hε(p, q, I, ϕ, s) = ±p2 2+ cos q−1+I2 2+εh(q, ϕ, s) (1.1) consisting of a pendulum and a rotor plus a time periodic perturbation h(q, ϕ, s). A priori unstable Hamiltonian systems like the above one were introduced by [Loc92, CG94]. They consist on a rotor in the variables (I, ϕ) as an integrable Hamiltonian in action-angle variables, a pendulum in the variables (p, q) which carries out a separatrix associated to a saddle point, plus a small perturbation of size ε. For ε= 0, Hamiltonian (1.1) is integrable and, in particular, the action Iis constant. We want to describe the global instability in the variable Ifor |ε|non-zero but otherwise arbitrary small. For simplicity, we refer to global instability in this paper simply as Arnold diffusion. Nevertheless, it is worth remarking that originally the term Arnold diffusion was coined for a priori stable Hamiltonian systems, which are perturbations of integrable Hamiltonian systems written in action-angle variable. See [Ber10] for a careful exposition of a priori unstable and a priori stable Hamiltonian systems. For instance, replacing V(q) by εV (q), our Hamiltonian (1.1) becomes a priori stable. In that case, Arnold diffusion would consisting on finding trajectories with large deviations (p(T), I(T)) −(p(0), I(0)). This would be a much more difficult problem that the one considered here, because one has to confront to exponentially small splitting of invariant manifolds with respect to the parameter εas well as to the passage through double resonances in the action variables p, I. In particular, exponential large estimates of the time of diffusion with respect to εdue to Nekhoroshev [Nek77, LM05, BM11] would apply. The main characteristic of an a priori unstable Hamiltonian system with 2+1/2 degrees of freedom is that there exists a 3D Normally Hyperbolic Invariant Manifold (NHIM) which is a large invariant object with 4D unstable and stable invariant manifolds. Inside this NHIM there exists an inner dynamics given by a Hamiltonian system with 1+1/2 degrees of freedom. This Hamiltonian possesses 2D invariant tori which prevent global instability inside the 3D NHIM. For ε= 0, the stable and unstable invariant manifold coincide along a huge separatrix filled with homoclinic orbits to the NHIM. For small |ε| 6= 0, the unstable and stable manifolds of the NHIM in general do not coincide, but otherwise intersect transversely along 3D homoclinic invariant manifolds. Through each point on each 3D homoclinic manifold, there exists a homoclinic orbit which begins in a point of the NHIM and finishes on another point of the NHIM, not necessarily the same one. This assignment between an initial and the final point on the NHIM is called the scattering map. In practice, one must select an adequate domain for any scattering map. 5 Under the action of a scattering map, the variable Ican increase (or decrease). The geometric mechanism of global instability consists on looking for trajectories of the scattering map with a large change on the variable I. Standard shadowing arguments provide the existence of nearby trajectories of Hamiltonian (1.1) with a large change on the variable I. We are going to assume that the perturbation h(q, ϕ, s) depends on two harmonics in the variables (ϕ, s): h(q, ϕ, s) = f(q)g(ϕ, s), f(q) = cos q, g(ϕ, s) = a1cos(k1ϕ+l1s) + a2cos(k2ϕ+l2s),(1.2) with k1, k2, l1, l2∈Z. One of the main goals of this thesis is to prove that for any non-trivial perturbation a1a26= 0 depending on any two independent harmonics k1k2 l1l26= 0, there is global instability of the action Ifor any ε > 0 small enough. Our first result is that the global instability happens for any arbitrary perturbation (1.2). Theorem 1. Assume that a1a26= 0 and k1l2−k2l16= 0 in Hamiltonian (1.1)-(1.2). Then, for any I∗>0, there exists ε∗=ε∗(I∗, a1, a2)>0such that for any ε,0< ε < ε∗, there exists a trajectory (p(t), q(t), I(t), ϕ(t)) such that for some T > 0 I(0) ≤ −I∗< I∗≤I(T). Remark 2. For a rough estimate of ε∗∼exp(−πI∗/2) at least for |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) for the diffusion time, see 2.4. Analogous estimates could be obtained for all the other values of the parameters. The proof is based on the geometrical method introduced in [DLS06] and relies on the concrete computation of several scattering maps. A scattering map is a map of transverse homoclinic orbits to a NHIM. For Hamiltonian (1.1), the NHIM turns out to be simply ˜ Λε=˜ Λ = (0,0, I, ϕ, s):(I, ϕ, s)∈R×T2.(1.3) In the unperturbed case, i.e., ε= 0, for any I∗>0 the NHIM ˜ Λ possesses a 4D separatrix, that is to say, coincident stable and unstable invariant manifolds W0˜ Λ = (p0(τ), q0(τ), I, ϕ, s) : τ∈R, I ∈[−I∗, I∗],(ϕ, s)∈T2, where (p0, q0) are the separatrices to the saddle equilibrium point of the pendulum (p0(t), q0(t)) = ±2 cosh t,4 arctan e±t. In the perturbed case, i.e., for small ε > 0, Wu(˜ Λε) and Ws(˜ Λε) do not coincide (this is the so-called splitting of separatrices), and every local transversal intersection between 6 them gives rise to a (local) scattering map which is simply the correspondence between a past asymptotic motion in the NHIM to the corresponding future asymptotic motion following a homoclinic orbit. Since the NHIM has also an inner dynamics, an adequate combination of these two dynamics on the NHIM, the inner one and the outer one provided by the scattering map, generates the Arnold diffusion as long as the outer dynamics does not preserve the invariant objects of the inner dynamics. Necessity of the assumptions If the determinant ∆ := k1l2−k2l1or some coefficient a1,a2vanishes, for instance, if there is only one harmonic in g, there is no global instability for the action I. Indeed, looking at the equations associated to Hamiltonian (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 this is clear for k1=k2= 0, since in this case Iis a constant of motion. If k1or k26= 0, say k16= 0, the change of variables ¯ϕ=k1ϕ+l1s, r ¯ϕ−¯s=k2ϕ+l2s, ¯ I=k1I+l1, where r=k2/k1can be assumed to satisfy 0 ≤r≤1 without loss of generality, casts system (1.4) into ˙q=±p˙p= [±1 + ε(a1cos ¯ϕ+a2cos(r¯ϕ−¯s))] sin q ˙ ¯ϕ=¯ I˙ ¯ I=εk2 1cos q(a1sin ¯ϕ+ra2sin(r¯ϕ−¯s)) ˙ ¯s= ∆/k1 which is a Hamiltonian system with the Hamiltonian given by ¯ Hε(p, q, ¯ I, ¯ϕ, ¯s) = ±p2 2+ cos q−1+¯ I2 2 +εk2 1cos q(a1cos ¯ϕ+a2cos(r¯ϕ−¯s)) . (1.5) If ∆ = 0 Hamiltonian (1.5) is autonomous with 2 degrees of freedom, and therefore a global drift for the action Iis not possible. Only drifts of size √εare possible due to KAM theorem. Analogously one easily checks that for a1a2= 0 Hamiltonian (1.1) is integrable or autonomous. Reduction of the harmonic types Under the hypothesis (k1l2−k2l1)a1a26= 0 of Theorem 1, the case k2= 0 of Theorem 1 is proved in Chapter 2. Indeed, k2= 0 implies r:= k2/k1= 0 and it turns out from (1.5) 7 that Hamiltonian (1.1) is equivalent to the one with k1= 1, k2= 0, l1= 0, l2= 1: Hε(p, q, I, ϕ, t) = ±p2 2+ cos q−1+I2 2+εcos q(a1cos ϕ+a2cos s),(1.6) which is just the Hamiltonian studied in Chapter 2. In Chapter 3 we prove Theorem 1 for k1k26= 0 or equivalently for r∈(0,1]. For the sake of clarity we will explain in full detail and prove Theorem 1 along Section 3.3 just for r= 1, which by (1.5) is equivalent to the case k1= 1, k2= 1, l1= 0, l2=−1: Hε(p, q, I, ϕ, t) = ±p2 2+ cos q−1+I2 2+εcos q(a1cos ϕ+a2cos(ϕ−s)) .(1.7) To finish the proof of Theorem 1, in Section 3.3 we will sketch the modifications needed for the case r∈(0,1). Scattering map types By the definition given at Section 2.2.2, a scattering map is in principle only locally defined, that is, for a small ball of values of the variables (I, ϕ, s) or (I, θ =ϕ−Is), since it depends on a non-degenerate critical point τ∗=τ∗(I, ϕ, s) of a real function (2.6), depending smoothly on the variables (I, ϕ, s), already introduced in [DLS06]. In the study carried out in Section 3.2, it will be described whether, in terms of the parameter µ:= a1/a2 and the variable I, a local scattering map can or cannot be smoothly defined for all the values of the angles (ϕ, s) or θ=ϕ−Is, becoming thus a global or extended scattering map. This description will depend essentially on a geometrical characterization of the function τ∗(I, ϕ, s) in terms of the intersection of crests and NHIM lines, following [DH11]. Any degeneration of the critical point τ∗=τ∗(I, ϕ, s) may give rise to more non-degenerate critical points and a bifurcation to multiple local scattering maps or to a non global scattering map. Different critical points τ∗=τ∗(I, ϕ, s) give rise to different local scattering maps, and putting together different local scattering maps, one can sometimes obtain piecewise smooth global scattering maps, which are very useful to design paths of instability for the action I, and are simply called diffusion paths. For instance, in Chapter 2 devoted to the Hamiltonian (1.6), it will be proven that for 0 < µ =a1/a2<0.625, there exist two different global scattering maps. Among the different kinds of associated orbits of these scattering maps, there will appear two of them called highways, where the drift of the action Iwas very fast and simple. As will be described in Section 3.2, such highways do not appear for Hamiltonian (1.7). Nevertheless, as will be proven in Section 3.4, there exist piecewise smooth global scattering maps, and the possible diffusion along the discontinuity sets opens the possibility of applying the theory of piecewise smooth dynamical systems [Fil88]. About the model chosen and related work Hamiltonian (1.1) is a standard example of an a priori unstable Hamiltonian system [CG94] formed by a pendulum, a rotor and a perturbation. It is usual in the literature 8 has a non degenerate critical point τ∗=τ∗(I, ϕ, s), where L(I, ϕ, s) = Z+∞ −∞ (f(q0(σ))g(ϕ+Iσ, s +σ; 0) −f(0)g(ϕ+Iσ, s +σ; 0)) dσ. Then, for 0<|ε|small enough, there exists a unique transversal homoclinic point ˜zto ˜ Λε, which is ε-close to the point ˜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)tWs(˜ Λε).(2.7) The function Lis called the Melnikov potential of Hamiltonian (1.1). In our case, from (2.2) and (2.3) L(I, ϕ, s) = A00 +A10(I) cos ϕ+A01 cos s, (2.8) where 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 Melnikov potential, µ=a10/a01 = 0.6 and I= 1. We now look for the critical points of (2.6) which indeed are the solutions of ∂L ∂τ (I, ϕ −Iτ, s −τ) = 0. Equivalently, τ∗=τ∗(I, ϕ, s) satisfies I A10(I) sin(ϕ−I τ∗) + A10 sin(s−τ∗) = 0.(2.10) From a geometrical view-point, for any (I, ϕ, s)∈[−I∗, I∗]×T2, finding τ∗=τ∗(I, ϕ, s) satisfying (2.10) is equivalent to looking for the extrema of Lon the NHIM line R(I, ϕ, s) = {(I, ϕ −Iτ, s −τ), τ ∈R},(2.11) 15 which corresponds to the unperturbed trajectory of Hamiltonian H0through (I, ϕ, s) along the unperturbed NHIM. Thus we can define the scattering map as in [DH11]. Let Wbe an open subset of [−I∗, I∗]×T2such that the map (I, ϕ, s)∈W7→ τ∗(I, ϕ, s), where τ∗(I, ϕ, s) is a critical point of (2.6) or, equivalently, a solution of (2.10), is well defined and C2. Therefore, there exists a unique ˜zsatisfying (2.7). Let Γ = {˜z(I, ϕ, s;ε),(I, ϕ, s)∈ W}. For any ˜z∈Γ there exist unique ˜x+,−= ˜x+,−(I, ϕ, s;ε)∈˜ Λεsuch that ˜z∈ Ws ε(˜x−)∩Wu ε(˜x+). Let H+,−=[{˜x+,−(I, ϕ, s;ε),(I, ϕ, s)∈W}. We define the scattering map associated to Γ as the map S:H−−→ H+ ˜x−7−→ S(˜x−) = ˜x+. By the geometric properties of the scattering map (it is an exact symplectic map [DLS08]) we have, see [DH09] and [DH11], that the scattering map has the explicit form 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) where L∗(I, ϕ, s) = L(I, ϕ −I τ∗(I, ϕ, s), s −τ∗(I, ϕ, s)).(2.13) The new variable θ=ϕ−Is Notice that if τ∗(I, ϕ, s) is a critical point of (2.6), τ∗(I, ϕ, s)−σis a critical point of 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 critical point of the right hand side of (2.14), by the uniqueness in Wwe can conclude that 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 particular for σ=s, L∗(I, ϕ −Is, 0) = L∗(I, ϕ, s). Introducing the new variable θ=ϕ−Is, we define the Reduced Poincar´e function L∗(I, θ) := L∗(I, ϕ −Is, 0) = L∗(I, ϕ, s).(2.16) We can write the scattering map on the variables (I, θ). From (I0, ϕ0, s0) = S(I, ϕ, s), we have that 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 that θ0=θ−ε∂L∗ ∂I (I, θ)+O(ε2) and I0=I+ε∂L∗ ∂θ (I, θ)+O(ε2). Then, in the variables (I, θ), the scattering map takes the simple form S(I, θ) = I+ε∂L∗ ∂θ (I, θ) + O(ε2), θ −ε∂L∗ ∂I (I, θ) + O(ε2),(2.17) so up to O(ε2) terms, S(I, θ) is the −εtimes flow of the autonomous Hamiltonian L∗(I, θ). In particular, the iterates under the scattering map follow the level curves of L∗up to O(ε2). Remark 6. We notice that the variable θis periodic in the variable ϕand quasi-periodic in the variable s. Fixing s, then θbecomes periodic. Remark 7. Note that if for some values of (I, θ) we have that ∇L∗(I, θ) = O(ε), then ε∂L∗/∂θ(I, θ) = O(ε2) and ε∂L∗/∂I(I, θ) = O(ε2). In this case, the level curves of L∗(I, θ) do not provide the dominant part of the scattering map S. Therefore, we will be able to describe properly the scattering map through the level curves of the Reduced Poincar´e function on the set of (I, θ) such that k∇L∗(I, θ)k  ε. 17 Remark 8. Using Eq.(2.15) and setting s=σ, we have that τ∗(I, ϕ−Is, 0) = τ∗(I, ϕ, s)− s. So we can define τ∗(I, θ) := τ∗(I, ϕ, s)−s(2.18) and from (2.13) and (2.16) we can write L∗as L∗(I, θ) = L(I, θ −Iτ∗(I, θ),−τ∗(I, θ)).(2.19) Remark 9. In the variables (I, θ), the variable sdoes not appear at all in the expression (2.17) for the scattering map, at least up to O(ε2). However, sdoes appear in the expression (2.12) in the original variables (I, ϕ), so we have in (2.12) a family of scattering maps parameterized by the variable s. Playing with the parameter s, we can have scattering maps with different properties. See Lemma 14 for an application of this phenomenon. The crests For the computation of the scattering maps, we use an important geometrical object introduced in [DH11], the crests. Definition 10. Fixed I, we define by crests C(I) the curves on {(I, ϕ, s),(ϕ, s)∈T2}, satisfying I∂L ∂ϕ(I, ϕ, s) + ∂L ∂s (I, ϕ, s)=0. In our case I A10(I) sin ϕ+A01 sin s= 0.(2.20) Note that a point (I, ϕ, s) belongs to a crest C(I) if it is a minimum or maximum, or more generally, a critical point of Lalong a NHIM line (2.11), that is, τ∗(I, ϕ, s) = 0 in (2.10), see Fig. 2.4. Fig. 2.4: Level curves of Lfor µ=a10/a01 = 0.5 and I= 1.2. Crests (dashed) in blue and green and the NHIM lines in black. 18 Remark 11. Note that any critical point of L(I, ·,·) belongs to the crest C(I). In general we have two curves satisfying Eq.(2.20), the maximum crest CM(I), and the minimum crest Cm(I). The maximum crest contains the point (I, ϕ = 0, s = 0), and the minimum crest the point (I, ϕ =π, s =π). For a10 >0, a01 >0, the Melnikov function L(I, ·,·) given in (2.8) has a maximum point at the point (I, ϕ, s)=(I, 0,0), and a minimum at (I, π, π), and the function (2.6) has a maximum on CM(I), and a minimum on Cm(I). For other combinations of signs of a10, a01, the location of maxima and minima changes, but for simplicity, we have preserved the name of maximum and minimum crest. We now proceed to study the crests. By (2.9) we can rewrite Eq. (2.20) as µα(I) sin ϕ+ sin s= 0,(2.21) where α(I) = IA10(I) µA01 =sinh(π 2)I2 sinh(π I 2)and µ=a10 a01 .(2.22) Note that if |µα(I)|<1 we can write sas a function of ϕfor any value of ϕ. On the other hand, if |µα(I)|>1 we can write ϕas a function of s. So, we have two different kinds of crests: •For |α(I)|<1/|µ|, the two crests are horizontal, see Fig. 2.5(a), with CM,m(I) = {(I, ϕ, ξM,m(I, ϕ)) : ϕ∈T}, ξM(I, ϕ) = −arcsin(µα(I) sin ϕ) mod 2π(2.23) ξm(I, ϕ) = arcsin(µα(I) sin ϕ) + πmod 2π. (a) Horizontal crests: µ=a10/a01 = 0.6 and I= 1.2. (b) Vertical crests: µ=a10/a01 = 1.2 and I= 1. Fig. 2.5: Types of crests. •For |α(I)|>1/|µ|, the two crests are vertical, see Fig. 2.5(b), with CM,m(I) = {(I, ηM,m(I, s), s) : s∈T}, ηM(I, s) = −arcsin(sin s/ (µα(I))) mod 2π(2.24) ηm(I, s) = arcsin(sin s/ (µα(I))) + πmod 2π. 19 Remark 12. The case |α(I)|= 1/|µ|is singular, since both crests are piecewise NHIM lines and they touch each other at the points (ϕ, s)=(π/2,3π/2) ,(3π/2, π/2). See Fig. 2.6. Fig. 2.6: Singular case: Crests for I= 1 and µ= 1. We can describe the relation between the crests C(I) and the NHIM lines R(I, ϕ, s) through the following Proposition: Proposition 13. Consider the crest C(I)defined by (2.21) and the NHIM line R(I, ϕ, s) defined in (2.11). a) For |µ|<0.625 the crests are horizontal and the intersections between any crest and any NHIM line is transversal. b) For 0.625 ≤ |µ| ≤ 0.97 the two crests C(I)are still horizontal, but for some values of Ithere exist two NHIM lines R(I, ϕ, s)which are quadratically tangent to the crests. c) For |µ|>0.97, the same properties as stated in b) hold, except that for |µα(I)|>1, the crests C(I)are vertical. Proof. The “horizontality” of a) and b) and the “verticality” of c) are due the upper bound of |µ|. Since |α(I)|<1/0.97 (see Fig.2.7), for |µ| ≤ 0.97, the crests are horizontal, that is, they can be expressed by equations (2.23). The condition of transversality is proved in [DH11]. Essentially, the proof is to observe that |Iα(I)|<1.6 and that there exists a ϕsuch that ∂ξ(I, ϕ)/∂ϕ = 1/I if, only if, |Iα(I)|<1/|µ|(we will prove it in a slightly different context, see the proof of Proposition 20.) About the amount of NHIM lines tangents to C(I), the proof is given in subsection 2.2.2. In Figs. 2.5(a) and 2.5(b) we have displayed a segment of the the NHIM line R(I, ϕ, s), |τ|< π, and we see that it intersects each crest CM(I) and Cm(I) transversally, giving rise to two values τ∗ Mand τ∗ m, therefore to two different scattering maps. We denote by τ∗ Mthe τwith minimum absolute value such that given (I, ϕ, s), (I, ϕ −Iτ, s −τ)∈CM(I) and τ∗ m is defined analogously when (I, ϕ −Iτ, s −τ)∈Cm(I) (see [DH11]). 20 Fig. 2.7: Graph of |α(I)| Scattering maps and crests Note that τ∗ mand τ∗ Mare associated to different homoclinic points to the NHIM ˜ Λ, and consequently, to different homoclinic connections. From this we build different scattering maps. The most natural way is to associate one scattering map to each crest. And we will do this on the variables (I, ϕ, s) and (I, θ), where θ=ϕ−Is. Before, we make some considerations about the NHIM lines defined in (2.11). Note that θ:= ϕ−Is = (ϕ−Iτ)−I(s−τ), that is, θis constant on each NHIM line R(I, ϕ, s), so we will also introduce another notation for 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 if I∈Q, whereas it is a dense line on T2if I /∈Q. In this case, R(I, ϕ, s) intersects the crests C(I) along an infinite number of points. Recall (see Remark 6) that θis quasi-periodic in the variable s∈T. To avoid monodromy with respect to this variable, we are going to consider from now on, in this Chapter, sas a real variable in an interval of length 2π,−π/2< s ≤3π/2. Under this restriction, the NHIM line R(I, ϕ, s) defined in (2.11) becomes a NHIM segment 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 written as Rθ(I) = {(I, ϕ, s) : ϕ−Is =θ, (ϕ, s)∈T×(−π/2,3π/2]}.(2.26) From now on, when we refer to R(I, ϕ, s) and Rθ(I), they will be these line segments. Notice that θ∈T. We begin to consider the primary scattering map SMassociated to the maximum crest CM, that is, we look only at the intersections between the segment R(I, ϕ, s) given in (2.25) and CM(I), parameterized 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 Equation (2.27) motivates us to introduce a new variable ψ=ϕ−Iτ∗ M(I, ϕ, s) that will be useful in many contexts. The variable ψ: a variable on the crest. Let C(I) be a crest such that it can be parameterized by ξ(I, ϕ) as in (2.23). Since τ∗(I, ϕ, s) is the value of τsuch that R(I, ϕ, s), given in (2.25), intersects C(I), we define ψ:= ϕ−Iτ∗(I, ϕ, s).(2.29) By (2.18) we can also write ψin terms of the variable 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 particular, for s= 0, ξ(I, ψ) = −τ∗(I, ϕ, 0) = −τ∗(I, θ) again by (2.18) and from (2.30) we have the expression of θin terms of ψ: θ=ψ−Iξ(I, ψ).(2.32) All the relations between the variables (ϕ, s), θand ψare written in Table 2.1 and are displayed in Fig. 2.8. By the definitions of L∗(I, ϕ, s) in (2.13), and L∗(I, θ) in (2.16) and (2.19), we have that L∗(I, θ) = L∗(I, ϕ, s) = L(I, ψ, ξ(I, ψ)),(2.33) So we can define the reduced Poincar´e function in terms of (I, ψ) simply as the restriction of the Melnikov potential L(I, ϕ, s) on the crest C(I) = {(I, ψ, ξ(I, ψ), ψ ∈T)}, i.e., L∗(I, ψ) := L(I, ψ, ξ(I, ψ)),(2.34) which in our case takes the simple and computable form L∗(I, ψ) = A00 +A10(I) cos ψ+A01 cos ξ(I, ψ),(2.35) for a horizontal crest (3.21). Therefore, as (I, ψ, ξ(I, ψ)) are points on the crest, the domain of L∗(I, ·,·) is a subset of C(I). So, if there exist different subsets where L∗(I, ·,·) can be well defined, we can build different scattering maps associated to C(I). Denote 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)) from (2.33) and (2.34). We state the following lemma Lemma 14. a) The Poincar´e Reduced functions L∗ M(I, ψ)and L∗ M(I, θ)are even functions in the variable I, that is, L∗ M(I, ψ) = L∗ M(−I, ψ)and L∗ M(I, θ) = L∗ M(−I, θ), and consequently SM(I, θ)is symmetric in this variable I. The same happens for Sm(I, θ), that is, for the scattering map associated to 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 three variables on the 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: Relation between variables. b) The scattering map for a value of µand s=π, associated to the intersection between Rθ(I)and Cm(I)has the same geometrical properties as the scattering map for −µ and s= 0, associated to the intersection between Rθ(I)and CM(I), i.e., Sµ,m(I, ϕ, π) = S−µ,M(I, ϕ, 0) = S−µ,M(I, θ) Proof. a) This is an immediate consequence of the fact that function A10(I) is even and ξM(I, ϕ) is odd in the variable I, see (2.9) and (2.23). b) First, we look for τ∗ msuch that the NHIM segment Rθ(I) intersects the crest Cm(I). If we fix s=π, we have 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 have by (2.10) IA10(I) sin(ϕ−Iτ∗ m) + A01 sin(π−τ∗ m)=0, which, introducing µ(2.22), is equivalent to µα(I) sin(ϕ−Iτ∗ m) + sin(π−τ∗ m)=0,(2.37) or −µα(I) sin(ϕ−Iτ∗ m) + sin(−τ∗ m)=0.(2.38) By (2.31) and (2.23) we have that π−τ∗ m=ξm(I, ϕ −Iτ∗ m) for π/2≤ξm≤3π/2 and therefore −π/2≤ −τ∗ m≤π/2. By looking at (2.37) and (2.38), τ∗ m(I, ϕ, π) for µis solution of the same equation as τ∗ M(I, ϕ, 0) for −µ, and lies in the same interval −π/2≤ −τ∗ M≤π/2. Therefore τ∗ m(I, ϕ, π) for µis equal to τ∗ M(I, ϕ, 0) for −µ. From (2.36), L∗ µ,m(I, ϕ, π) satisfies 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, their derivatives too and this implies that Sµ,m(I, ϕ, π) = S−µ,M(I, ϕ, 0) = S−µ,M(I, θ). The importance of the part b) of this lemma is that, concerning diffusion, the study for a positive µusing SM(I, θ) is equivalent to the study for −µusing Sm(I, ϕ, π), i.e., if we ensure the diffusion for a positive µ, we can ensure it for a negative one (just changing the scattering map). Besides, since SM(I, θ) symmetric in the variable I(from the first part of the lemma), from now on we will consider always I≥0, µ > 0 and SM. Now we are going to describe the influence of the intersections between the crests and the NHIM segments with respect to the parameter µdescribed in Proposition 13 on the scattering map associated to such crests. Single scattering map: µ < 0.625 As in [DH11], assuming µ < 1/1.6 = 0.625, the crests are horizontal and there is no tangency between Rθ(I) and CM(I), so that τ∗ M(I, θ) is well defined and by (2.19) and (3.11) the reduced Poincar´e function takes the form L∗ M(I, θ) = A00 +A10(I) cos(θ−Iτ∗ M(I, θ)) + A01 cos(−τ∗ M(I, θ)),(2.39) and therefore SM(I, θ) takes the form (2.17). Example To illustrate this construction, we fix µ= 0.6. In this case the crests are horizontal for all I, and we display CM(I) parametrized by ξM(see (3.21)) in Fig.2.9 for I= 1.2. We can see how Rθ(I) intersects transversally CM(I), as well as the phase space of scattering map SMgenerated by this intersection given by the level curves of L∗ M(I, θ). Remark 15. Recall from Remark 9 that sdoes not appear in the expression (2.17) for S(I, θ) and is a parameter in the expression (2.12) for S(I, ϕ, s). Computationally, one difference is that in expression (2.12), once fixed a value of s, one throws from any “initial point” (ϕ, s) the NHIM segment R(I, ϕ, s) until it touches the crest C(I) after a time τ∗(I, ϕ, s), obtaining a value for L∗(I, ϕ, s) given by (2.13), while in expression (2.17), sis fixed equal to 0 or, equivalently, the initial point to throw the NHIM segment Rθ(I) is of the form (θ, 0) (see Fig. 2.8). Multiple scattering maps: 0.625 ≤µ≤0.97 As said before, for µ < 1/1.6 = 0.625 and any value of I, the two crests CM(I) and Cm(I) are horizontal, and the NHIM segment Rθ(I) intersects transversely each of them, giving rise to a unique scattering map SMand Smassociated to each crest. We will now explore larger values of µto detect tangencies between C(I) and Rθ(I), that is, when there exists (ϕ, I) such that ∂ξ ∂ϕ(I, ϕ) = 1/I, 24 •Case 1 |µ|<0.625, that is, 1/|µ|>1.6. Then, by (2.47) and (2.48), α(I)≤1.03 <1 µand β(I)≤1.6<1 |µ|, for all I, that is, for I > 0, B= [0,+∞). •Case 2 Note that for 0.625 ≤ |µ|<0.97 α(I)≤1.03 = 1/0.97 <1 |µ|≤1.6 = β(Iβ), and by (2.47), A= [0,+∞). But, now β(Ib)≥1/|µ|. Then there exist two values I∈Asuch that β(I)=1/|µ|. Define I+= min {I:β(I) = 1/|µ|} and I++ = max {I:β(I)=1/|µ|}.(2.49) By the characterization (2.46) of the set Bwe have B= [0, I+)∪(I++,+∞). For 0.97 ≤ |µ| ≤ 1, there exist Ia< I¯asuch that α(Ij) = 1/|µ|,j∈ {a, ¯a}and A= [0, Ia)∪(I¯a,+∞). Analogously, there exist Ib< I¯ bsuch that β(Ij) = 1/|µ|, j∈ {b,¯ b}. As Ib≤Iaand I¯a< I¯ b, we have B= [0, Ib)∪(I¯ b,+∞), see Fig. 2.12. But this the equivalent to B= [0, I+)∪(I++,+∞), where I+and I++ are given by (2.49). •Case 3 This case is similar to the Case 2 for 0.97 ≤ |µ| ≤ 1. But now, as |µ| ≥ 1, we have Ia≤Ib. So, in this case we have B= [0, Ia)∪(I¯ b,∞), or B= [0, I+)∪(I++,+∞), where I+= min {I:α(I) = 1/|µ|} and I++ = max {I:β= 1/|µ|}. Finally, we see that L∗ M(I, θ) = A00 +A01 is composed by two curves in rectangles (θ, I)∈((0, π)∪(π, 2π)) ×B. This is equivalent to prove that the derivative of this curve with respect to the variable θis different from 0 for all Iin B. For any I∈B, we compute the expression for ∂L∗ M/∂θ(I, θ) which using (2.10) and the change of variables (2.45) takes the form ∂L∗ M ∂θ (I, θ) = −A10(I) sin(ψ),(2.50) and never vanishes for ψ∈(0, π)∪(π, 2π), or equivalently, for θ∈(0, π)∪(π, 2π). Then L∗ M(I, θ) = A00 +A01 is composed by two vertical curves on B. As we have seen in Lemma 54, L∗(−I, θ) = L∗(I, θ). Then, the level curve L∗ M(I, θ) = A00 +A01 is also defined for I < 0, which concludes the proof. Remark 21. Using the expressions above for I+and I++ one can check that I+∼π 2|µ|sinh(π/2) and I++ ∼2 πlog(|2 sinh(π/2)µ|),as |µ| → +∞. 31 Definition 22. We call highways the two curves Hl⊂(0, π)×Tand Hr⊂(π, 2π)×Tsuch that L∗(I, θ) = A00 +A01. By Proposition 20, they exist at least for I∈(−∞,−I++)∪ (−I+, I+)∪(I++,+∞) for |µ| ≥ 0.615 and for any value Ifor |µ|<0.625. If a10 >0, by (2.50), ∂L∗/∂θ is positive (respectively negative) along the highway Hr(resp. Hl). If a10 <0, change Hlto Hr. Proposition 23. Consider the Hamiltonian Hε(p, q, I, ϕ, s) = ±p2 2+ cos q−1+I2 2+εcos q(a1cos ϕ+a2cos s), a1a26= 0.The highways take the form θh(I) =    arccos A2(1−f(I)) A1(I)+Iarccos(f(I)), I ≤0; arccos A2(1−f(I)) A1(I)−Iarccos(f(I)), I > 0; and θH(I) =   −arccos A2(1−f(I)) A1(I)−Iarccos(f(I)), I ≤0; −arccos A2(1−f(I)) A1(I)+Iarccos(f(I)), I > 0; where θh∈(0, π)and θH∈(π, 2π). Proof. From (2.20), (2.33) and the definition of the highways, we have the following two equations A1(I) cos(θ−Iτ∗) + A2cos(−τ∗) = A2(2.51) IA1(I) sin(θ−Iτ∗) + A2sin(−τ∗) = 0. Multiplying by Ithe first equation we obtain IA1(I) cos(θ−Iτ∗) + IA2(cos(−τ∗)−1) = 0 IA1(I) sin(θ−Iτ∗) + A2sin(−τ∗) = 0. or equivalently IA1(I) cos(θ−Iτ∗) = −IA2(cos(−τ∗)−1) IA1(I) sin(θ−Iτ∗) = −A2sin(−τ∗). We sum these two equations squared and we obtain I2A2 1(I)=[IA2(cos(−τ∗)−1)]2+A2 2sin2(−τ∗). After some arithmetical manipulations we obtain the following equation of second degree in cos(−τ∗) (I2−1)A2 2cos2(−τ∗)−2I2A2 2cos(−τ∗) + A2 2(I2+ 1) −I2A2 1(I)=0. 32 Solving this equation we have cos(−τ∗) = 2I2A2 2±p4I4A4 2−4(I2−1)A2 2[A2 2(I2+ 1) −I2A2 1(I)] 2(I2−1)A2 2 . After more arithmetical manipulation and considering that −1≤cos(−τ∗)≤1 we have cos(−τ∗) = I2A2−pA2 2+ (I2−1)I2A2 1(I) (I2−1)A2 . In order to simplify the notation we define f(I) := I2A2−pA2 2+ (I2−1)I2A2 1(I) (I2−1)A2 And therefore, ⇒ −τ∗(I, θ) = ±arccos(f(I)). Remember that we have two highways. This explains why we have found two different values for the function τ∗. Then we can rewrite the first equation of (2.51) as A1(I) cos(θ±Iarccos(f(I))) + A2f(I) = A2. This immediately implies θ=±arccos A2(1 −f(I)) A1(I)∓Iarccos(f(I)). From the four possibilities, by comparing with numerical results we obtain θh(I) =    arccos A2(1−f(I)) A1(I)+Iarccos(f(I)), I ≤0; arccos A2(1−f(I)) A1(I)−Iarccos(f(I)), I > 0; and θH(I) =   −arccos A2(1−f(I)) A1(I)−Iarccos(f(I)), I ≤0; −arccos A2(1−f(I)) A1(I)+Iarccos(f(I)), I > 0; . 2.3.2 Results about global instability Now we are going to prove two results about existence of the diffusion phenomenon in our model. The first one is a direct application of the geometrical Proposition 20 just proved and describes the diffusion that takes place close to the highways. The second is a more general type of diffusion, valid also for the values of the action Iwhere there are no highways. 33 Fig. 2.13: Highways in black for µ= 0.6. Diffusion close to highways Theorem 24. Assume that a10 a01 6= 0 in the Hamiltonian (2.1)+(2.3). Then, for any I∗there exists ε∗=ε∗(I∗)>0such that for 0< ε < ε∗, there exists a trajectory (p(t), q(t), I(t), ϕ(t)) such that for some T > 0 I(0) ≤ −I∗;I(T)≥I∗, where the admissible values for I∗=I∗(µ)satisfy •For |µ|<0.625,I∗is arbitrary I∗∈(0,+∞). •For 0.625 ≤ |µ| ≤ 1,I∗∈(0, I+), where I+= min{I > 0 : I3sinh(π/2)/sinh(πI/2) = 1/|µ|}. •For |µ| ≥ 1,I∗∈(0, I+), where I+={I > 0 : I2sinh(π/2)/sinh(πI/2) = 1/|µ|}. Proof. Recall that the reduced Poincar´e function, given in (2.39), is L∗ M(I, θ) = A00 +A10(I) cos(θ−Iτ∗ M(I, θ)) + A01 cos(−τ∗ M(I, θ)). During this proof, we denote τ∗ M(I, θ) simply by τ∗ M. For εsmall enough, the scattering map SM(I, θ) takes the form (2.17) for L∗=L∗ M, so that orbits under the scattering map are contained in the level curves of the reduced Poincar´e function L∗ M, up to error of O(ε2). Proposition 20 ensures the existence of the highways as two vertical level curves L∗ M(I, θ) = A00 +A01 for Iin •(−∞,+∞) for |µ|<0.625. •(−I+, I+),where –I+= min{I > 0 : I3sinh(π/2)/sinh(πI/2) = 1/|µ|} for 0.625 ≤ |µ| ≤ 1; –I+= min{I > 0 : I2sinh(π/2)/sinh(πI/2) = 1/|µ|} for |µ| ≥ 1. 34 Take a10 >0. Then given I∗>0 (with the restriction I∗< I+if |µ|>0.625), ∂L∗ M>0 along the highway Hr. Note that (I0, θ0) := (0,3π/2) ∈Hr. Taking any (Ii, θi)∈Hr,Ii>0, its image under the scattering map (e Ii+1,e θi+1) = SM(Ii, θi) satisfies e Ii+1−Ii=O(ε)>0 and is O(ε2)-close to Hr. Using the inner map on ˜ Λ, we find (Ii+1, θi+1) = φti+1 (e Ii+1,e θi+1)∈Hr with Ii+1 −Ii=O(ε)>0. Continuing recursively in this way, we get a pseudo-orbit {(Ii, θi), i = 0, . . . , N} ⊂ Hrwith IN≥I∗formed by applying successively the scattering map and the inner map. Using the symmetry of Hr, introducing Ii=−Iifor i < 0, we have the pseudo-orbit {(Ii, θi),|i| ≤ N} ⊂ Hr. Using standard shadowing results in [FM00, FM03] based on the existence of transverse heteroclinic orbits between nonresonant tori (changing slightly Iito obtain an irrational frequency of the inner map, if necessary) or newer results like the corollary 3.5 of [GLS14] where the recurrence property of the inner dynamics is also used, there exists a trajectory of the system such that for some T,I0≤ −I∗and I(T)≥I∗. If a10 <0, changing Hrto Hlall the previous reasoning applies. Fig. 2.14: The diffusion trajectory in SMfor µ= 0.6. The general diffusion Now we present a theorem that ensures the diffusion for all values of the parameter a10, a01 (as long as a10a01 6= 0) and for any value of I∗. Besides, we prove it using the geometrical properties of the scattering map that we have explored up to now. Theorem 25. Assume that a10 a01 6= 0 in the Hamiltonian (2.1)+(2.3). Then, for any I∗>0, there exists ε∗=ε∗(I∗)>0such that for any ε,0< ε < ε∗, there exists a trajectory (p(t), q(t), I(t), ϕ(t)) such that for some T > 0 I(0) ≤ −I∗< I∗≤I(T). Proof. Our proof consists on showing the existence of adequate orbits under several scattering maps, whose orbits will be given approximately by the level curves of the corresponding reduced Poincar´e functions, in such a way the value of Iwill be increasing. Later on, we will combine them with orbits under the inner map to produce adequate pseudo-orbits for shadowing. 35 We begin with the simplest case. Assume |µ|<0.625. In this case the highways, by Proposition 20, are defined for any value of I∈Rand Theorem 24 ensures the diffusion phenomenon. We now assume 0.625 ≤ |µ| ≤ 0.97. In this case for some value of Ithere may exist tangencies between the crests CM(I) and the NHIM lines Rθ(I). Again by Proposition 20, in this case the highways are defined for all I∈(−∞,−I++)∪(−I+, I+)∪(I++,+∞) where 0 < I+≤I++. The case I∗∈(0, I+) is contained in the result of Theorem 24. So, we are going to consider I∗∈[I+,+∞). As before, we have one SM-orbit contained in one highway where Iis increasing. We have to study the region of Iwhere the highways are not defined. Our strategy is proving the existence of a scattering map in the side of θwhere the Iis increasing, that is, for θ∈(0, π) or θ∈(π, 2π) (this depends on sign(a10)) where ∂L∗ M/∂θ is positive. Then, we will use the inner map (or another scattering map S0) for changing of pseudo-orbit (level curve) of L∗ M. In this way, we continue the growth of I. For any I∈(−I++,−I+)∪(I+, I++), there exist tangencies between CM(I) and Rθ(I), i.e., there exists ψsuch that ∂ξM/∂ψ = 1/I, and therefore there exist three different scattering maps. Consider the case with µ > 0. As we have seen in Subsection 2.2.2, ψ∈T7→ θ∈Tgiven in (2.41) is no longer a change of variables, but we have three bijections θi:Di(I)→T, i∈ {A,B,C}(see (2.42)). And for each bijection we have a scattering map associated to it. Among these three scattering maps, we will chose only one for the diffusion. Consider first the case a10 >0 (recall that the highway Hrgoes from −I+toward I+). We chose for instance, the scattering map associated to the reduced Poincar´e function 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 for ψ∈DA(I)∩(π, 2π) and therefore the iterates under the scattering map SM.A(I, θ) (2.17) associated to L∗ M,A(I, θ) increase the values of Ifor θ∈(π, 2π). Notice that by definition of DA(I) for ψ∈ DA(I)∩(π, 2π) = (ψ2,2π) with ψ2∈(π, 3π/2) (see Subsection 2.2.2) there are no tangencies between the crest and the NHIM segment. We can now proceed in the following way. We first construct a pseudo-orbit {(Ii, θi) : i= 0, . . . , N1} ⊂ Hrwith I0= 0 and IN=I+, as in the proof of Theorem 24. Note that all these points lie in the same level curve of L∗ M, that is, L∗ M(Ii, θi) = A00 +A01,i= 0, . . . , N1. Applying the inner dynamics, we get (IN1+1, θN1+1) = φtN1(IN1, θN1) with θN1+1 ∈ (θA(ψ2(IN1)),2π) and then we construct a pseudo-orbit {(Ii, θi) : i=N1+1, ..., N1+M1} ⊂ L∗ M,A(IN1+1, θN1+1) = lN1+1 with θi∈(θN1+1,2π), 2π−θN1+M1=O(ε2). Applying the inner dynamics, we get (IN1+M1+1, θN1+M1+1) = φtN1+M1(IN1+M1, θN1+M1) with θN1+M1+1 ∈ θA(ψ2(IN1+M1),2π)). Recursively, we construct pseudo-orbit {(Ii, θi) : i= N1+ 1, ..., N2} such that IN2≥I++.We finally follow the highway from I++ to I∗constructing a pseudoorbit {Ii, θi) : i= N2, ..., IN3} ⊂ Hrwith IN3=I∗. Using the symmetry properties (see Lemma 54) introducing Ii=−Iifor i < 0 we have a pseudo-orbit {(Ii, θi) : |i| ≤ N3}with I−N3=−I∗,IN3=I∗. Using now the same 36 shadowing techniques as in the proof of Theorem 25, there exists a diffusion trajectory. If a10 <0, changing Hrto Hlall the previous reasoning applies. Remark 26. For the proof of this theorem we have chosen a simple pseudo-orbit, just choosing the scattering map SM,A when it was not unique. Of course, there is a lot of freedom in choosing pseudo-orbits, and we do not claim that the one chosen here is the best one concerning minimal time of diffusion. (a) SMfor µ= 1.5 (b) SMcombined with inner map (in red) Fig. 2.15: For µ= 1.5, highways are not preserved. Inner map and scattering map can be adequately combined Remark 27. A rough estimate for ε∗=ε∗(I∗)of Theorem 25 . The scattering map S(I, θ) (2.17) is the −εtime map of the Hamiltonian L∗(I, θ) given in (2.39), up to order O(ε2). Therefore, as already noticed in Remark 7, if |∂L∗/∂θ(I, θ)| ≤ εor |∂L∗/∂I(I, θ)| ≤ ε, the level curves of L∗(I, θ) are not useful enough to describe the orbits of S. It is easy to check that ∇L∗(I, θ) only vanishes for I= 0, θ= 0, π mod 2πand that k∇L∗(I, θ)k. 8π|a10I|e−π|I|/2for |I| → +∞. Thus, in general one has to avoid small neighborhoods of (I, θ) = (0,0),(0, π) and take care in regions where |I|is very large. In particular, the highways Hl, Hrare far from (I, θ) = (0,0),(0, π) and on them k∇L∗(I, θ)k ≥ A10(I)(1 − O(β(I)µ)) &4π|a10I|e−π|I|/2for large |I|, from which we get an upper bound for ε∗(I∗), which is exponentially small in |I∗|for large |I∗|: ε∗(I∗)<4π|a10||I∗|exp(−π|I∗|/2). For smaller values of I∗, one can compute numerically the level curves of k∇L∗(I, θ)k=ε and obtain ε∗> ε∗(I∗) such that k∇L∗(I, θ)k=ε∗implies |I|>|I∗|. See Table 2.2 for some values of I∗, and µ= 0.9. 2.4 The time of diffusion In this section we will provide an estimate of the diffusion time. For simplicity, we are going to estimate the time for a diffusion using a highway (see Definition 22) as a guide, 37 I∗1 2 3 4 ε∗(I∗) 1.4 0.75 0.25 0.07 Table 2.2: Estimates of ε∗for µ= 0.9 that is, we are going to construct a pseudo-orbit close a the highway. This implies to iterate the scattering map using as initial point a point on a highway. As we have seen before, see Subsection 2.2.2, one iterate of SM(I, θ) is approximated by −εtime map of the Hamiltonian L∗ M(I, θ) up to O(ε2). However, if we iterate the scattering map a number n of times, it generates a propagated error with respect to the level curve of L∗ M(I, θ). So, first we study the error generated by niterates of the scattering map. Later, we will estimate the time of diffusion along the highway combining the scattering and the inner maps. 2.4.1 Accuracy of the scattering map Equation (2.17) for the scattering map Sis good enough up to an error of O(ε2) for understanding one iterate of S. But if we consider Sn, that is, n-iterates of S, some problems appear. These problems are related with the lack of precision of the equation (2.17): •Equation (2.17) of the scattering map has a relative error of order O(ε) and an absolute error O(ε2). Therefore, for n-iterates, when nis large, the error is propagated in a such way that it cannot be discarded. •Highways are unstable, i.e., the nearby level curves of L∗move away from highways (see instance Fig.2.9.b). Now, our goal is to show how we can control these errors along a region Uin the phase space (I, θ) close to a highway. Basically, the control is to choose a good moment and interval to apply the inner map to come back to the highway and to maintain the errors small enough. The propagated error After iterating ntimes formula (2.17) for the scattering map, one gets for (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) From now on, in this section, we will use the following notation: 38 • S(I, θ) is the scattering map, see (2.17). •ST(I, θ) = (I+ε ∂L∗/∂θ(I, θ), θ −ε ∂L∗/∂I(I, θ)) is the truncated scattering map. •S0,t(I, θ)=(I(t), θ(t)) is the solution of the Hamiltonian system ˙ I(t) = ∂L∗ ∂θ (I(t), θ(t)) ˙ θ(t) = −∂L∗ ∂I (I(t), θ(t)),(2.53) with initial condition (I(0), θ(0)) = (I, θ). Let (Ih, θh) be a point in the highway. The error between the scattering map and the level curve of the reduced Poincar´e function after n-iterates is given by kSn(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih, θh)k,(2.54) where ∆Iand ∆θare small. Note that we can rewrite (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 proceed to study each subtraction. •We begin with Sn(Ih+ ∆I, θh+ ∆θ)−Sn T(Ih+ ∆I, θh+ ∆θ). From (2.52), we can readily obtain by induction that Sn(Ih+ ∆I, θh+ ∆θ)−Sn T(Ih+ ∆I, θh+ ∆θ) = O(nε2).(2.55) •Now we consider Sn T(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih+ ∆I, θh+ ∆θ). By the definition of STwe have that Sn Tis the n-step of the Euler method with step size εin each coordinate for solving the system (2.53). It is not difficult to check the standard bound (see, for instance, [SB02]) kSn T(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih+ ∆I, θh+ ∆θ)k ≤ Lε 2[(1 + εK)n−1] ,(2.56) where 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 the Hessian matrix of L∗(I, θ). •Now we look for the last subtraction S0,nε(Ih+ ∆I, θh+ ∆θ)) −S0,nε(Ih, θh). Applying Gr¨ onwall’s inequality on the variational equation associated to the Hamiltonian vector field −∇L∗(I, θ), one gets kS0,εn(Ih+ ∆I, θh+ ∆θ)) −S0,εn(Ih, θh)k≤k(∆I, ∆θ)keKεn.(2.57) 39 We can now conclude from (2.55), (2.56) and (2.57), that the propagated error is kSn(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih, θh)k≤O(nε2) + Lε 2[(1 + εK)n−1] + k(∆I, ∆θ)keKεn To avoid large propagated errors, one has to choose nsuch that nε 1. For instance, taking n=ε−c,(2.58) with 0 < c < 1 (which implies nε 1) and k(∆I, ∆θ)k=εa,a > 0, one gets kSn(Ih+ ∆I, θh+ ∆θ)−S0,nε(Ih, θh)k=O(ε2−c, εa).(2.59) 2.4.2 Estimate for the time of diffusion In this section our goal is to estimate the time of diffusion along the highway. We have three different types of estimates associated to the time of diffusion. •The total number of iterates Nsof the scattering map. This is the number of iterates that scattering map spends to cover a piece of a level curve of the reduced Poincar´e function L∗. •The time under the flow along the homoclinic invariant manifolds of e Λ. This is the time spent by each application of the scattering map following the concrete homoclinic orbit to e Λ up to a distance δof e Λ. This time is denoted by Th=Th(δ). •The time under the inner map. This time appears if we use the inner map between iterates of the scattering map (it is sometimes called ergodization time) and we denoted it by Ti. For each iterate of the scattering map we have to consider the time Th. Besides, we have seen in the previous subsection that to control the propagated error, we iterate successively the scattering map just a number n=ε−cof times, 0 < c 1. From now on we denote this number nby Nss. So, after Nss iterates of the scattering maps we apply the inner dynamics during some time Tito come back to a distance εato the highway. Therefore, the total time spent under the inner map is bNs/NsscTi. We estimate that the diffusion time along the highway is thus Td=NsTh+bNs/NsscTi.(2.60) Theorem 28. The time of diffusion Tdclose to a highway of Hamiltonian (2.1)+(2.3) between −I∗to I∗, for any 0< I∗< I+, with I+given in Proposition 20, satisfies the following asymptotic expression Td=Ts ε2 log C ε+O(εb),for ε→0,where 0<b<1, 40 Consider the autonomous extended Hamiltonian K(I, A, ϕ, s) = I2 2+A+ε(a1cos ϕ+a2cos(ϕ−s)) ,(3.4) with associated differential equations ˙ϕ=I˙ I=ε(a1sin ϕ+a2sin(ϕ−s)) ˙s=1 ˙ A=−εa2sin(ϕ−s). This system is equivalent to the system represented by (3.2)+(3.3). We wish to eliminate the dependence on the angle variables. Consider a change of variables ε-close to the identity (ϕ, s, I, A) = g(φ, σ, J, B)=(φ, σ, J, B) + O(ε) such that it is the one-time flow for a Hamiltonian εG, i.e., g=gt=1, where gtis solution of dgt dt =J2∇εG ◦gt,where J2is the symplectic matrix 0 1 −1 0. Composing Kwith gand expanding in a Taylor series around t= 0, one obtains K◦g=K+K, εG+1 2K, εG, εG+. . . , where {·} is the Poisson bracket. Using the expansion (3.4) of K, the equation above can be written 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 want to find Gsuch that a1cos φ+a2cos(φ−σ) + nJ2 2+B, Go= 0, or equivalently, J∂G ∂φ +∂G ∂σ =a1cos φ+a2cos(φ−σ). Given a < b < 1, consider any function Ψ ∈C∞(R) satisfying Ψ(x) = 1 for x∈[−a, a] and Ψ(x) = 0 for |x| ≥ band introduce G(J, B, φ, σ) := a1 J(1 −Ψ(J)) sin φ+a2 J−1(1 −Ψ(J−1)) sin(φ−σ), Substituting the above function G(J, B, φ, σ) in (3.5) we have K◦g=J2 2+B+O(ε2),(3.6) 47 for J, J −1/∈[−b, b]. For J∈[−a, a], K◦g=J2 2+B+εa1cos φ+O(ε2).(3.7) Finally, for J−1∈[−a, a], K◦g=J2 2+B+εa2cos(φ−σ) + O(ε2).(3.8) From (3.7) and (3.8), one sees that on J= 0 and J= 1 there are resonances of first order in εwith a pendulum-like behavior. Coming back to the original variables, three kinds of invariant tori are obtained. For the first order resonance I= 0, there is a positive asuch that the invariant tori are given by F0(I, ϕ, s) = constant with F0(I, ϕ, s) = I2 2+εa1cos ϕ+O(ε2).(3.9) for I∈[−a, a]. Analogously, for the first order resonance I= 1, with F1(I, ϕ, s) = (I−1)2 2+εa2cos(ϕ−s) + O(ε2), for I−1∈[−a, a]. Remark 32. As commented in [DLS06], there exists a secondary resonance in I= 1/2, but the size of the gap in its resonant region is much smaller than the size of gaps in resonant regions associated to I= 0 and I= 1. Remark 33. For Hamiltonian (1.5) with r6= 1, the resonances take place in I= 0 and I= 1/r. From (3.6), on the non-resonant region the invariant tori has equations Fnr(I) = constant with Fnr(I) = I2 2+O(ε2). An illustration of the inner dynamics is displayed in Figure 3.1. 3.2 Scattering map We are going to explore the properties of the scattering maps of Hamiltonian (3.1). The notion of scattering map on a NHIM was introduced in [DLS00]. Let Wbe an open set of [−I∗, I∗]×T2such that the invariant manifolds of the NHIM ˜ Λ introduced in (1.3) intersect 48 Fig. 3.1: Plane ϕ×Iof inner dynamics for µ= 0.75 and ε= 0.01. transversally along a homoclinic manifold Γ = {˜z(I, ϕ, s;ε),(I, ϕ, s)∈W}so that for any ˜z∈Γ there exist unique ˜x+,−= ˜x+,−(I, ϕ, s;ε)∈˜ Λ such that ˜z∈Ws ε(x−)∩Wu ε(˜x+). Let H+,−=[{˜x+,−(I, ϕ, s;ε) : (I, ϕ, s)∈W}. The scattering map associated to Γ is the map S:H−−→ H+ ˜x−7−→ S(˜x−) = ˜x+. For the characterization of the scattering maps, it is required to select the homoclinic manifold Γ and this is done using the Poincar´e-Melnikov theory. From [DH11, DLS06], we have the following proposition (compare with Chapter 2, Prop. 5) Proposition 34. Given (I, ϕ, s)∈[−I∗, I∗]×T2, assume that the real function τ∈R7−→ L(I, ϕ −I τ, s −τ)∈R(3.10) has a non degenerate critical point τ∗=τ∗(I, ϕ, s), where L(I, ϕ, s) := Z+∞ −∞ (f(q0(σ)) −f(0)) g(ϕ+Iσ, s +σ; 0)dσ. Then, for 0< ε small enough, there exists a unique transversal homoclinic point ˜zto ˜ Λε of Hamiltonian (1.1), which is ε-close to the point ˜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)tWs(˜ Λε). The function Lis called the Melnikov potential of Hamiltonian (1.1). For the concrete Hamiltonian (3.1) it takes the form L(I, ϕ, s) = A1(I) cos ϕ+A2(I) cos(ϕ−s),(3.11) 49 where A1(I) = 2πIa1 sinh(πI/2) and A2(I) = 2π(I−1)a2 sinh(π(I−1)/2). The homoclinic manifold Γ is characterized by the function τ∗(I, ϕ, s). Once a τ∗(I, ϕ, s) is chosen, which under the conditions of Proposition 34, is locally smoothly well defined, by the geometric properties of the scattering map, see [DH09, DH11, DLS08], the scattering map has the explicit local form S(I, ϕ, s) = I+ε∂L∗ ∂ϕ (I, ϕ, s) + O(ε2), ϕ −ε∂L∗ ∂I (I, ϕ, s) + O(ε2), s, where L∗(I, ϕ, s) = L(I, ϕ −Iτ∗(I, ϕ, s), s −τ∗(I, ϕ, s)).(3.12) Notice that the variable sis fixed under the scattering map. As a consequence, see [DH11], introducing the variable θ=ϕ−Is and defining the reduced Poincar´e function L∗(I, θ) := L∗(I, ϕ −Is, 0) = L∗(I, ϕ, s),(3.13) in the variables (I, θ), the scattering map has the simple form S(I, θ) = I+ε∂L∗ ∂θ (I, θ) + O(ε2), θ −ε∂L∗ ∂I (I, θ) + O(ε2), so up to O(ε2) terms, S(I, θ) is the εtimes flow of the autonomous Hamiltonian −L∗(I, θ). In particular, the iterates under the scattering map follow the level curves of L∗up to O(ε2). 3.2.1 Crests and NHIM lines We have seen that the function τ∗plays a central role in our study. Therefore, we are interested in finding the critical points τ∗=τ∗(I, ϕ, s) of function (3.10). For our concrete case (3.11), τ∗is a solution of IA1(I) sin(ϕ−Iτ∗)+(I−1)A2(I) sin(ϕ−s−(I−1)τ∗)=0.(3.14) This equation can be viewed from two equivalently geometrical viewpoints. The first one is that to find τ∗=τ∗(I, ϕ, s) satisfying (3.14) for any (I, ϕ, s)∈[−I∗, I∗]×T2is the same as to look for the extrema of Lon the NHIM line R(I, ϕ, s) = {(I, ϕ −Iτ, s −τ) : τ∈R}.(3.15) Remark 35. Since (ϕ, s)∈T2,R(I, ϕ, s) is a closed line if I∈Qand it is a dense line on {I}×T2if I /∈Q. 50 The other viewpoint is that, fixing (I, ϕ, s), a solution τ∗of (3.14) is equivalent to finding intersections between a NHIM line (3.15) and a curve defined by IA1(I) sin ϕ+ (I−1)A2(I) sin(ϕ−s)=0. These curves are called crests, and in a general way can be defined as follows. Definition 36. [DH11] We define by Crests C(I) the curves on (I, ϕ, s), (ϕ, s)∈T2, such that ∂L ∂τ (I, ϕ −Iτ, s −τ)|τ=0 = 0,(3.16) or equivalently, I∂L ∂ϕ(I, ϕ, s) + ∂L ∂s (I, ϕ, s)=0. As in our case L(I, ϕ −Iτ, s −τ) = A1(I) cos(ϕ−Iτ) + A2(I) cos(ϕ−s−(I−1)τ), equation (3.16) takes the form (3.16). Introducing σ=ϕ−s, (3.17) equation (3.16) can be rewritten as µα(I) sin ϕ+ sin σ= 0,(3.18) for I6= 1, where µ=a1 a2 and α(I) = I2sinh(π 2(I−1)) (I−1)2sinh(πI 2).(3.19) From now on, when we refer to crests C(I) we mean the set of points (I, ϕ, σ) satisfying equation (3.18). See an illustration in Fig. 3.3. Remark 37. In Chapter 2 the crests were described on the plane (ϕ, s), whereas now such curves lie on the plane (ϕ, σ). Besides, differently from the cases studied in [DH11] and in Chapter 2, the function α(I) introduced in (3.19) is not defined for all I. More precisely, it is not defined for I= 1. For this value of I, equation (3.18) is not adequate, and one has to use (3.16) to check that for I= 1 the crests are just two vertical straight lines on the plane (ϕ, σ) given by ϕ= 0 and ϕ=π. Remark 38. For Hamiltonian (1.5) and r∈(0,1), αr(I) is not defined for I= 1/r and is given by αr(I) = I2sinh π 2(rI −1) (rI −1)2sinh πI 2.(3.20) 51 We are interested in understanding the behavior of these crests because, as we have seen in[DH11] and Chapter 2, their intersection with the NHIM lines determine the existence and behavior of scattering maps. From (3.18), when |α(I)|<1/|µ|,σcan be written as a function of ϕfor all ϕ∈Ton the crest C(I). On the other hand, if |α(I)|>1/|µ|,ϕcan be written as a function of σfor all σ∈T.These two conditions give us two kinds of crests: horizontal for |α(I)|<1/|µ| and vertical for |α(I)|>1/|µ|. These names are due to their forms on the plane (ϕ, σ). We consider the same characterization used in Chapter 2: •For |α(I)|<1/|µ|, there are two horizontal crests σ=ξM,m(I, ϕ) CM,m(I) = {(I, ϕ, ξM,m(I, ϕ)) : ϕ∈T}, ξM(I, ϕ) = −arcsin(µα(I) sin ϕ) mod 2π(3.21) ξm(I, ϕ) = arcsin(µα(I) sin ϕ) + πmod 2π. •For |α(I)|>1/|µ|, there are two vertical crests ϕ=ηM,m(I, σ) CM,m(I) = {(I, ηM,m(I, σ), σ) : σ∈T}, ηM(I, σ) = −arcsin(sin σ/ (µα(I))) mod 2π ηm(I, σ) = arcsin(sin σ/ (µα(I))) + πmod 2π. Remark 39. |α(I)|= 1/|µ|is a singular or bifurcation case. In this case, the crests are straight lines and are not differentiable in ϕ=π/2 and ϕ= 3π/2. See Fig. 2.6. Remark 40. The crest containing the point (ϕ, σ) = (0,0) will be denoted by CM(I) and the crest containing the point (ϕ, σ) = (π, π) by Cm(I). Note that the function |α(I)|is not bounded, indeed lim I→1|α(I)|= +∞. This implies that for any µthere exists a neighborhood Uof I= 1 such that for all I∈U the crests are vertical. On the other hand, since α(0) = 0 there exists a neighborhood V of I= 0 such that for all I∈Vthe crests are horizontal. We notice here a remarkable difference with the Hamiltonians studied in [DH11] and Chapter 2, where, for |µ| ≤ 0.97, all the crests are horizontal for all I. Now take a look at the properties of the function α(I) introduced in (3.19) to describe under which conditions in µthe crests are horizontal or vertical. First of all, observe that for I6= 1, α(I) is smooth and α0(I)6= 0,and for I= 1 α(I) is not bounded, indeed it has a vertical asymptote lim I→1−α(I) = −∞ and lim I→1+α(I) = +∞. 52 Given a µ6= 0, since α(0) = 0, there exists a unique Ic∈(0,1) such that |α(I)|= 1/|µ|. So, the crests are horizontal for I∈[0, Ic) and vertical for I∈(Ic,1). Others important limits are lim I→−∞ α(I) = exp(π/2) and lim I→+∞α(I) = exp(−π/2). The first limit implies that |α(I)|<exp(π/2) for I∈(−∞,0). Thus, if exp(π/2) ≤1/|µ| the crests are horizontal for I∈(−∞,0). Otherwise, if 1/|µ|<exp(π/2), there exists a unique Il∈(−∞,0) such that |α(I)|= 1/|µ|and the crests are vertical for I∈(−∞, Il) and horizontal for I∈(Il,0). The second limit implies that |α(I)|>exp(−π/2) for I∈(1,+∞). Then, if exp(−π/2) ≥ 1/|µ|, the crests are vertical for I∈[1,+∞). if exp(−π/2) <1/|µ|, there exists a unique Ir∈(1,+∞), such that the crests are vertical for any Iin [1, Ir) and horizontal for I∈(Ir,+∞). Summarizing, for 1/|µ| ≥ exp(π/2), crests are horizontal for I∈(−∞, Ic)∪(Ir,+∞) and vertical for I∈(Ic, Ir). For exp(−π/2) <1/|µ|<exp(π/2), crests are horizontal for I∈(Il, Ic)∪(Ir,+∞) and vertical for I∈(−∞, Il)∪(Ic, Ir). Finally, if 1/|µ|<exp(−π/2), crests are horizontal for I∈(Il, Ic) and vertical for I∈(−∞, Il)∪(Ic,+∞). Remark 41. For r∈(0,1), αr(I) (3.20) is not bounded on a neighborhood of the resonance I= 1/r, i.e., limI→1/r−αr(I) = −∞ and limI→1/r+αr(I)=+∞. The same behavior takes place for r= 1 and close to I= 1. On the other hand, for I→ ±∞,αr(I) has the same behavior as in the case for r= 0, limI→±∞ αr(I) = 0. This implies that for any value of µ, for Iclose enough to I= 1/r the crests are vertical, and for |I|large enough the crests are horizontal. Example To illustrate this discussion, we present a concrete example. Taking µ= 0.5, we have exp(−π/2) <1/µ = 2 <exp(π/2). In this case we have Il≈ −1.807, Ic≈0.701 and Ir≈1.367. The crests are horizontal in (−1.807,0.701) ∪(1,367,+∞) and vertical in (−∞,−1.807) ∪(0.701,1.367). We emphasize that this scenario is very different from the case in Chapter 2. There, for µ= 0.5 the crests are horizontal for all I. Now, we are going to focus on the transversality of the intersection between NHIM lines R(I, ϕ, s) and crests C(I). On the plane (ϕ, σ) the NHIM lines can be written as RI(ϕ, σ) = {(ϕ−Iτ, σ −(I−1)τ), τ ∈R},(3.22) so that its slope is (I−1)/I in such plane. Therefore, there exists an intersection between NHIM lines and crests that is not transversal if, and only if, there exists a tangent vector of C(I) at a point that is parallel to (I, I −1), or, using the parameterizations, ∂ξ ∂ϕ(I, ϕ) = I−1 Ior ∂η ∂σ(I, σ) = I I−1. 53 Considering a horizontal parameterization of C(I), the tangency condition is equivalent to ±α(I)µcos ϕ p1−µ2α2(I) sin2ϕ=I−1 I. Therefore, there exists a ϕsatisfying the above condition if, and only if, |β(I)| ≥ 1 |µ|,where β(I) = Iα(I) I−1 and ϕtakes the form ϕ=±arctan sβ(I)2−(1/µ)2 (1/µ)2−α(I)2!. In an analogous way, for a vertical parameterization η(I, σ), there are tangencies if, and only if, |β(I)| ≤ 1 |µ|with σ=±arctan  I−1 Is(1/µ)2−β(I)2 α(I)2−(1/µ)2!. Remark 42. Observe that in both cases, horizontal and vertical crests, there are tangencies if, and only if, |α(I)|− 1 |µ||β(I)|− 1 |µ|<0. The function |β(I)|is smooth in R\{1}and d|β(I)|/dI = 0 only for I= 0. Besides, we have (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). Therefore, there are three possibilities: •for 1/|µ| ≥ exp(π/2), there exist I0∈(1/2,1) and I+∈(1,+∞) such that I0and I+are solutions of |β(I)| − 1/|µ|= 0. Besides, |β(I)|<1/|µ|for I∈(−∞, I0)∪ (I+,+∞) and |β(I)|>1/|µ|for I∈(I0,1) ∪(1, I+). •for exp(−π/2) <1/|µ|<exp(π/2), there exist I−∈(−∞,0), I0∈(0,1) and I+∈(1,+∞) such that I−,I0and I+are solutions of |β(I)|−1/|µ|= 0. Besides, |β(I)|<1/|µ|for I∈(I−, I0)∪(I+,+∞) and |β(I)|>1/|µ|for I∈(−∞, I−)∪ (I0,1) ∪(1, I+). •For 1/|µ| ≤ exp(−π/2), there exist I−∈(−∞,0) and I0∈(0,1/2) such that I−and I0are solutions of |β(I)|− 1/|µ|= 0. Besides, |β(I)|<1/|µ|for I∈(I−, I0) and |β(I)|>1/|µ|for I∈(−∞, I−)∪(I0,1) ∪(1,∞). 54 Putting together this description of |β(I)|with the study about vertical and horizontal crests and adding that |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 state the proposition below. Proposition 43. Consider the two crests C(I)defined by (3.18) and the NHIM line RI(ϕ, σ)defined in (3.15) for Hamiltonian (3.1). •For |µ| ≤ exp(−π/2), there exist Ib< Ia< IA< IBsuch that –for I < Ibor IB< I,C(I)are horizontal and intersect transversally any RI(ϕ, σ); –for Ib≤I < Iaor IA< I ≤IB, the crests C(I)are horizontal, but now, there exist tangencies between C(I)and two NHIM lines RI(ϕ, σ); –for Ia< I < IA, the crests C(I)are vertical and intersect transversally any RI(ϕ, σ). •For exp(−π/2) <|µ|<exp(π/2) there exist Ib< Ia< Ic≤IC< IA< IBsuch that –for I < Ibor IC< I < IA,C(I)are vertical and intersect transversally any RI(ϕ, σ); –for Ib≤I < Ia, the crests C(I)are vertical and there exist tangencies between C(I)and two NHIM lines RI(ϕ, σ); –for Ia< I < Icor IB< I,C(I)are horizontal and intersect transversally any RI(ϕ, σ); –for IA≤I≤IB, the crests C(I)are horizontal and there exist tangencies between C(I)and two NHIM lines RI(ϕ, σ); –for Ic≤I≤IC, if Ic<1/2, the crests C(I)are vertical and there exist tangencies between C(I)and RI(ϕ, σ). If Ic= 1/2, from the properties of α(I)and β(I)this interval is just one point. If Ic>1/2, the crests C(I)are horizontal and there exist tangencies. •For |µ| ≥ exp(π/2) there exist Ib< Ia< IA< IBsuch that –for I < Ibor IB< I,C(I)are vertical and intersect transversally any RI(ϕ, σ); –for Ib≤I < Iaor IA< I ≤IB, the crests C(I)are vertical and there exist tangencies between C(I)and two NHIM lines RI(ϕ, σ); –for Ia< I < IA, the crests C(I)are horizontal and intersect transversally any RI(ϕ, σ). Remark 44. Note that we are not considering the singular case |α(I)|= 1/|µ|described in Remark 39. 55 Example Again, to illustrate this proposition, we take the case with µ= 0.5, see Fig. 3.2(a). In this case, we have |β(I)|= 1/µ for I≈ −2.942,0.595,1.85 and •for I∈(−∞,−2.942) ∪(0.701,1) ∪(1,1.367) ⇒|α(I)|>1/|µ| ⇒ vertical crests |β(I)|>1/|µ| ⇒ no tangencies •for I∈[−2.942,−1.807) ⇒|α(I)|>1/|µ| ⇒ vertical crests |β(I)| ≤ 1/|µ| ⇒ tangencies •for I∈(−1.807,0.595) ∪(1.85,+∞)⇒|α(I)|<1/|µ| ⇒ horizontal crests |β(I)|<1/|µ| ⇒ no tangencies •for I∈[0.595,0.701) ∪(1.367,1.85] ⇒|α(I)|<1/|µ| ⇒ horizontal crests |β(I)| ≥ 1/|µ| ⇒ tangencies Once more, we compare with the Hamiltonian (1.6) studied in Chapter 2. For Hamiltonian (1.6) and µ= 0.5 there is no tangency, but for Hamiltonian (3.1) we can find tangencies for horizontal and vertical crests. Indeed, for Hamiltonian (1.6) and any 0 <|µ|<0.625 there is no tangency, whereas for any µ6= 0 there are tangencies for Hamiltonian (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) |αr(I)|and |βr(I)|:µ= 0.5 and r= 0.5. Fig. 3.2: |α(I)|and |β(I)|: Behavior of the crests and tangencies. Remark 45. For r∈(0,1) in Hamiltonian (1.5), βr(I) is defined by βr(I) = Iαr(I)/(rI − 1). In this case, limI→1/r |βr(I)|= +∞and limI→±∞ |βr(I)|= 0. In Fig. 3.2(b), a comparison between the functions αr(I), βr(I) and the straight line 1/|µ|for r= 1/2 is displayed. For each crest, where it is well defined, there exists, at least, a value τ∗such that (ϕ−Iτ∗, σ −(I−1)τ∗)=(ϕ−Iτ∗, ξ(I, ϕ −Iτ∗)) or (η(I, σ −(I−1)τ∗), σ −(I−1)τ∗), which means that RI(ϕ, σ)∩ C(I)6=∅. This intersection is intrinsically associated to a homoclinic orbit to the NHIM. To make a choice about how to take such τ∗is to choose in which homoclinic manifold Γ the homoclinic points ˜z∗lie. Even more, it is to choose what scattering map we are going to use. 56 c) For I > 1, one more time α(I)>0 and sin ξ1(I, ϕ) sin(ϕ)<0, but now 0 < m = 1−1/I < 1. We first fix θ= 3π/2 and search for Isuch that 3π 2−Iτ∗(I, 3π/2) = 0 3π 2−(I−1)τ∗(I, 3π/2) = π. We obtain I= 3/2, so θ−Iτ∗ 1(I, θ)∈(0, π) for any I≥3/2 and θ∈(π, θ+= 3π/2). Consequently, sin(θ−Iτ∗ 1(I, θ)) >0 and ˙ I > 0. For the values of I∈(1,3/2) we change the strategy. We look for θ∗such that θ−Iτ∗(I, θ) = 0 θ−(I−1)τ∗(I, θ) = π. We have θ∗=πI and θ−Iτ∗ 1(I, θ∗)∈(0, π) for any I∈(1,3/2) and θ∈(π, θ∗), so ˙ I > 0. Note that θ∗<3π/2 and we can define θ+:= θ∗. Observe that for I= 1 the crests are vertical, and for I= 0, θ=θ−Iτ∗ 1(I, θ), and ˙ I > 0 for θ∈(π, 3π/2). Consider now the case of vertical crests (|α(I)µ|>1). a) For I < 0, sin η1(I, σ) sin σ=−µα(I) sin2σ≤0 and m > 1. We fix θ= 3π/2 and look for Isuch that 3π/2π−Iτ∗=π 3π/2−(I−1)τ∗(I, 3π/2) = 0. We obtain I=−1/2 and therefore, sin(θ−(I−1)τ∗ 1(I, θ)) >0 for I∈(−∞,−1/2) and θ∈(π, 3π/2). Consequently, ˙ I > 0 from (3.27). For I∈(−1/2,0), we have that θ+= (1 −I)πsatisfies θ−Iτ∗(I, θ+) = π θ+−(I−1)τ∗(I, θ+) = 0. Therefore, sin(θ−(I−1)τ∗ 1)(I, θ)>0 and ˙ I > 0 for any θ∈(π, θ+). b) For 0 < I < 1 sin η1(I, σ) sin σ≥0 and m < 0. θ+= (I+ 1)πsatisfies 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 for any θ∈(π, θ+). Note that θ+<3π/2 for I∈(0,1/2). c) Finally, for I > 1, sin η1(I, σ) sin σ≤0. We have that θ−(I−1)τ∗ 1(I, θ)∈(π, 2π), so sin(θ−(I−1)τ∗ 1(I, θ)) <0 and ˙ I > 0 for any θ∈(π, 3π/2). 63 For I= 0 the crests are horizontal. For I= 1, θ=θ−(I−1)τ∗ 1(I, θ), so ˙ I > 0 for θ∈(π, 2π). Remark 52. If a1<0, we have that there exists a θ−such that ˙ I > 0 for any θ∈(θ−, π). Remark 53. An analogous proposition holds for S2(I, θ), the scattering map associated to the graphs of ξ2and η2of C2(I). In such case, there is a θ+such that ˙ I≥0 for any θ∈(θ+,2π) where θ≥3π/2 for I∈(1/2,3/2). Note that this proposition leads us to ensure the diffusion in an analogous way to the one used to prove Theorem 25. Next, the diffusion mechanism is stated and the Arnold diffusion is proven. 3.3 Arnold Diffusion In this section we are going to complete our goal proving the existence of global instability or Arnold diffusion, that is, Theorem 1. We begin by presenting some general geometrical properties of the scattering maps that we have to take into account to prove the theorem of diffusion. The first one reduces the study of scattering maps to positive values of µ. More precisely, we have the lemma below Lemma 54. The scattering map for a value of µand s=π, associated to the intersection between R(I, ϕ, s)and Cm(I)(CM(I)) has the same geometrical properties as the scattering map for −µand s= 0, associated to the intersection between 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) Proof. First, we look for τ∗ msuch that the NHIM segment R(I, ϕ, s) intersects the crest Cm(I). If we fix s=π, we have from (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∗satisfies µα(I) sin(ϕ−Iτ∗ m) + sin(ϕ−π−(I−1)τ∗ m) = 0, or −µα(I) sin(ϕ−Iτ∗ m) + sin(ϕ−(I−1)τ∗ m) = 0. We have that ϕ−π−(I−1)τ∗ m(mod 2π) = ξm(I, ϕ −Iτ∗ m) with π/2≤ξm≤3π/2. Then, for each τ∗ mthere exists a K∈Zsuch that 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. Therefore, ϕ−(I−1)τ∗ m(mod 2π)<π 2or ϕ−(I−1)τ∗ m(mod 2π)>3π 2. We can conclude that ϕ−(I−1)τ∗ m(mod 2π) = ξM(I, ϕ −Iτ∗ m). Therefore τ∗ m(I, ϕ, π) for µis equal to τ∗ M(I, ϕ, 0) for −µ. From (3.29), L∗ µ,m(I, ϕ, π) satisfies 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, their derivatives too and this implies that Sµ m(I, ϕ, π) = S−µ M(I, ϕ, 0) = S−µ M(I, θ). From now on, just to simplify the exposition, a1and a2are considered positive. The same strategy used in Chapter 2, Section 2.3, is applied to prove the existence the diffusion: we combine the scattering map in an interval of θwhere ˙ I > 0 and the inner map to build a diffusion pseudo-orbit. Then we apply shadowing results to get the existence of a diffusion orbit. Since I= 0 and I= 1 are resonance values, the application of the inner map must be more careful, because in these resonance regions, for some orbits, the value of Idecreases in order O(√ε), i. e., the tori cannot be considered flat. We study the transversality between the foliations of invariant sets of the inner and the scattering map in resonant and non-resonant regions and its image under the scattering map S. For more details and a more general case, the reader is referred to [DH09]. Consider the resonant region associated to I= 0. In such region, the tori can be approximated by F0(I, ϕ) given in (3.9). The tranversality between invariant sets of the inner and the scattering map holds if the gradient vectors of the level curves of F0and L∗ are not parallel vectors, or equivalently, F0(I, θ),L∗(I, θ)6= 0, where {,}is the Poisson bracket, F0,L∗=∂F0 ∂θ ∂L ∂I −∂F0 ∂I ∂L ∂θ . From (3.9), the partial derivatives of F0are ∂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 have the partial derivatives given 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∗). Note that if |I|>O(ε), ∂F0/∂I dominates ∂F0/∂θ, so the Poisson bracket above can be reduced to F0,L∗≃ −∂F0 ∂I ∂L ∂θ =−IA1(I) sin(θ−Iτ∗) I−1 Expanding sin(θ−Iτ∗) in Taylor’s series around I= 0, we have sin(θ−Iτ∗) = sin θ+O(I), which implies {F0,L∗}= 0 if, and only if, θ≈0, π, assuming that O(I) is small enough. Now, we consider I=O(ε) and look at the intersections between the NHIM lines and the graph of ξ1. Note that as the value of Iis close to 0 we can assume that the crests are horizontal. Using Taylor’s series we can write 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) Taylor expanding the functions A1(I), A0 1(I) and A0 2(I) around I= 0, we obtain A1(I) = 4a1+O(I2), A0 1(I) = O(I) and A0 2(I) = a2π(πcoth π/2−2)csch(π/2) + O(I) Plugging these expressions in (3.30), we set F0,L∗=−4a1Isin θ I−1−εa1sin θ[a2π(πcoth π/2−2)csch(π/2) +4a1(π−θ) sin θ] + O(I2, Iε). Therefore, F0,L∗= 0 ⇔a1sin θ−4I I−1−εa2ππcoth π 2−2csch π 2) +ε4(π−θ) sin θ] = 0. 66 In other words, we do not have transversality if, and only if, θ= 0, π or satisfies (π−θ) sin θ=I εa1 +π(coth π/2−2)cschπ/2) 4, which is not an horizontal curve in the plane (θ, I) and is transversal to an invariant torus of the inner dynamics. For the other resonant region I= 1, F1is very similar. Assuming I−1 = O(ε), we have F1,L∗=a2sin θ4I−1 I−ε[πa1(2 −πcoth(π/2))csch(π/2) + 4a2sin θ]. Applying the same methodology, we obtain an analogous result for the other resonant region F1. In short, we conclude that the image S(Ti) of an invariant torus Tiof the inner map under the scattering map intersects tranversally another invariant torus Ti+1 of the inner map. Finally, in the non-resonant region, we notice that {Fnr,L∗}=−∂Fnr ∂I ∂L∗ ∂θ =−IA1(I) sin(θ−Iτ∗) I−1, just the same expression as the one for the resonance I= 0, so the transversality between invariant sets of the inner and the scattering map follows. Now, a constructive proof of Theorem 1 is presented. This proof is similar to the proof presented in Subsection 2.3.2 of Chapter 2, but now, there is no any piece of “highway” or fast vertical lines where |I|is large. So, the inner map is applied more times. 3.3.1 Proof of Theorem 1 Proof. We consider r= 1 in Hamiltonian (1.5). First of all we have to choose what scattering map we use. This choice depends on the sign of µas explained in Lemma 54. Assuming µ > 0, we take S1(I, θ), the global scattering map associated to the graphs of ξ1 and η1. If a1>0, by Proposition 51 for any Ithere exists an interval θ∈(π, θ+) where ˙ I > 0. Define Hrthe set (ρ, θ+)×[−I∗, I∗], where ρ=π+δis such that π < ρ < θ+ and the transversality between NHIM lines and L∗ 1holds. We first construct a pseudoorbit {(Ii, θi) : i= 0, . . . , N1} ⊂ Hrwith I0=−I∗and θN1as close as possible to ρ. Note that all these points lie in the same level curve of L∗ 1, that is, L∗ 1(I0, θ0) = L∗ 1(Ii, θi), i= 1, . . . , N1. Applying the inner dynamics, we get (IN1+1, θN1+1) = φtN1(IN1, θN1) with θN1+1 ∈(ρ, θ+) and then we construct a pseudo-orbit {(Ii, θi) : i=N1+ 1, . . . , N1+M1} ⊂ L∗ 1(IN1+1, θN1+1) = lN1+1 with θi∈(ρ, θN1+1), θ+−θN1+M1=O(ε2). Applying the inner dynamics, we get (IN1+M1+1, θN1+M1+1) = φtN1+M1(IN1+M1, θN1+M1) with θN1+M1+1 ∈(ρ, θ+). Recursively, we construct a pseudo-orbit {(Ii, θi) : i= N1+ 1,...,N2}such that IN2≥I∗. In the same way, as in the proof of Teorem 25, we can apply shadowing techniques of [FM00, FM03, GLS14], due to the fact that the inner dynamics is simple enough to satisfy 67 the required hypothesis of these references, to prove the existence of a diffusion trajectory. If a10 <0, changing Hrto Hl= (θ+, π) all the previous reasoning applies. Considering Remark 33, Remark 38, Remark 41 and Remark 45, for any r∈(0,1), an equivalent diffusion result is readily obtained. And, finally, the case for r= 0 is proved in Theorem 25 in Chapter 2. 3.4 Piecewise smooth global scattering maps In this section, the geometric freedom of the choice of τ∗is explored. Until now, only two different scattering maps have been used to build a global one, and this was enough to ensure diffusion. But, with this approach, finding a diffusion pseudo-orbit is not always easy enough and this pseudo-orbit can be also complicated. This depends simply on the “aspect” of the scattering map obtained. We now suggest a new criterion to choose τ∗: to take the minimal value for |τ∗|for any (θ, I). This provides us with a piecewise smooth global scattering map with a good property: the phase space of this scattering map which is O(ε2)-close to the level sets of the reduced Poincar´e function L∗(I, θ) associated to the chosen τ∗is simpler and“cleaner”than the phase spaces of other scattering maps displayed up to now. By a cleaner scattering map, we mean that we can easily identify and understand the orbits of the scattering maps, except for a small region which contains the tangency locus. Besides, the zones where the value of Iis increased or decreased under the scattering map is well behaved. Idecreases for θ∈(0, π) (the red region on all pictures in Fig. 3.6) and Iincreases for θ∈(π, 2π) (the green region on all pictures in Fig. 3.6). So it is easy to infer that for finding a diffusion pseudo-orbit it is enough to build a combination between the inner map and this scattering map restricted to (π, 2π), for example if an increased value of Iis wished. The same idea used in the proof of Theorem 1. Observe that the scattering maps we are now considering are a mix of the scattering maps studied previously. As an example, we illustrate the scattering map obtained for µ= 0.9. Such scattering map can be divided into three regions and in each region, the scattering map coincides with a scattering map studied before. In Fig. 3.7, for regions I (0 < θ < π/2), II (π/2< θ < 3π/2) and III (3π/2< θ < 2π) the scattering map has the following correspondence: I Extended scattering map S0(I, θ) associated to the horizontal CM(I) “under” σ=ϕ. II Extended scattering map S1(I, θ) associated to the horizontal Cm(I). III Extended scattering map S2(I, θ) associated to the horizontal CM(I) “over” σ=ϕ. If extended scattering maps are not considered and we just use scattering maps associated to horizontal and vertical crests, one can see that these scattering maps can be divided into 6 regions, i.e., they can be viewed as a combination of up to 6 scattering maps. 68 (a) Piecewise scattering map for µ= 0.3. (b) Piecewise scattering map for µ= 0.5. (c) Piecewise scattering map for µ= 0.9. (d) Piecewise scattering map for µ= 1.5. Fig. 3.6: Examples of piecewise smooth global scattering maps. The orbits of scattering maps are represented by the blue lines. In the red zones the values of Ion such orbits decrease, in the green one the values of Iincrease. Another property of these scattering maps is the loss of differentiability on the straight lines θ=π/2 and θ= 3π/2. The vector field associated to the Hamiltonian −L∗ idefined around these discontinuity lines behaves as the vector fields studied in non-smooth dynamics theory. More precisely, we can find regions with slide and unstable slide behavior [Fil88]. In a future work, we envisage to design special pseudo-orbits along these discontinuity lines using such theory. Note that these pseudo-orbits would be very similar to the “highways” defined in 2.3 of Chapter 2, so in principle, one can expect fast and simple diffusion along these discontinuity lines. 69 Fig. 3.7: A piecewise smooth global scattering map divided into 3 regions. The vertical black lines are the boundaries of the domains of smooth scattering maps. 70 Chapter 4 A case of 3+1/2 degrees of freedom After a study about an a priori unstable Hamiltonian system with 2 + 1/2 degrees of freedom, a natural question is what happens for a similar system with more degrees of freedom. In this chapter, we try to answer, at least partially, this question. Partially because we consider a particular case for 3 + 1/2 degrees of freedom. We are going to consider a generalization of the Hamiltonian considered in Chapters 2 and 3, which is given by the a priori unstable Hamiltonian with 3+1/2 degrees of freedom Hε(p, q, I1, I2, ϕ1, ϕ2, s) = ±p2 2+ cos q−1+h(I1, I2) + εf(q)g(ϕ1, ϕ2, s),(4.1) where f(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), with k= (k1, k2)∈Z2and (ϕ1, ϕ2)∈T2. Remark 55. In [DLS16], the authors dealt with k= (1,1) as an example for their results. In this thesis, we restrict our attention to the case with k= (0,0). There are two main reasons for this restriction: First, this system is a direct generalization of Hamiltonian (2.1)+(2.3) in Chapter 2. So, for this Hamiltonian, we can expect to find a similar behavior of the crests, the existence of global scattering maps and, moreover, the existence of highways. Besides, we have a well-known case to compare with the new results obtained. The second reason is that it is much easier to handle it because we reduced the number of parameters and its inner dynamics is simplified. Therefore, from now on, we are always assume g(ϕ1, ϕ2, s) = a1cos ϕ1+a2cos ϕ2+a3cos s. (4.2) For a simpler notation, we denote I= (I1, I2) and ϕ= (ϕ1, ϕ2). 71 4.1 Unperturbed case In the unperturbed case (ε= 0), such system is the Hamiltonian system with Hamiltonian H0(p, q, I, ϕ, s) = ±p2 2+ cos q−1+h(I), and equations ˙q=p˙p= sin q ˙ϕ1=ω1˙ I1= 0 ˙ϕ2=ω2˙ I2= 0 ˙s= 1, where ωi= ΩiIi,i= 1,2. This system consists of a pendulum plus two rotors. From the equations above, I1and I2are constants and the flow has the form Φt(p, q, I, ϕ)=(p(t), q(t), I, ϕ +tω), where ω= (ω1, ω2). And we have an invariant set (on the extend phase space) TI={(0,0, I, ϕ, s); ϕ, s ∈T3}. In this case, the NHIM is ˜ Λ = {(0,0, I, ϕ, s):(I, ϕ, s)∈R2×T3},(4.3) 4.2 Inner dynamics The inner dynamics is derived from the restriction of the Hamiltonian (4.1) and its equations to ˜ Λ, given in (4.3), i.e., Kε(I, ϕ, s) = h(I) + ε(a1cos ϕ1+a2cos ϕ2+a3cos s) and its equations ˙ϕ1=ω1˙ I1=εa1sin ϕ1 ˙ϕ2=ω2˙ I2=εa2sin ϕ2 ˙s= 1. Note that the inner dynamics is integrable, with first integrals F1(I1, ϕ1) = Ω1I2 1 2+a1(cos ϕ1−1) and F2(I2, ϕ2) = Ω2I2 2 2+a2(cos ϕ2−1) in involution. The inner dynamics is just the product in the spaces (I1, ϕ1), (I2, ϕ2) of the dynamics described in Fig. 2.2, so there are two resonances centered at I1= 0 and I2= 0. 72 From (4.6), it is to verify that Aiis an even function, and therefore, L∗(−I, θ) = A1cos(θ1+ω1τ∗(−I, θ))+A2cos(θ2+ω2τ∗(−I, θ))+A3cos(−τ∗(−I, θ)). (4.17) From (4.12) and (4.12), τ∗(I, θ) is the solution of ω1A1sin(θ1−ω1τ) + ω2A2sin(θ2−ω2τ) + A3sin(−τ)=0.(4.18) Analogously, τ∗(−I, θ) is the solution of −ω1A1sin(θ1+ω1τ)−ω2A2sin(θ2+ω2τ) + A3sin(−τ) = 0. Note that the above equation can be written as ω1A1sin(θ1−ω1(−τ))ω2A2sin(θ2−ω2(−τ)) + A3sin(−(−τ)) = 0.(4.19) From the local uniqueness of the solution of (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 this equality in (4.17), we obtain L∗ 0(−I, θ) = A1cos(θ1−ω1τ∗ 0(I, θ)) + A2cos(θ2−ω2τ∗ 0(I, θ)) + A3cos(τ∗ 0(I, θ)). Therefore, L∗ 0(I, θ) = L∗ 0(−I, θ). Let (I+, θ+) = S0(I, θ) and (I−, θ−) = S−1 0(−I, θ), where S−1 0is the inverse image of the scattering map. We are going to prove that I+=−I−and θ+=θ−. From (4.9) we have I+=I+ε∂L∗ 0 ∂θ (I, θ) + O(ε2) and θ+=θ−ε∂L∗ 0 ∂I (I, θ) + O(ε2). On the other hand, it is easy to verify that 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 the fact that 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 to the above case, τ∗(I, 2π−θ) is the solution of ω1A1sin(2π−θ1−ω1τ) + ω2A2sin(2π−θ2−ω2τ) + A3sin(−τ)=0. 79 Or, equivalently, ω1A1sin(θ1−ω1(−τ)) + ω2A2sin(θ2−ω2(−τ)) + A3sin(−(−τ)) = 0. This implies τ∗(I, θ) = −τ∗(I, 2π−θ). As the related scattering map depends on which interval the function τ∗(I, θ) belongs, we can write τ∗ j(I, θ) = −τ∗ −j(I, 2π−θ), j∈Z. Therefore, we have τ∗ 0(I, θ) = −τ∗ 0(I, 2π−θ). Using this equality and the 2πperiodicity of the 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). Let (I+, θ+) = S0(I, 2π−θ) and (I−, θ−) = S−1 0(I, θ), where S−1 0is the inverse image of the scattering map. We want to prove I−=I+and θ+= 2π−θ−. From (4.9) we have 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 the 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−. Theorem 62 (The general diffusion).Consider the Hamiltonian (4.1)+(4.2). Assume a1a2a36= 0 and |a1/a3|+|a2/a3|<0.625. Then, for every δ < 1there exists ε0>0such that for every 0<|ε|< ε0, given I±∈ I∗\{(0,0)}, there exists an orbit ˜x(t)and T > 0, such that |I(˜x(0)) −I−| ≤ Cδ |I(˜x(T)) −I+| ≤ Cδ Proof. Consider first the case that I2−=I2+ so that I−,I+are joined by a horizontal line γ: [0, t∗]→R2such that γ(0) = I−,γ(t∗) = I+,I1−< I1+ and γ(t)6= (0,0) for t∈[0, t∗]. Given a positive δ, define the finite open covering of the image of the curve γ N [ i=0 Bδ(γ(ti)), 80 where Bδ(γ(ti)) = {p∈R2:kγ(ti)−pk∞< δ}. Let Ii∈Bδ(γ(ti)) and Ii/∈Bδ(γ(ti+1)). This implies Ii 1< γ1(ti)< γ1(ti+1). We take the vector ui=γ(ti+1)−Ii. We want to find a vector visatisfying vi 1ui 1>0 and vi 2ui 2>0. Assume ui 2>0 (ui 1=γ1(ti+1)−Ii 1>0 ). In this case we wish vj>0, j={1,2}. This implies Ii 2< γ2(ti) = γ2(ti+1). Since vi=˙ 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, vj>0 if, and only if, θ∗ j−ωi jτ∗(Ii, θ∗)∈(π, 2π). Since the initial values (Ii, θi), we want to use the inner dynamics to displace θito a point θ∗∈(π, 2π)2. The inner dynamics is very simple and as Chapter 2 we are going to assume that it is horizontal, i.e., it is described by the equations ˙ Ij= 0 and ˙ϕj=ωj, j = 1,2. And therefore, ϕ(t) = ωt +ϕ(0). So, we wish to prove the existence of a t∗such that θ(t∗)−ωiτ∗(Ii, θ(t∗)) ∈(π, 2π)2, where θ(t) = θi+ωit. Define ψj(t) = θi j(t)−ωi jτ∗(Ii, θ(t)). Without loss of generality we can assume ωi 1≥ωi 2, we have ψ2=ωi 2 ωi 1 ψ1+¯ ψ, where ¯ ψ=θi 2−ωi 2θi 1/ωi 1. For ωi 2/ωi 1∈R\Q, (ψ1, ψ2(ψ1)) is dense in T2, then there exists at∗such that (ψ1(t∗), ψ2(t∗)) ∈(π, 2π). For ωi 2/ωi 1=p/q ∈Q,q, p ∈Z, assume without loss of generality q p > 0, this implies 0< p/q ≤1. Now, we look at ψ2(ψ1) = 2πpψ1/q +¯ ψas a rotation by the angle 2πp/q of Con the S1. So, we write rl(¯ ψ) = 2πp ql+¯ ψ. We want to prove that for any ¯ ψthere exists a l∈Nsuch that rl(¯ ψ)∈(π, 2π], so that the straight lines (ψ1, ψ2(ψ1)) intersects (π, 2π)2. Suppose by contradiction that rl(¯ ψ)∈(0, π], l∈N. Note that rl(¯ ψ) is a q-periodic function. This implies that there exists a l0∈N\0 such that rl0(¯ ψ) = ¯ ψ. Therefore, if ¯ ψ∈(π, 2π] we obtain a contradiction. So, assume ¯ ψ∈(0, π] and consider the orbit O=0, r1(¯ ψ), . . . , rq−1(¯ ψ). For q6= 1, if we sort the points of the orbit we have to obtain qequidistant points in S1. Impossible if rl(¯ ψ)∈(0, π], l∈ {0, . . . , q −1}. For q= 1, ωi 1=ωi 2and it is easy to verify that (ψ1, ψ2(ψ1)) does not intersect (π, 2π]2 only for ¯ ψ=π. We first prove the case that it does not happen. 81 We want to prove for any k∈ {0, . . . , N},Ikis δ-close to the curve γ. We have Ii+1 =Ii+εvi+O(ε2).(4.20) So, Ii+1 is δ-close to γif the following conditions are satisfied Ii+1 2< γ2(ti) + δand Ii+1 1< γ1(ti+1) + δ. From (4.20) and if we consider only the terms of the first order, these conditions are equivalent to ε < γ2(ti)−Ii 2+δ vi 2 and ε < γ1(ti+1)−Ii 1+δ vi 1 . Note that γ2(ti)−Ii 2< δ and γ(ti+1)−Ii 1> δ. Besides, vi 1, vi 2≤ kvik∞. Therefore, it is enough to require ε < γ2(ti)−Ii 2+δ kvik∞ .(4.21) Define εi= sup nε:ε < γ2(ti)−Ii 2+δ kvik∞o,we obtain for any 0 <  ≤εi,Ii+1 is δ-close to γ. For u2≤0, (4.21) takes the form ε < Ii 2−γ2(ti) + δ kvik∞ . Now we wish to obtain a similar result for any iterate of scattering map. Observe that kvik∞<4a, for any iand a= max {a1, a2}. Therefore, the result is hold if we consider ε < δ 4a. That is, we take ε0= sup ε: 0 <ε< δ 4a, and thus for any ε<ε0we obtain a pseudo-orbit δ-close to γ. Now we come back to the case where ωi 1=ωi 2and ¯ ψ=π. In this case (ψ1, ψ2(ψ1)) intersects just ((0, π)×(π, 2π)) S((π, 2π)×(0, π)). Now, we consider a finite open cover of the image of the straight line γgiven by N [ i=0 Bδ/2(γ(ti)), where Bδ/2(γ(ti)) = {p∈R2:kγ(ti)−pk∞< δ/2}. As ui 1, ui 2>0, we take vi 1>0 and vi 2<0. The image of the scattering map in the variable Iis given by Ii+1 1=Ii 1+εvi 1+O(ε2) and Ii+1 2=Ii 2+εvi 2+O(ε2).(4.22) The problem is when Ii+1 2< γ2(ti)−δ. From (4.22), Ii+1 2> γ2(ti)−δ⇔δ > γ2(ti)−Ii 2−vi 2εi 2+O(ε2). 82 As Ii∈Bδ(γ(ti)), we have γ2(ti)−Ii 2< δ/2. Besides, kvik∞<4a. Therefore, Ii+1 2> γ2(ti)−δ⇔δ/2>4aε. Or explicitly, Ii+1 2< γ2(ti)−δfor any εsatisfying ε < δ 8a. We prove now that this situation is not invariant, we mean, it is not possible in our purpose to obtain ωi+1 1=ωi+1 2and θi+1 2−θi+1 1=π. We have θi+1 2−θi+1 1=θi 2−εvi 2−θi 1+εvi 1+O(ε2) =π−εvi 2−vi 1+O(ε2). Then, θi+1 2−θi+1 1= 0 if, and only if, −ε(vi 2−vi 1) + O(ε2)=2πK,K∈Z. Since vi 1vi 2<0, K6= 0. From the definition of vi, we have vi 1=−A1(I1) sin(θ1−ω1τ∗(Ii, θi)) and vi 2=−A2(I2) sin(θ2−ω2τ∗(Ii, θi)) From ωi 1=ωi 2and θi 2=θi 1+π, we obtain A2(I1) = a2A1(I1)/a1and vi 2−vi 1=a2+a1 a1A1(I1) sin(θ1−ω1τ∗(I, θ)). Therefore εvi 2−vi 1<δ 8a4(|a1|+|a2|)< δ. So, −ε(vi 2−vi 1) + O(ε2) = 2πK is satisfied only for a delta satisfying δ > 2π+O(ε2). But this δis too big and it is out our interest. For vertical lines, the same result can be stated mutatis mutandis. For a more general case, that is, C1-curve γ: [0, t∗]→R2such that γ(0) = I−, γ(t∗) = I+, we take a stairstep curve γstep, a combination of horizontal and vertical lines, in a such way that γstep is a good enough approximation of γ, where “good enough” we mean, the result is hold for γapplying the above results (for horizontal and vertical lines) for γstep. Using the shadowing lemmas of [FM00, FM03, GLS14] we obtain the desired orbit. 4.4 Highways In analogy with Definition 22, in Chapter 2, we define a Highway as an invariant set H={(I, Θ(I))}of the Hamiltonian given by the reduced Poincar´e function L∗(I, θ) which 83 is contained in the level energy L∗(I, θ) = A3. It is therefore a Lagrangian manifold, that is, Θ(I) is gradient function, i.e., there exists a function F(I) such that Θ(I) = ∇F(I). As Θ is a gradient function, it has to satisfy the following condition ∂Θ1 ∂I2 =∂Θ2 ∂I1 . This condition is equivalent to ∂2F ∂I2∂I1 =∂2F ∂I1∂I2 . Proposition 63. Consider the Hamiltonian (4.1)+(4.2). Assume a1a2a36= 0 and |a1/a3|+ |a2/a3|<0.625. For I1and I2close to infinity, the function Ftakes the asymptotic form 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) Proof. Assume a candidate of a function F(I) given by (4.23), such that Θ = ∇F(I). Θ(I) has to satisfy the energy level for highways in the reduced Poincar´e function 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 to satisfy the equation of the crest ω1A1(I1) sin(Θ1−ω1τ∗(I, Θ)) + ω2A2(I2) sin(Θ2−ω2τ∗(I, Θ)) +A3sin(−τ∗(I, θ)) = 0.(4.25) We want to write their version for I1and I2close to infinity. Using (4.23) we notice that Θi= Θi(I) takes the form 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∗ ∞), where τ∗ ∞is an asymptotic approximation of τ∗that we are going to estimate below. First, we notice that the functions A1(I1) and A2(I2) can be approximated by Ai(Ii) = 4πaiωie−πωi/21 + e−2πωi+. . . = 4πaiωie−πωi/2+Oωie−5πωi/2. 84 From (4.13), the function τ∗(I, Θ) satisfies −τ∗(I, Θ) = −arcsin A1(I1)ω1 A3 sin(Θ1−ω1τ∗(I, Θ)) + A2(I2)ω2 A3 sin(Θ2−ω2τ∗), and therefore, τ∗ ∞≈X i=1,2 2aisinh(π/2)ω2 ie−πωi/2+Oωie−5πωi/2. Applying these estimates in Eq. (4.24) we obtain that the left hand of Eq. (4.24) satisfies 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 the same way, applying in Eq. (4.25) the estimates obtained, we have that the left hand of Eq. (4.25) satisfies −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). Therefore, up to order O(ω2 1ω2 2e−π(ω1+ω2)/2), the equation of the crest and the energy level of the reduced Poincar´e function are satisfied. We finish this chapter with an explicit equation of the highway in a special case. Proposition 64. (Highways in a very special case) Consider the Hamiltonian (4.1)+(4.2) and a1=a2=asatisfying 2|a/a3|<0.625 and Ω1= Ω2= Ω. Let O=(I0, θ0),...,(IN, θN)be an orbit in a highway, N∈Nsuch that I0 1=I0 2and θ0 1=θ0 2. Then, Ii 1=Ii 2=¯ Iiand θi 1=θi 2=¯ θifor any i∈ {0, . . . , N}and can be described by ¯ θh(¯ I) =    arccos A3(1−f−(¯ I)) A(¯ I)+ ¯ωarccos(f−(¯ I)),¯ I≤0; arccos A3(1−f−(¯ I)) A(¯ I)−¯ωarccos(f−(¯ I)), I > 0; or ¯ θH(I) =   −arccos A3(1−f−(¯ I)) A(¯ I)−¯ωarccos(f−(¯ I)),¯ I≤0; −arccos A3(1−f−(¯ I)) A(¯ I)+ ¯ωarccos(f−(¯ I)),¯ I > 0; , where f−(¯ I) = ¯ωA3−pA2 3+ (¯ω−1)¯ I2A2(¯ I)/[A3(¯ω2−1)] and ¯ω=¯ IΩ1. 85 Proof. We have that the trajectories of the scattering map are given by the ε-time flow of the Hamiltonian −L∗(I, θ) up to order O(ε2). And such flow is given by the following differential equations: ˙ 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))) , for i= 1,2. Assuming Ω1= Ω2=: Ω and a1=a2=: aand taking initial conditions satisfying I(0) = I0and θ(0) = θ0where I0 1=I0 2and θ0 1=θ0 2, the solution (I(t), θ(t)) of (4.26) satisfies θ1(t) = θ2(t) and I1(t) = I2(t). Let O=(I0, θ0),(I1, θ1),...,(IN, θN)be an orbit of the scattering map in a ε-time flow of the Hamiltonian −L∗(I, θ) up to order O(ε2), N∈N. Therefore, Il 1=Il 2and θl 1=θl 2for any l∈ {0, . . . , N}. We simply denote I1=I2and θ1=θ2. If the orbit Ois a highway, it has to satisfy two equations: the equation of the crests given in (4.12) and L∗(I, θ) = A3. But now, as I1=I2=: ¯ I,θ1=θ2=: ¯ θ, Ω1= Ω2and a1=a2, these equations can be rewritten 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, where ¯ω:= ω1=ω2and A(¯ I)=4π¯ωa/ sinh(π¯ω/2). From a similar approach used in Proposition 23, we obtain the crests are described by ¯ θh(¯ I) =    arccos A3(1−f−(¯ I)) A(¯ I)+ ¯ωarccos(f−(¯ I)),¯ I≤0; arccos A3(1−f−(¯ I)) A(¯ I)−¯ωarccos(f−(¯ I)), I > 0; and ¯ θH(I) =   −arccos A3(1−f−(¯ I)) A(¯ I)−¯ωarccos(f−(¯ I)),¯ I≤0; −arccos A3(1−f−(¯ I)) A(¯ I)+ ¯ωarccos(f−(¯ I)),¯ I > 0; , where f−(¯ I) = ¯ωA3−pA2 3+ (¯ω−1)¯ I2A2(¯ I)/[A3(¯ω2−1)]. 86 Chapter 5 Some open questions 5.1 Highways in piecewise smooth global scattering maps As we showed in Section 3.4, in the piecewise smooth global scattering maps there exist two lines of discontinuity in the vector field of the scattering map. It seems that we can define two special orbits using the theory developed by [Fil88], such that these orbits lie on the lines of discontinuity and behave like the highways defined in Chapter 1. In a future work we plan to perform numerical experiments to verify whether it is possible to find real orbits of the Hamiltonian behaving like these special orbits in this region of the phase space. After that we wish to exploit these orbits to obtain fast and simple diffusion. 5.2 About the case with 3 + 1/2 degrees of freedom For the case studied in this thesis, i.e., the Hamiltonian system given by (4.1)+(4.2), in Proposition 63 we obtain an asymptotic approximation of the highways. A next step is to check that this approximation is good enough in order to continue globally those highways to obtain a global description. Besides, here we have presented results for a restricted set of values of a1and a2, more precisely, for a1and a2satisfying |a1|+|a2| ≤ 0.625. And we have obtained similar results to the part of the results in Chapter 2. The next step is to eliminate this restriction over the values of a1and a2and to study the bifurcation of crests, the bifurcation of the scattering maps and the existence of the highways. Finally, we expect to study the case of a complementary perturbation with respect to (4.2) to cover the complete family case, in an analogous way that we have done in Chapter 3. 87 5.3 About Shadowing lemmas Taking into account our numerical experiments, the geometrical mechanisms used in our proof and our estimates for the time, we wish to understand the real role played by the inner map in this mechanism. Our main question is: Is it really necessary to use the inner dynamics in the building of a pseudo-orbit to guarantee the existence of real orbit of the system? In our theorems we were able to use the results of [FM00, FM03, GLS14]. In [GLS14] they proved a shadowing lemma for pseudo-orbits built by using a number of iterates of scattering map. But the result appears not to be very practical for fast diffusion. In the ongoing work we plan to use numerical experiments to verify the existence of real orbits close to pseudo-orbits of a scattering map (or a combination of multiple scattering maps). Besides, in the future we wish to carry out an analytic approach as well. 5.4 Relation between the formulas of the scattering and separatrix maps The separatrix map was introduced by Zaslavskii and Filonenko in [ZF68], and has been studied and developed in [Tre98, Tre02, Pif06, PT07, GKZ16, DT16]. Under certain conditions we believe that the formulas obtained in [Tre02] can be improved 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+¯ t∂hω0 λlog κω0 λ+O1, where λ, κ and µare functions of I∗,¯ tis an integer. s+¯ t+∂hω0 λlog κω0 λ< c−1 and O1=O(ε1 4)(ε7 8) log2ε,O2=O(ε1 4)(ε5 4) log2εand L∗is a reduced Poincar´e function. Since the Scattering map takes the explicit form Sε(I, θ) = I+ε∂ ∂θL∗(I, θ) + O(ε2), θ −ε∂ ∂I L∗(I, θ) + O(ε2). We expect to verify analytically and numerically these equations. 88