Chapter 1. Introduction 1 Chapter 2. Heuristic discussion of the mechanism 2.1. Integrable systems, resonances, secondary tori 2.2. Heuristic description of the mechanism Chapter 3. A simple model Chapter 4. Statement of rigorous results Chapter 5. Notation and definitions, resonances Chapter 6. Geometric features of the unperturbed problem Chapter 7. Persistence of the normally hyperbolic invariant manifold and its stable and unstable manifolds 7.1. Explicit calculations of the perturbed invariant manifold Chapter 8. The dynamics inΛ ε 37 8.1. A system of coordinates forΛ ε 39 8.2. Calculation of the reduced Hamiltonian 8.3. Isolating the resonances (resonant averaging) 8.3.1. The infinitesimal equations for averaging 8.3.2. The main averaging result, Theorem 8.9 8.3.3. Proof of Theorem 8.9 8.4. The non-resonant region (KAM theorem) 8.4.1. Some results on Diophantine approximation 8.4.2. The KAM Theorem for twist maps 8.5. Analyzing the resonances 8.5.1. Resonances of order 3 and higher 8.5.2. Preliminary analysis of resonances of order one or two 8.5.3. Primary and secondary tori near the first and second order resonances 62 8.5.4. Proof of Theorem 8.30 and Corollary 8.31 8.5.5. Existence of stable and unstable manifolds of periodic orbits Chapter 9. The scattering map 9.1. Some generalities about the scattering map 9.2. The scattering map in our model: definition and computation v vi CONTENTS Chapter 10. Existence of transition chains 97 10.1. Transition chains 99 10.2. The scattering map and the transversality of heteroclinic intersections 99 10.2.1. The non-resonant region and resonances of order 3 and higher 10.2.2. Resonances of first order 10.2.3. Resonances of order 2 10.3. Existence of transition chains to objects of different topological types Chapter 11. Orbits shadowing the transition chains and proof of theorem 4.1 Chapter 12. Conclusions and remarks 12.1. The role of secondary tori and the speed of diffusion 12.2. Comparison with [DLS00] 12.3. Heuristics on the genericity properties of the hypothesis and the phenomena 12.4. The hypothesis of polynomial perturbations 12.5. Involving other objects 12.6. Variational methods 12.7. Diffusion times Chapter 13. An example Acknowledgments Bibliography

We consider fast quasiperiodic perturbations with two frequencies (1=" =") of a pendulum, where is the golden mean number. The complete system has a t wo-dimensional invariant torus in a neighbourhood of the saddle point. We study the splitting of the three-dimensional invariant manifolds associated to this torus. Provided that the perturbation amplitude is small enough with respect to ", and some of its Fourier coe cients (the ones associated to Fibonacci numbers), are separated from zero, it is proved that the invariant manifolds split and that the value of the splitting, which turns out to be exponentially small with respect to ", is correctly predicted by the Melnikov function.

The splitting of separatrices for Hamiltonians with 1 1 2 degrees of freedom h(x t=") = h 0 (x) + " p h 1 (x t=") is measured. We assume that h 0 (x) = h 0 (x 1 x 2) = x 2 2 =2 + V (x 1) has a separatrix x 0 (t), h 1 (x) i s 2-periodic in , and " > 0 are independent small parameters, and p 0. Under suitable conditions of meromorphicity f o r x 0 2 (u) and the perturbation h 1 (x 0 (u)), the order`of the perturbation on the separatrix is introduced, and it is proved that, for p `, the splitting is exponentially small in ", and is given in rst order by the Melnikov function.

We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+¿j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I¿I¿Rd,f¿TdI¿I¿Rd,f¿Td, p,q¿Rnp,q¿Rn, t¿T1t¿T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in , and and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0

The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet or an asteroid) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum.The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold P + (resp. P − ). In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity P∞, which we call the manifold at parabolic infinity.On P∞ it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. Since the inner dynamics inside P∞ is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion.2000 Mathematics Subject Classification: Primary 37J40, 70F15. Keywords: Elliptic Restricted Three Body problem, parabolic motions, Manifold of parabolic motions at infinity, Arnold diffusion, splitting of separatrices, Melnikov integral. Main result and methodologyThe restricted planar elliptic three body problem (RPETBP) describes the motion q of a massless particle (a comet) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter ) with mass ratio µ revolving around their center of mass on elliptic orbits with eccentricity J . In this paper we search for trajectories of motion which show a large variation of the angular momentum G = q ×q. In other words, we search for global instability ("diffusion"This is one of the reasons why we are going to restrict ourselves to the region G ≥ C large enough and J G ≤ c small enough along this paper to get the diffusive orbits.Among the harmonics L 0, of 0 order in s, by (42), the harmonic L 0,0 appears to be the dominant one, but we will also estimate L 0,1 to get information about the variable α, and bound the rest of harmonics L 0, for ≥ 2. Among the harmonics of first order L 1,k , again by (42), the and the error functions satisfyTo bound the integral (81) for m ≥ 0 we will consider two different cases: 0 ≤ q ≤ m and 0 ≤ m < q. Let us first consider the case 0 ≤ q ≤ m. By the analyticity and periodicity of the integral we change the path of integration from (E) = 0 to E = ln(2a 2 / J )so that e iE = e iu−ln(2a 2 / J) = J 2a 2 e iu and then, by (78), (79) and (83a), (83b), 23Therefore along the complex path...