Global instability in the elliptic restricted three body problem

• Autores: Abraham de la Rosa Ibarra
• Directores de la Tesis: María Teresa Martínez-Seara i Alonso (dir. tes.), Amadeu Delshams i Valdés (dir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2014
• Idioma: inglés
• Tribunal Calificador de la Tesis: Carles Simó (presid.), Pau Martín de la Torre (secret.), Marcel Guardia Munarriz (voc.), Regina Martínez Barchino (voc.), Marian Gidea (voc.)
• Materias:
  • Matemáticas
    • Análisis y análisis funcional
      • Ecuaciones diferenciales ordinarias
• Enlaces
  • Tesis en acceso abierto en: TDX
• Resumen

  • The goal of this thesis is to show global instability or Arnold's diffusion in the elliptic restricted three body problem (ERTBP) by proving the existence of pseudo-trajectories diffusing along the phase space for certain ranges of the eccentricity of the primaries (e), the angular momentum of the comet (G) and the parameter of mass (µ). More precisely, the results presented in his thesis, are valid for G big enough, eG bounded and µ small enough. The thesis is divided in two chapters and two appendices. The chapter one, contains all the main results. After introducing the ERTBP, we use McGehee coordinates to define the infinity manifold, which turn to be a three dimensional invariant manifold in the extended phase space which behaves topologically as a Normally Hyperbolic Invariant Manifold (NHIM), although it is of parabolic type. This means that the rate of approach and departure from it along its invariant manifolds is polynomial in time, instead of exponential-like as happens in a standard NHIM. On the other hand, the inner dynamics is trivial, since it is formed by a two-parameter family of 2p-periodic orbits in the 5D extended phase space which correspond to constant solutions in the 4D phase space. As a consequence, the stable and unstable manifold of the infinity manifold are union of the stable and unstable manifolds of its periodic orbits, and as long as these manifolds intersect along transversal heteroclinic orbits, the scattering map can be defined, as De la Llave, Seara and Delshams did. Unfortunately, since the inner dynamics of the infinity manifold is so simple, the classical mechanisms of diffusion, consisting of combining the inner and outer dynamics, do not work here. Instead, as a novelty, we will be able to find two different scattering maps which will be combined in a suitable way to provide orbits whose angular momentum increases. The asymptotic formula of the scattering map relies entirely in the computation of the so called Menikov potential as defined in the works of Delshams, Gutiérrez and Seara. The first derivative of the Melnikov potential gives the first order approximation of the distance between the stable and unstable invariant manifolds of the infinity manifold whenever the parameter of mass is exponentially small. Given this setting, a series of lemmas and propositions will lead to a formula of the dominant terms of this Melnikov potential. The key idea is to compute its Fourier coefficients which will be exponentially small when the angular momentum is large and an explicit formula will be not possible, therefore and effective computation will be necessary. To do so the product eG will play a key role which lead to theorems 1.5 and 1.6, the former gives an asymptotic formula for the Melnikov potential whenever eG is samll, and the latter whenever eG is finite. Both of them requires µ to be exponentially small with respect to G, and G to be big enough. These theorems naturally produce asymptotic formulas for the scattering maps in both cases and are the base for theorems 1.15 and 1.16 which formulate the existence of pseudo-trajectories in the ERTBP. In chapter two, we provide the details and the proofs of the results concerning the asymptotic formulas, given in chapter one, for the Melnikov potential and the scattering maps, including effective bounds of every error function involved. The appendices have the more technical results needed to complete in a rigorous way every proof, but because of its nature, can be relegated to the end, to make easier to follow up the main proofs.