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September  2008, 10(2&3, September): 455-483. doi: 10.3934/dcdsb.2008.10.455

Computing the scattering map in the spatial Hill's problem

Amadeu Delshams 1, , Josep J. Masdemont 2, and  Pablo Roldán 1,

1. 

Departament de Matemàtica Aplicada I, ETSEIB-Universitat Politècnica de Catalunya, Diagonal 647, E-08028 Barcelona, Spain
2. 

IEEC & Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av Diagonal 647, ETSEIB, 08028 Barcelona, Spain

Received  November 2006 Revised  July 2007 Published  June 2008

Let $A_1$ and $A_2$ be two normally hyperbolic invariant manifolds for a flow, such that the stable manifold of $A_1$ intersects the unstable manifold of $A_2$ transversally along a manifold Γ. The scattering map from $A_2$ to $A_1$ is the map that, given an asymptotic orbit in the past, associates the corresponding asymptotic orbit in the future through a heteroclinic orbit. It was originally introduced to prove the existence of orbits of unbounded energy in a perturbed Hamiltonian problem using a geometric approach.
    We recently computed the scattering map in the planar restricted three body problem using non-perturbative techniques, and we showed that it is a (nontrivial) integrable twist map.
    In the present paper, we compute the scattering map in a problem with three degrees of freedom using also non-perturbative techniques. Specifically, we compute the scattering map between the normally hyperbolic invariant manifolds $A_1$ and $A_2$ associated to the equilibrium points $L_1$ and $L_2$ in the spatial Hill's problem.
    In the planar problem, for each energy level (in a certain range) there is a unique Lyapunov periodic orbit around $L_{1,2}$. In the spatial problem, this periodic orbit is replaced by a three-dimensional invariant manifold practically full of invariant 2D tori. There are heteroclinic orbits between $A_1$ and $A_2$ connecting these invariant tori in rich combinations. Hence the scattering map in the spatial problem is more complicated, and it allows nontrivial transition chains.
    Scattering maps have application to e.g. mission design in Astrodynamics, and to the construction of diffusion orbits in the spatial Hill's problem.
Keywords: homoclinic and heteroclinic trajectories, Restricted Three-Body Problem, scattering maps, normally hyperbolic invariant manifolds, quasi-periodic motions, Arnold diffusion., normal forms, invariant tori.
Mathematics Subject Classification: Primary: 70F07, 70K44; Secondary: 70K43, 70K45, 37J4.
Citation: Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455
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