Title: 
Three Dimensional p-q Resonant Orbits Close to Second Species Solutions
Authors: 
Esther Barrabés
Departament d'Informātica i Matemātica Aplicada,
Universitat de Girona, 
Av. Lluís Santaló, s/n, 17071 Girona, Spain. 
barrabes@ima.udg.es

Gerard Gómez 
IEEC & Departament de Matemātica Aplicada i Anālisi,
Universitat de Barcelona,
Gran Via 585, 08007 Barcelona, Spain.}
gomez@cerber.mat.ub.es

Abstract:
      The purpose of this paper is to study, for small values of $\mu$, the
      three dimensional $p$-$q\;$resonant orbits that close to periodic
      Second Species Solutions (SSS) of the Restricted Three Body Problem.
      The work is based on an analytic study of the in-- and out--maps.  These
      maps are associated to follow, under the flow of the problem, initial
      conditions on a sphere  of radius $\mu^{\alpha}$ around the small
      primary, and consider the images of those initial points on the same
      sphere.  The out--map is associated to follow the flow forward in time
      and the in--map backwards.  For both mappings we give analytical
      expressions, in powers of the mass parameter. Once these
      expressions are obtained, we proceed to the study of the matching
      equations between both, obtaining initial conditions of orbits that
      will be ``periodic'' with an error of the order $\mu^{1-\alpha}$, for
      some $ \alpha \in (1/3,1/2)$.  Since, as $\mu \rightarrow 0$, the {\sl
      inner solution} and the {\sl outer solution} will collide with the
      small primary, these orbits will be close to SSS.