  In this paper, two models of interest for Celestial Mechanics are presented
and analysed, using both analytic and numerical techniques, from the point of
view of the possible presence of regular and/or chaotic motion, as well as the
stability of the considered orbits. The first model, presented in a Hamiltonian
formalism, can be used to describe the motion of a satellite around the Earth,
taking into account both the non-spherical shape of our planet and the
third-body gravitational influence of Sun and Moon. Using semi-analytical
techniques coming from Normal Form and Nekhoroshev theories it is possible to
provide stability estimates for the orbital elements of its geocentric motion.
The second dynamical system presented can be used as a simplified model to
describe the motion of a particle in an elliptic galaxy having a central
massive core, and is constructed as a refraction billiard where an inner
dynamics, induced by a Keplerian potential, is coupled with an external one,
where a harmonic oscillator-type potential is considered. The investigation of
the dynamics is carried on by using tools of ODEs' theory and is focused on
studying the trajectories' properties in terms of periodicity, stability and,
possibly, chaoticity.
