TITLE:
On the vertical families of two-dimensional tori near the triangular
points of the Bicircular problem

AUTHORS:
Enric Castella^(1,2), Angel Jorba^(2)

(1) Departament de Matem\`atica Aplicada I,
    Universitat Polit\`ecnica de Catalunya,
    Diagonal 647, 08028 Barcelona (Spain).

(2) Departament de Matem\`atica Aplicada i An\`alisi,
    Universitat de Barcelona,
    Gran Via 585, 08007 Barcelona (Spain).

E-mails: enric@maia.ub.es, angel@maia.ub.es

ABSTRACT:
This paper focuses on some aspects of the motion of a small particle
moving near the Lagrangian points of the Earth-Moon system. The model
for the motion of the particle is the so-called Bicircular problem
(BCP), that includes the effect of Earth and Moon as in the spatial
Restricted Three Body Problem (RTBP), plus the effect of the Sun as a
periodic time-dependent perturbation of the RTBP. Due to this periodic
forcing coming from the Sun, the Lagrangian points are no longer
equilibrium solutions for the BCP. On the other hand, the BCP has
three periodic orbits (with the same period as the forcing) that can
be seen as the dynamical equivalent of the Lagrangian points.

In this work, we first discuss some numerical methods for the accurate
computation of quasi-periodic solutions, and then we apply them to the
BCP to obtain families of 2-D tori in an extended neighbourhood of the
Lagrangian points. These families start on the three periodic orbits
mentioned above and they are continued in the vertical ($z$ and
$\dot{z}$) direction up to a high distance. These (Cantor) families
can be seen as the continuation, into the BCP, of the Lyapunov
family of periodic orbits of the Lagrangian points that goes in the
$(z,\dot{z})$ direction.

These results are used in a forthcoming work to find regions
where trajectories remain confined for a very long time. It is
remarkable that these regions seem to persist in the real system.