TITLE:
Invariant Manifolds near $L_1$ and $L_2$ in the Sun-Jupiter
Elliptic Restricted Three-Body Problem I

AUTHORS:
Gladston Duarte^(1) and Angel Jorba^(2,3)

(1) Dipartimento di Ingegneria Meccanica e Aerospaziale,
    Politecnico di Torino,\newline
    Via Duca degli Abruzzi 24, 10129, Torino, Italy.
(2) Departament de Matem{\`a}tiques i Inform{\`a}tica,
    Universitat de Barcelona,\newline
    Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain.
(3) Centre de Recerca Matemàtica, Edifici C, Campus Bellaterra,
     08193 Bellaterra, Spain

E-mails: gladston.duarte@polito.it, angel@maia.ub.es

ABSTRACT:
In this paper, we present a way of combine the computation of
invariant tori and their stable and unstable manifolds with the
multiple shooting technique. We start by showing some of the results
of [Jorba01] that should be modified in order to introduce the
multiple shooting technique in these computations. After that it is
introduced, by a direct application in the Planar Elliptic Restricted
Three-Body Problem (PERTBP), how to modify the equations and methods
to compute the above-mentioned objects. It is shown, in particular,
the structure of the (systems of) equations and matrices involved in
these computations. An application of these computations can be found
in [DuarteJ23b], where the dynamics of comet 39P/Oterma is modelled as
a PERTBP.