TITLE:
Stability of homographic solutions of the planar three-body problem with
homogeneous potentials

AUTHORS:
Regina Mart\'inez^(1), Anna Sam\`a^(1), Carles Sim\'o^(2)

(1) Departament de Matem\`atiques, Facultat de Ci\`encies, Univ. Aut\`onoma
de Barcelona, 08193 Bellaterra, (Spain)

(2) Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona,
Gran Via, 585, 08071 Barcelona (Spain)

ABSTRACT:
Collinear and triangular homographic solutions of the planar three-body
problem with potentials of the form $r^{-\alpha},\, \alpha\in(0,2)$ are 
considered. Given the masses, $m_i,i=1,2,3$ normalised by $\sum_{i=1}^3m_i=1$
and a value of the energy, these solutions depend on an additional parameter,
either the angular momentum $c$ or a (generalised) eccentricity $e$. The
purpose is to describe the stability properties and the bifurcations for any
values of $m_i,e$ and $\alpha.$ After several reductions, it is enough to
study a three-parameter family of four-dimensional linear differential
equations depending periodically on time. We are interested in the domains
where the linear stability is of type EE (elliptic-elliptic), EH 
(elliptic-hyperbolic), HH (hyperbolic-hyperbolic) or CS (complex-saddle), as
well as in the boundaries of these domains.  The bifurcations emerging from
$e=0$ are studied by using normal forms. These provide also approximations
for the boundaries. For $e$ close to 1 (the value $e=1$ corresponding to
homographic solutions with a triple collision) a blow up method, applied to
the variational equations,is the key tool used to study the bifurcations. For
the intermediate region the bifurcation diagrams are completed numerically.