TITLE:
Analysis of the stability of a family of singular--limit linear periodic systems
in $ \R^4.$ Applications

AUTHORS:
Regina Mart\'{\i}nez^(1), Anna Sam\`a^(1) and Carles Sim\'o^(2)

(1) Dept. de Matem\`atiques Universitat Aut\`onoma de Barcelona,
    Bellaterra 08193, Barcelona, Spain.

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

E-mail: reginamb@mat.uab.es, sama@mat.uab.es, carles@maia.ub.es

ABSTRACT
In this paper we consider a 4D periodic linear system depending on a small
parameter $\delta>0$. We assume that the limit system has a singularity at $t=0$
of the form $\frac{1}{c_1+c_2 t^2+\ldots}$, with $c_1,c_2>0$ and $c_1\to 0$ as
$\delta \to 0.$ Using a blow up technique we develop an asymptotic formula for
the stability parameters as $ \delta $ goes to zero. As an example we consider
the homographic solutions of the planar three body problem for an homogeneous
potential of degree $\alpha\in (0,2) $. Newtonian three-body problem is obtained
for $ \alpha=1$. The parameter $\delta$ can be taken as $1-e^2$ being $e$ the
eccentricity (or a generalised eccentricity if $\alpha\ne 1$). The behaviour of
the stability parameters predicted by the formula is checked against numerical
computations and some results of a global numerical exploration are displayed.