  The dynamics of a preinflacionary phase of the universe, and its exit to
inflation, is discussed. This phase is modeled by a closed
Friedmann-Robertson-Walker geometry, the matter content of which is radiation
plus a scalar field minimally coupled to the gravitational field. The simple
configuration, with two effective degres of freedom only, presents a very
complicated dynamics connected to the existence of critical points of
saddle-center type and saddle type in phase space of the system. Each of these
critical points is associated to an extremum of the scalar field potential. The
Topology of the phase space about the saddle-center is characterized by
homoclinic cylinders emanating from unstable periodic orbits, and the
transversal crossing of the cylinders, due to the non-integrability of the
system, results in a chaotic dynamics. The topology of the homoclinic cylinders
provides an invariant characterization of chaos. The model exhibits one or more
exits to inflation, associated to one or more strong asymptotic de Sitter
attractors present in phase space, but the way out from the initial singularity
into any of the inflationary exits is chaotic. We discuss possible mechanisms,
connected to the spectrum of inhomogeneous fluctuations in the models, which
would allow us to distinguish physically the several exits to inflation.
