  We apply dynamical systems techniques to investigate cosmological models
inspired in scalar-tensor theories written in the Einstein frame. We prove that
if the potential and the coupling function are sufficiently smooth functions,
the scalar field almost always diverges into the past. The dynamics of two
important invariant sets is investigated in some detail. By assuming some
regularity conditions for the potential and for the coupling function, it is
constructed a dynamical system well suited to investigate the dynamics where
the scalar field diverges, i.e. near the initial singularity. The critical
points therein are investigated and the cosmological solutions associated to
them are characterized. We find that our system admits scaling solutions. Some
examples are taken from the bibliography to illustrate the major results. Also
we present asymptotic expansions for the cosmological solutions near the
initial space-time singularity, which extend in a way previous results of other
researchers.
