On the past asymptotic dynamics of non-minimally coupled dark energy

We apply dynamical system 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 are investigated in some detail. By assuming some regularity conditions for the potential and the coupling function, a dynamical system is constructed, well suited to investigating the dynamics where the scalar field diverges, i.e. near the initial singularity. The critical points therein are investigated and the cosmological solutions associated with 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 spacetime singularity, which extend, in a way, the previous results of other researchers.