On the Past Asymptotic Dynamics of Non-minimally Coupled Dark Energy
  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. 
Genly Leon