  In this work we have studied a cosmological model based on a quintom dark
energy model non-minimally coupled with gravity, endowed with a specfic
potential energy of the exponential squared type. For this specific type of
potential energy and non-minimal coupling, the dynamical properties are
analyzed and the corresponding cosmological effects are discussed. Considering
the linear stability method, we have investigated the dynamical properties of
the phase space structure, determining the physically acceptable solutions. The
analysis showed that in this model we can have various cosmological epochs,
corresponding to radiation, matter domination, and de Sitter eras. Each
solution is investigated from a physical and cosmological point of view,
obtaining possible constraints of the model's parameters. In principle the
present cosmological setup represent a possible viable scalar tensor theory
which can explain various transitional effects related to the behavior of the
dark energy equation of state and the evolution of the Universe at large
scales.
