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In this work we have studied a cosmological model based on a quintom darkenergy model non-minimally coupled with gravity, endowed with a specficpotential energy of the exponential squared type. For this specific type ofpotential energy and non-minimal coupling, the dynamical properties areanalyzed and the corresponding cosmological effects are discussed. Consideringthe linear stability method, we have investigated the dynamical properties ofthe phase space structure, determining the physically acceptable solutions. Theanalysis showed that in this model we can have various cosmological epochs,corresponding to radiation, matter domination, and de Sitter eras. Eachsolution is investigated from a physical and cosmological point of view,obtaining possible constraints of the model's parameters. In principle thepresent cosmological setup represent a possible viable scalar tensor theorywhich can explain various transitional effects related to the behavior of thedark energy equation of state and the evolution of the Universe at largescales.

Dynamical description of a quintom cosmological model nonminimally coupled with gravity