Non-minimally coupled scalar field dark sector of the universe: in-depth (Einstein frame) case study

In this study, motivated by recent results from DESI DR2 indicating a possible preference for evolving dark energy in some combined data analyses, we analyze spatially flat FLRW interacting scalar-tensor cosmological models with non-minimal coupling (NMC) between the scalar field (SF) and matter/cosmological dust in the Einstein conformal frame. By using modified expansion normalized variables that account for negative values of the scalar field potential, we derive cosmological dynamical system equations and expressions for physical variables. Five specific scalar field models (axions/ALPs, cyclic ekpyrotic, exponential with a constant, quintessence, and SFDM) are examined in depth to determine how they evolve, as they serve as representative candidates used in the literature for the evolving dark energy in the late Universe. Using appropriate mathematical methods (e.g. linear stability, center manifold and Poincar\'e sphere), we present critical points along with their character and physical interpretation with respect to the possible evolution of the Universe. Considering both positive and negative values of the coupling parameter allows us to examine the transfer of energy from the scalar sector to dust and from matter to the scalar field. Considering four representative absolute values of this parameter provides a comprehensive analysis that incorporates all significant types of dynamical system evolution (from mathematical and physical perspectives). The initial conditions for the in-depth numerical analysis were constructed analytically from observationally motivated Planck 2018 and DESI DR2 CPL parametrizations.