Assertion of the nature and stability of solutions of field equations recast in terms of an autonomous set of variables in a so-called phase space is a crucial requirement for all dynamical systems including the cosmological
ones, particularly those which lead to an effective dark universe picture, i.e., with dark energy (DE) and dark matter
(DM), and plausible interactions thereof. The latter can be perceived naturally in the equivalent scalar-tensor
(ST) formulations of various modified gravity theories, under conformal transformations. Viable cosmologies are
indeed shown to emerge from the ST formulations for exponential type scalar field potentials in the conformal
(Einstein) frame. Keeping our attention on these potentials, we carry out a rigorous phase space analysis from
the perspective of generic exponents as well as in a specific case-by-case manner, by working out the autonomous
first-order coupled differential equations from the ST cosmological equations in the Einstein frame, after defining
suitable phase space variables. Thesecase studies turn out to be significant not only in working out numerical
solutions of the autonomous equations for the physically relevant parametric ranges (which we assert by inspection)
and in identifying the attractors in the resulting phase portraits, but also in determining which of the equilibrium
solutions change their roles to the stable ones (sink or spiral) in various sub-domains of such parametric ranges.
Finally, we demonstrate the robustness of the analysis from a precise match of the stable solutions with the exact
analytic ones, as well as from a comparison with the dynamics of the effectively non-interacting cosmological
systems, in the respective cases of fixed exponents in the scalar field potential. On the whole we make a concrete
step towards establishing stable solutions in the phase space of autonomous variables for a fairly generic scalar-
tensor cosmological system which culminates to an interacting cosmic dark sector.