  In this paper we study in detail the phase space of a cosmological system
consisting of two coupled fluids, namely a dark energy fluid coupled with a
superfluid dark matter fluid. The dark matter fluid is assumed to have a
superfluid equation of state, hence it is not pressureless and our aim is to
find the impact of this non-trivial equation of state on the phase space of the
coupled system. We shall use two theoretical contexts, namely that of classical
cosmology and that of loop quantum cosmology. In the classical case, we
investigated the existence and stability of fixed points, and as we will show,
no de Sitter fixed points occur, however matter and radiation domination fixed
points occur, which are hyperbolic and unstable. We also show that there exist
limited sets of initial conditions for which singular solutions occur in the
phase space. With regard to the loop quantum cosmology case, we demonstrate
that stable de Sitter fixed points exist, for some values of the free
parameters of the theory, and interestingly enough, for the same values,
singular solutions corresponding to general sets of initial conditions occur.
To our knowledge this feature does not occur so frequently in loop quantum
cosmological frameworks. We also demonstrate that non-singular solutions
corresponding to a general set of initial conditions occur, however these occur
when the dark matter superfluid has negative pressure, so it is a rather
physically unappealing situation.
