  In this work we study the occurrence of finite-time cosmological
singularities in a cosmological system comprising from three fluids.
Particularly, the system contains two dark fluids, namely that of dark energy
and dark matter, which are interacting, and of a non-interacting baryonic
fluid. For the study we adopt the phase space approach by constructing the
cosmological dynamical system in such a way so that it rendered to be an
autonomous polynomial dynamical system, and in order to achieve this, we
appropriately choose the variables of the dynamical system. By employing a
rigid mathematical framework, that of dominant balances analysis, we
demonstrate that there exist non-singular solutions of the dynamical system,
which correspond to a general set of initial conditions, which proves that no
Big Rip or Type III finite-time singularities occur in this LQC multifluid
dynamical system. Thus the new feature of this work is that we are able to do
this using an analytic technique instead of adopting a numerical approach. In
addition, we perform a fixed point analysis of the cosmological dynamical
system, and we examine the behavior of the total effective equation of state
parameter, at the fixed points, as a function of the free parameters of the
system. Finally, we investigate the phenomenological implications of the dark
energy equation of state which we assumed that it governs the dark energy
fluid.
