  Recently, a Friedmann-Robertson-Walker universe filled with various
cosmological fluids has been considered by S.D. Odintsov et al. in [30] from
phase space vantage point where various expressions for the Equation-of-State
(EoS) parameter were studied. Since these types of EoS parameters are
generative of appreciable results in the Hilbert-Einstein model, hence we
intend to investigate all the cases in a homogeneous $F(T)$-gravity ($T$ is the
torsion) through phase space analysis in precise detail. In short, three viable
models of interaction between dark matter and dark energy, including
usual-type, power-law type, and oscillating type, are investigated
comprehensively. It is indicated that the power-law interaction in the related
dynamical systems should be of increasing nature with time to get more critical
points. Due to the failure of the oscillating model of ref. [30] in
$F(T)$-gravity, four modified models are suggested and examined in both $F(T)$
and Hilbert-Einstein models. As to be seen, the modified models not only are
generative of critical points equivalent to that of ref. [30], but also give
rise to further critical points covering crucial stages of the evolution of the
universe. In the context of these four models, such as the old one, at early
times the interactions are negligible and they commence to grow as the cosmic
time approaches the late-time in which the unification of early inflation and
late acceleration is obtained. Using an indirect method, it is shown that the
oscillating models have substantial roles in transitions between eras.
