  In this work we shall investigate the occurrence of future cosmological
finite-time singularities in the dynamical system corresponding to two
cosmological theories, namely that of vacuum $f(R)$ gravity and that of three
fluids. The vacuum $f(R)$ gravity is an example for which the variables we will
choose to quantify the phase space dynamics, do not necessarily blow-up near a
cosmological singularity. After appropriately choosing the variables, we shall
investigate the behavior of the corresponding dynamical system near some types
of cosmological finite-time singularities, for some limiting cases in which we
can produce analytic solutions for the dynamical variables. The most
interesting case from both a mathematical and physical point of view, is the
Big Rip case, and particularly in the limiting case of a very strong
singularity. The physically appealing outcome is that the resulting
non-autonomous dynamical system is attracted asymptotically to an accelerating
attractor solution, with equation of state parameter $w_{eff}=-1$. Our analytic
results, show that an extremely strong Big Rip singularity in vacuum $f(R)$
gravity theories is always related to an accelerating solution, or tends to
acceleration. The converse statement though may not be true. The second
cosmology we shall study is a multifluid cosmology, consisting of three fluids,
the interacting dark matter and dark energy fluids, and the baryonic fluid. By
appropriately choosing the variables, we will show that the dynamical system
can become an autonomous polynomial dynamical system, in which case, by using a
dominant balance analysis, we shall investigate the occurrence of finite-time
singularities in this system.
