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In this work we shall investigate the occurrence of future cosmologicalfinite-time singularities in the dynamical system corresponding to twocosmological theories, namely that of vacuum $f(R)$ gravity and that of threefluids. The vacuum $f(R)$ gravity is an example for which the variables we willchoose to quantify the phase space dynamics, do not necessarily blow-up near acosmological singularity. After appropriately choosing the variables, we shallinvestigate the behavior of the corresponding dynamical system near some typesof cosmological finite-time singularities, for some limiting cases in which wecan produce analytic solutions for the dynamical variables. The mostinteresting case from both a mathematical and physical point of view, is theBig Rip case, and particularly in the limiting case of a very strongsingularity. The physically appealing outcome is that the resultingnon-autonomous dynamical system is attracted asymptotically to an acceleratingattractor solution, with equation of state parameter $w_{eff}=-1$. Our analyticresults, show that an extremely strong Big Rip singularity in vacuum $f(R)$gravity theories is always related to an accelerating solution, or tends toacceleration. The converse statement though may not be true. The secondcosmology we shall study is a multifluid cosmology, consisting of three fluids,the interacting dark matter and dark energy fluids, and the baryonic fluid. Byappropriately choosing the variables, we will show that the dynamical systemcan become an autonomous polynomial dynamical system, in which case, by using adominant balance analysis, we shall investigate the occurrence of finite-timesingularities in this system.

Dynamical Systems Perspective of Cosmological Finite-time Singularities in $f(R)$ Gravity and Interacting Multifluid Cosmology