Fixed points and FLRW cosmologies: Flat case

We use a phase space approach to study possible consequences of fixed points in a single fluid flat Friedmann-Lemaitre-Robertson-Walker (FLRW) models with pressure p(H), where H is the Hubble parameter. One of these consequences is that a fluid with a differentiable pressure, i.e., a finite adiabatic speed of sound, reaches a fixed point in an infinite time and has no finite-time singularities of types I, II, and III described by Nojiri, Odintsov, and Tsujikawa [Phys. Rev. D 71, 063004 (2005)]. It is impossible for such a fluid to cross the phantom divide in a finite time. We show that a divergent dp/dH, or the speed of sound, is a necessary but not sufficient condition for phantom crossing. We use pressure properties, such as asymptotic behavior and fixed points, to qualitatively describe the entire behavior of a solution in flat FLRW models. We discuss FLRW models with bulk viscosity eta similar to rho(r), in particular, solutions for r = 1 and r = 1/4 cases, which can be expressed in terms of the Lambert-W function. The last solution behaves as either a nonsingular phantom fluid or a unified dark fluid. Using causality and stability constraints, we show that the universe must end as a de Sitter space. Relaxing the stability constraint leads to a de Sitter universe, an empty universe, or a turnaround solution that reaches a maximum size and then recollapses.