Fixed points and FLRW cosmologies: Flat case

We use phase space method to study possible consequences of fixed points in flat FLRW models. One of these consequences is that a fluid with a finite sound speed, or a differentiable pressure, reaches a fixed point in an infinite time and has no finite-time singularities of types I, II and III described in hep-th/0501025. It is impossible for such a fluid to cross the phantom divide in a finite time. We show that a divergent $dp/dH$, or a speed of sound is 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 $\eta \sim \rho^r$, in particular, solutions for $r=1$ and $r=1/4$ cases, which can be expressed in terms of Lambert-W function. The last solution behaves either as 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, then recollapses.