  We construct a compact phase space for flat FLRW spacetimes with standard
matter described by a perfect fluid with a barotropic equation of state for
general f(R) theories of gravity, subject to certain conditions on the function
f. We then use this framework to study the behaviour of the phase space of
Universes with a non-negative Ricci scalar in R + {\alpha}R^n gravity. We find
a number of interesting cosmological evolutions which include the possibility
of an initial unstable power-law inflationary point, followed by a curvature
fluid dominated phase mimicking standard radiation, then passing through a
standard matter (CDM) era and ultimately evolving asymptotically towards a
de-Sitter-like late-time accelerated phase.
