  In this paper, we propose a class of cosmological models based on the
gravitational theory $f(R)=R+\zeta R^2$, and on some nonideal fluids obeying to
the thermodynamical equations of state of Redlich-Kwong, (Modified) Berthelot,
and Dieterici. We investigate the dynamics of such models by applying dynamical
system techniques showing that our models can account for both the inflationary
and accelerated late-time epochs, with a radiation dominated phase and a
nonsingular bounce in between, where the latter is discovered by appropriately
compactifying the phase space. The infinity of the compactified phase space
will be interpreted thermodynamically as a region in which the non-ideal
behaviors of the cosmic fluids are suppressed, as long as the three previously
mentioned modelings are assumed. Although the dimensionless variables we adopt
were already known in the literature, we will derive some physical constraints
on them restricting the available region of the phase space by demanding the
theory to be free from ghost and tachyonic instabilities. A novel cosmological
interpretation of such variables is also proposed by connecting them to the
deceleration, jerk, and snap cosmographic parameters. It will be shown that the
stability nature of the equilibrium points depends only on the parameter of the
cosmic matter equations of state which describes their ideal limits, whilst a
nonzero parameter accounting for the nonlinearities within the fluids is
necessary for having a second de Sitter epoch. These properties of the
cosmological dynamics are interpreted to constitute a generic behavior because
they are identified in all the three fluid modelings. On the other hand,
differences between the three models are critically examined by showing that
the Redlich-Kwong scenario can admit a second radiation-dominated epoch and a
Big Rip Singularity.
