  The time-evolution dynamics of two nonlinear cosmological real gas models has
been reexamined in detail with methods from the theory of Hamiltonian dynamical
systems. These examples are FRWL cosmologies, one based on a gas, satisfying
the van der Waals equation and another one based on the virial expansion gas
equation. The cosmological variables used are the expansion rate, given by the
Hubble parameter, and the energy density. The analysis is aided by the
existence of global first integral as well as several special (second)
integrals in each case. In addition, the global first integral can serve as a
Hamiltonian for a canonical Hamiltonian formulation of the evolution equations.
The conserved quantities lead to the existence of stable periodic solutions
(closed orbits) which are models of a cyclic Universe. The second integrals
allow for explicit solutions as functions of time on some special trajectories
and thus for a deeper understanding of the underlying physics. In particular,
it is shown that any possible static equilibrium is reachable only for infinite
time.
