  Using the dynamical system approach, properties of cosmological models based
on the Horava-Lifshitz gravity are systematically studied. In particular, the
cosmological phase space of the Horava-Lifshitz model is characterized. The
analysis allows to compare some key physical consequences of the imposition (or
not) of detailed balance. A result of the investigation is that in the detailed
balance case one of the attractors in the theory corresponds to an oscillatory
behavior. Such oscillations can be associated to a bouncing universe, as
previously described by Brandenberger, and will prevent a possible evolution
towards a de Sitter universe. Other results obtained show that the cosmological
models generated by Horava-Lifshitz gravity without the detailed balance
assumption have indeed the potential to describe the transition between the
Friedmann and the dark energy eras. The whole analysis leads to the plausible
conclusion that a cosmology compatible with the present observations of the
universe can be achieved only if the detailed balance condition is broken.
