  A detailed qualitative analysis and numerical modeling of the evolution of
cosmological models based on nonlinear classical and phantom scalar fields with
self-action are performed. Complete phase portraits of the corresponding
dynamical systems and their projections onto the Poincar\'e sphere are
constructed. It is shown that the phase trajectories of the corresponding
dynamical systems can, depending on the parameters of the model of the scalar
field, split into bifurcation trajectories along 2, 4, or 6 different dynamical
flows. In the phase space of such systems, regions can appear which are
inaccessible for motion. Here phase trajectories of the phantom scalar field
wind around one of the symmetric foci (centers) while the phase trajectories of
the classical scalar field can have a limit cycle determined by the zero
effective energy corresponding to a Euclidean Universe.
