  A detailed comparative qualitative analysis and numerical simulation of
evolution of the cosmological models based on classical and phantom scalar
fields with self-action was performed. The phase portraits of the dynamic
systems of classical and phantom fields and their projections to the Poincare
sphere were constructed. It was shown that the phase trajectories of the
corresponding dynamic systems can be split by bifurcation trajectories into 2,4
or 6 different dynamic streams corresponding to different pairwise symmetric
histories of the Universe depending on the parameters of the scalar field's
model. The phase space of such systems becomes multiply connected, the ranges
of negative total effective energy unavailable for motion, getting appear
there. In the case when attracting centers are situated inside these ranges,
the phase trajectories of the classical scalar field in the infinite future
tend to limit cycles, winding onto the boundaries of these ranges. The phase
trajectories of the scalar field, in turn, get away from the boundaries of
ranges with null effective energy and in the infinite future are wound onto one
of the symmetrical focuses (centers). Thus, the situations when the Universe,
in case of classical scalar field, begins its history with the inflation and
ends it up in the Euclidean future, or, in the case of phantom scalar field, in
opposite, has the Euclidian start and proceeds to inflation mode after
anomalous burst of the acceleration, both become possible. The potentials of
scalar fields on the surface of null curvature are distinct from zero and
thereby define a certain vacuum state.
