  A detailed comparative qualitative analysis and numerical simulation of
evolution of the cosmological models based on the doublet of classical and
phantom scalar fields with self-action. The 2-dimensional and 3-dimensional
projections of the phase portraits of the corresponding dynamic system are
built. Just as in the case of single scalar fields, the phase space of such
systems, becomes multiply connected, the ranges of negative total effective
energy unavailable for motion, getting appear there. The distinctive feature of
the asymmetrical scalar doublet is the time dependency of the prohibited
ranges' projections on the phase subspaces of each field as a result of which
the existence of the limit cycles with null effective energy depends on the
parameters of the field model and initial conditions. The numerical models
where the dynamic system has limit cycles on hypersurfaces of null energy, are
built. It is shown, that even quite weak phantom field in such model undertakes
functions of management of the dynamic system and can significantly change the
course of the cosmological evolution.
