  The paper provides a qualitative and numerical analysis-based investigation
of cosmological models founded on an asymmetrical scalar doublet comprising of
a classical and a phantom scalar fields. Presence of a phantom scalar field
allows one to consider also classical scalar fields with attraction of
like-charged particles which significantly extends a diversity of cosmological
models' behaviours. It is shown that a cosmological model based on an
asymmetric scalar doublet in the case of minimal interaction has 9 singular
points 2 of which are attractive and the rest are non-stable saddle ones. It is
also shown that a presence of even essentially weak phantom field significantly
changes the dynamics of a cosmological model.
