  We investigate the late-time evolution of the Universe within a cosmological
model in which dark matter and dark energy are identified with two interacting
scalar fields. Using methods of qualitative analysis of dynamical systems, we
identify all cosmological solutions of this model. We show that viable
solutions -- in the sense that they correspond to a cosmic evolution in which a
long enough matter-dominated era is followed by a current era of accelerated
expansion -- can be found in several regions of the parameter space. These
solutions can be divided into two categories, namely, solutions that evolve to
a state of everlasting accelerated expansion, in which the energy density of
the dark-matter field rapidly approaches zero and the evolution becomes
entirely dominated by the dark-energy field, and solutions in which the stage
of accelerated expansion is temporary and the ratio between the energy
densities of dark energy and dark matter tends, asymptotically, to a constant
nonzero value.
