  The present paper considers a three-fluid cosmological model consisting of
noninteracting dark matter, dark energy and baryonic matter in the background
of the Friedmann- Robertson- Walker- Lemaitre flat spacetime. It has been
assumed that the dark matter takes the form of dust whereas the dark energy is
a quintessence (real) scalar field with exponential potential. It has been
further assumed that the baryonic matter is a perfect fluid with barotropic
equation of states. The field equations for this model takes the form of an
autonomous dynamical system after some suitable changes of variables. Then a
complete stability analysis is done considering all possible parameter (the
adiabatic index of the baryonic matter and the parameter arising from the dark
energy potential) values and for both the cases of hyperbolic and
non-hyperbolic critical points. For non-hyperbolic critical points, the
invariant manifold theory (center manifold approach) is applied. Finally
various topologically different phase planes and vector field diagrams are
produced and the cosmological interpretation of this model is presented.
