  The present work is an example of the application of the dynamical system
analysis in the context of cosmology. Here cosmic evolution is considered in
the background of homogeneous and isotropic flat
Friedmann-Lema\^{i}tre-Robertson-Walker space-time with interacting dark energy
and varying mass dark matter as the matter content. The Dark Energy (DE) is
chosen as phantom scalar field with self-interacting potential while the Dark
Matter (DM) is in the form of dust. The potential of the scalar field and the
mass function of dark matter are chosen as exponential or power-law form or in
their product form. Using suitable dimensionless variables the Einstein field
equations and the conservation equations constitute an autonomous system. The
stability of the non-hyperbolic critical points are analyzed by using center
manifold theory. Finally, cosmological phase transitions have been detected
through bifurcation analysis which has been done by Poincar\'{e} index theory.
