  In this work, considering the background dynamics of flat
Friedmann-Lemaitre-Robertson-Walker(FLRW) model of the universe, we investigate
a non-canonical scalar field model as dark energy candidate which interacting
with the pressureless dust as dark matter in view of dynamical systems
analysis. Two interactions from phenomenological point of view are chosen: one
is depending on Hubble parameter $H$, another is local, independent of Hubble
parameter. In Interaction model 1, an inverse square form of potential as well
as coupling function associated with scalar field is chosen and a two
dimensional autonomous system is obtained. From the 2D autonomous system, we
obtain scalar field dominated solutions representing late time accelerated
evolution of the universe. Late time scaling solutions are also realized by the
accelerated evolution of the universe attracted in quintessence era. Center
Manifold Theory can provide the sufficient conditions on model parameters such
that the de Sitter like solutions can be stable attractor at late time in this
model. In the Interaction model 2, potential as well as coupling function are
considered to be evolved exponentially on scalar field and as a result of which
a four dimensional autonomous system is achieved. From the analysis of 4D
system, we obtain non-hyperbolic sets of critical points which are analyzed by
the Center Manifold Theory. In this model, de Sitter like solutions represent
the transient evolution of the universe.
