  In this paper, using a qualitative theory of dynamical systems, we study the
stability of a unified dark matter-dark energy framework known as quartessence
Chaplygin model (QCM) with three different equation-of-states within
ultraviolet (UV) deformed Friedmann-Robertson-Walker (FRW) cosmologies without
Big-Bang singularity. The UV deformation is inspired by the non-commutative
(NC) Snyder spacetime approach in which by keeping the transformation groups
and rotational symmetry there is a dimensionless, Planck scale characteristic
parameter $\mu_0$ with dual implications dependent on its sign that addresses
the required invariant cutoffs for length and momentum in nature, in a separate
manner. Our stability analysis is done in the $(H,\rho)$ phase space at a
finite domain concerning the hyperbolic critical points. According to our
analysis, due to constraints imposed on the signs of $\mu_0$ from the
phenomenological parameters involved in quartessence models $(\Omega_m^*,
c_s^2, \rho_*)$, for an expanding and accelerating late universe, all three
QCMs can be stable in the vicinity of the critical points. The requirement of
stability for these quartessence models in case of admission of a minimum
invariant length, can yield a flat as well as non-flat expanding and
accelerating universe in which Big-Bang singularity is absent. This feedback
also phenomenologically credits to braneworld-like framework versus loop
quantum cosmology-like one as two possible scenarios which can be NC Snyder
spacetime generators (correspond to $\mu_0<0$ and $\mu_0>0$, respectively). As
a result, our analysis show that between quartessence models with Chaplygin gas
equation-of-states and accelerating FRW backgrounds occupied by a minimum
invariant length, there is a possibility of viability.
