Growing evidence as the observations of the CMB (cosmic microwave background), galaxy clustering and high-redshift supernovae address a stable dynamically universe dominated by the dark components. 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-stateswithin 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 μ0 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, ρ) phase space at a finite domain concerning the hyperbolic critical points. According to our analysis, due to constraints imposed on the signs of μ0 from the phenomenological parameters involved in quartessence models (∗ m, c2 s , ρ∗), 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.
