Cosmological singularities in interacting dark energy models with an
  $\omega(q)$ parametrization
  Future singularities arising in a family of models for the expanding
Universe, characterized by sharing a convenient parametrization of the energy
budget in terms of the deceleration parameter, are classified. Finite-time
future singularities are known to appear in many cosmological scenarios, in
particular, in the presence of viscosity or non-gravitational interactions, the
last being known to be able to suppress or just change in some cases the type
of the cosmological singularity. Here, a family of models with a
parametrization of the energy budget in terms of the deceleration parameter are
studied in the light of Gaussian processes using reconstructed data from
$40$-value $H(z)$ datasets. Eventually, the form of the possible
non-gravitational interaction between dark energy and dark matter is
constructed from these smoothed $H(z)$ data. Using phase space analysis, it is
shown that a non-interacting model with dark energy $\omega_\mathrm{de} =
\omega_{0} + \omega_{1}q$ ($q$ being the deceleration parameter) may evolve,
after starting from a matter dominated unstable state, into a de Sitter
Universe (the solution being in fact a stable node). Moreover, for a model with
interaction term $Q = 3 H b \rho_\mathrm{dm}$ ($b$ is a parameter and $H$ the
Hubble constant) three stable critical points are obtained, what may have
important astrophysical implications. In addition, part of the paper is devoted
to a general discussion of the finite-time future singularities obtained from
direct numerical integration of the field equations, since they appear in many
cosmological scenarios and could be useful for future extended studies of the
models here introduced. Numerical solutions for the new models, produce
finite-time future singularities of Type I or Type III, or an
$\omega$-singularity, provided general relativity describes the background
dynamics.
