  Recently \citet{PEDE:2019ApJ} proposed a phenomenologically emergent dark
energy (PEDE) which consider that the dark energy density evolves as
$\widetilde{\Omega}_{\rm{DE}}(z)\,=\,\Omega_{\rm{DE,0}}\left[ 1 -
{\rm{tanh}}\left( {\log}_{10}(1+z) \right) \right]$ with the advantage that it
does not have degree of freedom. Later on, \citet{PEDE:2020} proposed a
generalized model by adding one degree of freedom to the PEDE model, encoded in
the parameter $\Delta$. Motivated by these proposals, we constrain the
parameter space ($h,\Omega_m$) and ($h,\Omega_m, \Delta$) for PEDE and
Generalized Emergent Dark Energy (GEDE) respectively, by employing the most
recent observational (non-) homogeneous Hubble data. Additionally, we
reconstruct the deceleration and jerk parameters and estimate yield values at
$z=0$ of $q_0 = -0.784^{+0.028}_{-0.027}$ and $j_0 = 1.241^{+0.164}_{-0.149}$
for PEDE and $q_0 = -0.730^{+0.059}_{-0.067}$ and $j_0 =
1.293^{+0.194}_{-0.187}$ for GEDE using the homogeneous sample. We report
values on the deceleration-acceleration transition redshift with those reported
in the literature within $2\sigma$ CL. Furthermore, we perform a stability
analysis of the PEDE and GEDE models to study the global evolution of the
Universe around their critical points. Although the PEDE and GEDE dynamics are
similar to the standard model, our stability analysis indicates that in both
models there is an accelerated phase at early epochs of the Universe.
