Revisiting pre-inflationary universe of family of $\alpha-$attractor in
  loop quantum cosmology
  In this work, we revisit the dynamics of pre-inflationary universe with a
family of $\alpha-$attractor potentials, in the framework of loop quantum
cosmology, in which the big bang singularity is generically resolved purely
with quantum geometric effects, and replaced by a quantum bounce. At the
bounce, the background evolution is divided into two distinct classes, the
first is dominated by the kinetic energy of the inflaton field and the second
by the potential energy. In both classes, we find the physically viable initial
conditions numerically that provide not only the slow-roll inflation, but also
sufficient e-folds to be compatible with observations. In the entire range of
kinetic energy dominated initial conditions (except some subsets of Models 2
and 4), the background evolution prior to reheating is always split into three
different phases: bouncing, transition and slow-roll inflation. In the bouncing
phase, the numerical evolution of the scale factor is independent not only of
the initial data, but also the inflationary potentials, as long as it is
dominated by the kinetic energy, and can be well approximated by an analytical
solution, whereas in the potential energy dominated case, such approximated
results do not exist. Moreover, we study the phase space analysis for a class
of $\alpha-$attractor potentials, and discuss the phase space trajectories for
physically viable initial conditions of the inflaton field.
