  In this paper, we study the pre-inflationary dynamics for the power-law
potential $(V(\phi) \propto \phi^n)$ with $n<2$ in the framework of loop
quantum cosmology. In the case where the kinetic energy of the inflaton
dominates at the initial, the evolution of the universe can always be divided
into three different phases prior to preheating: {\em bouncing, transition and
slow-roll inflation}. During the bouncing phase, the evolution of the expansion
factor is independent not only on the initial conditions but also the
inflationary potentials, and is given explicitly by an analytical solution. In
contrast, for the potential energy dominated initial conditions, this
universality is lost. We also obtain total number of e-folds during the
slow-roll inflation, whereby physically viable models are identified. In
addition, we present phase space analysis for the inflationary potentials under
consideration and compare our results with the ones obtained previously for
different potentials.
