  We focus on the dynamical feature for a $p$-wave superconductor model in
different parameter regions in terms of the appearance of gapless edge modes in
reel geometry. Firstly, we show the parameter region with gapless edge modes
versus a parameter and quasi-momentum. The parameter diagram can be reflected
by the expectations of Pauli matrices in global manners. In another view, the
dynamical feature of the excitation behave differently in the parameter regions
with topological gapless edge modes and not. And the cusps of dynamical return
rate vanish as the parameter pass the boundary in the parameter region slowly
enough. It is found that the dynamics in the parameter region with gapless edge
modes behaves differently to that without edge modes and related mostly to the
eigenenergy gap between the pre-and post-quench eigenstates. The cusps of the
dynamical return rate behave robustly against the noise in the lattice until
localization behavior dominates. This work benefits detecting topological edge
modes by dynamical manners.
