To verify the exact underlying mechanism of ultrafast vortex-core reversal as well as the vortex state stability, we conducted numerical calculations of the dynamic evolution of magnetic vortices in Permalloy cylindrical nanodots under an oscillating in-plane magnetic field over a wide range of the field frequency and amplitude. The calculated results reveal different kinds of the nontrivial dynamic responses of vortices to the driving external field, including the vortex-core reversal. In particular, the results offer insight into the 10 ps scale underlying physics of the ultrafast vortex-core reversal driven by small-amplitude (similar to 10 Oe) oscillating in-plane fields. This work also provides fundamentals of how to effectively manipulate the vortex dynamics as well as the dynamical switching of the vortex-core orientation.