Nonlinear dynamics of the magnetic vortex state in a circular nanodisk was studied under a perpendicular alternating magnetic field that excites the radial modes of the magnetic resonance. Here, we show that as the oscillating frequency is swept down from a frequency higher than the eigenfrequency, the amplitude of the radial mode is almost doubled to the amplitude at the fixed resonance frequency. This amplitude has a hysteresis vs. frequency sweeping direction. Our result showed that this phenomenon was due to a Duffing-type nonlinear resonance. Consequently, the amplitude enhancement reduced the vortex core-switching magnetic field to well below 10 mT. A theoretical model corresponding to the Duffing oscillator was developed from the Landau-Lifshitz-Gilbert equation to explore the physical origin of the simulation result. This work provides a new pathway for the switching of the magnetic vortex core polarity in future magnetic storage devices.