The nonlinear dynamic behavior of a magnetic skyrmion in circular nanodots was studied numerically by solving the Landau-Lifshitz-Gilbert equation with a classical spin model. We show that a skyrmion core reversal can be achieved within nanoseconds using a perpendicular oscillating magnetic field. Two symmetric switching processes that correspond to excitations of the breathing mode and the mixed mode (combination of the breathing mode and a radial spinwave mode) are identified. For excitation of the breathing mode, the skyrmion core switches through nucleation of a new core from a transient uniform state. In the mixed mode, the skyrmion core reverses with the help of spins excited both at the edge and core regions. Unlike the magnetic vortex core reversal, the excitation of radial spin waves does not dominate the skyrmion core reversal process.