We investigate the effect of the magnetic anisotropy ($K_z$) on the static and dynamic properties of magnetic vortices in small disks. Our micromagnetic calculations reveal that for a range of $K_z$ there is an enlargement of the vortex core. We analyze the influence of $K_z$ on the dynamics of the vortex core magnetization reversal under the excitation of a pulsed field. The presence of $K_z$, which leads to better resolved vortex structures, allows us to discuss in more details the role played by the in-plane and perpendicular components of the gyrotropic field during the vortex-antivortex nucleation and annihilation.
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