  We study the motion of a non-planar vortex in a circular easy-plane
ferromagnet, which imitates a magnetic nanodot. Analysis was done using
numerical simulations and a new collective variable theory which includes the
coupling of Goldstone-like mode with the vortex center. Without magnetic field
the vortex follows a spiral orbit which we calculate. When a rotating in-plane
magnetic field is included, the vortex tends to a stable limit cycle which
exists in a significant range of field amplitude B and frequency $\omega$ for a
given system size L. For a fixed $\omega$, the radius R of the orbital motion
is proportional to L while the orbital frequency $\Omega$ varies as 1/L and is
significantly smaller than $\omega$. Since the limit cycle is caused by the
interplay between the magnetization and the vortex motion, the internal mode is
essential in the collective variable theory which then gives the correct
estimate and dependency for the orbit radius $R\sim B L/\omega$. Using this
simple theory we indicate how an ac magnetic field can be used to control
vortices observed in real magnetic nanodots.
