Through numerical diagonalization of the spin equations of motion linearized about a non-uniformly magnetized state, we have found the spinwave spectrum for finite circular systems containing a single vortex at the center. The spectrum shows some important dynamical properties of individual magnetic vortices. The most significant feature is the presence of a particular mode [mode (a) in the Figures] whose frequency comes close to zero near , and whose rms radius comes close to one lattice constant at the same time. For just below , such a localized mode has a time-dependent spin structure with radial dependence of its amplitude very similar to the static out-of-plane vortex structure present for just above . This is suggested by comparing the mode (a) in Fig. 8 and the lowest curve () in Fig. 1. Thus, this mode is the precursor to the instability of an in-plane vortex to become an out-of-plane vortex. This idea is further supported by the fact that the rms radius of mode (a) becomes comparable to the ``vortex core'' radius of an out-of-plane vortex for , defined by, 
On the other hand, for far below , mode (a) bears a lot of similarity to the lowest mode when there is no vortex present. This can be seen by comparing Fig. 7a and Fig. 10a. However, in the absence of the vortex, this lowest mode undergoes no substantial changes as is increased, even through .