  The dispersion relations for collective magnon modes for square-planar arrays
of vortex-state magnetic dots, having closure magnetic flux are calculated. The
array dots have no direct contact between each other, and the sole source of
their interaction is the magnetic dipolar interaction. The magnon formalism
using Bose operators along with translational symmetry of the lattice, with the
knowledge of mode structure for the isolated dot, allows the diagonalization of
the system Hamiltonian giving the dispersion relation. Arrays of vortex-state
dots show a large variety of collective mode properties, such as positive or
negative dispersion for different modes. For their description, not only
dipolar interaction of effective magnetic dipoles, but non-dipolar terms common
to higher multipole interaction in classical electrodynamics can be important.
The dispersion relation is shown to be non-analytic as the value of the
wavevector approaches zero for all dipolar active modes of the single dot. For
vortex-state dots the interdot interaction is not weak, because, the dynamical
part (in contrast to the static magnetization of the vortex state) dot does not
contain the small parameter, the ratio of vortex core size to the dot radius.
This interaction can lead to qualitative effects like the formation of modes of
angular standing waves instead of modes with definite azimuthal number known
for the insolated vortex state dot.
