  After removal of an applied magnetic field, artificial square spin ice can be
left in a metastable remanent state, with nonzero residual magnetization and
excess energy above the ground state. Using a model of magnetic islands with
dipoles of fixed magnitude and local anisotropies, the remanent states are
precisely determined here, including all long-range dipole interactions. Small
deviations away from remanent states are analyzed and the frequencies of modes
of oscillation are determined. Some modes reach zero frequency at high symmetry
wave vectors, such that the stability limits are found, as determined by the
local anisotropy strength relative to the dipolar coupling strength.
