  We prove the existence of non-constant time periodic vortex solutions to the
Gross-Pitaevskii equations for small but \textit{fixed} $\varepsilon > 0.$ The
vortices of these solutions follow periodic orbits to the point vortex system
of ordinary differential equations \textit{for all time}. The construction uses
two approaches-- constrained minimization techniques adapted from \cite{GS} and
topological minimax techniques adapted from \cite{LinMinMax}, applied to a
formulation of the problem within a rotational ansatz.
