  The application of numerical techniques to the study of energy landscapes of
large systems relies on sufficient sampling of the stationary points. Since the
number of stationary points is believed to grow exponentially with system size,
we can only sample a small fraction. We investigate the interplay between this
restricted sample size and the physical features of the potential energy
landscape for the two-dimensional $XY$ model in the absence of disorder with up
to $N=100$ spins. Using an eigenvector-following technique, we numerically
compute stationary points with a given Hessian index $I$ for all possible
values of $I$. We investigate the number of stationary points, their energy and
index distributions, and other related quantities, with particular focus on the
scaling with $N$. The results are used to test a number of conjectures and
approximate analytic results for the general properties of energy landscapes.
