Frobenius-unstable bundles and $p$-curvature

We use the theory of $p$-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for $p$-curvature to analyze low-characteristic cases, and then using degeneration techniques to obtain an answer for a general curve by degenerating to an irreducible rational nodal curve, and applying the results of additional works by the author. We also apply our explicit formulas to give a new description of the strata of curves of genus 2 of different $p$-rank.