  This paper studies a class of $1\frac12$-degree-of-freedom Hamiltonian
systems with a slowly varying phase that unfolds a Hamiltonian pitchfork
bifurcation. The main result of the paper is that there exists an order of
$\ln^2\epsilon^{-1}$-many periodic orbits that all stay within an $\mathcal
O(\epsilon^{1/3})$-distance from the union of the normally elliptic slow
manifolds that occur as a result of the bifurcation. Here $\epsilon\ll 1$
measures the time scale separation. These periodic orbits are predominantly
unstable. The proof is based on averaging of two blowup systems, allowing one
to estimate the effect of the singularity, combined with results on asymptotics
of the second Painleve equation. The stable orbits of smallest amplitude that
are {persistently} obtained by these methods remain slightly further away from
the slow manifold being distant by an order $\mathcal
O(\epsilon^{1/3}\ln^{1/2}\ln \epsilon^{-1})$.
