In this paper, we revisit the folded node and the bifurcations of secondary canards at resonances μ∈N. In particular, we prove for the first time that pitchfork bifurcations occur at all even values of μ. Our approach relies on a time-reversible version of the Melnikov approach in wechselberger2002a, used in wechselberger_existence₂005 to prove the transcritical bifurcations for all odd values of μ. It is known that the secondary canards produced by the transcritical and the pitchfork bifurcations only reach the Fenichel slow manifolds on one side of each transcritical bifurcation for all 0<ε≪1
Nearby in the stack
. In this paper, we provide a new geometric explanation for this fact, relying on the symmetry of the normal form and a separate blowup of the fold lines. We also show that our approach for evaluating the Melnikov integrals of the folded node -- based upon local characterization of the invariant manifolds by higher order variational equations and reducing these to an inhomogeneous Weber equation -- applies to general, quadratic, time-reversible, unbounded connection problems in
R3
. We conclude the paper by using our approach to present a new proof of the bifurcation of periodic orbits from infinity in the Falkner-Skan equation and the Nosé equations.