On the pitchfork bifurcation of the folded node and other unbounded time-reversible connection problems in $\mathbb R^3$

In this paper, we revisit the folded node and the bifurcations of secondary canards at resonances $\mu\in \mathbb N$. In particular, we prove for the first time that pitchfork bifurcations occur at all even values of $\mu$. Our approach relies on a time-reversible version of the Melnikov approach in \cite{wechselberger2002a}, used in \cite{wechselberger_existence_2005} to prove the transcritical bifurcations for all odd values of $\mu$. 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 $0<\epsilon\ll 1$. 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 $\mathbb R^3$. 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 Nos\'e equations.