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In this paper, we revisit the folded node and the bifurcations of secondarycanards at resonances $\mu\in \mathbb N$. In particular, we prove for the firsttime that pitchfork bifurcations occur at all even values of $\mu$. Ourapproach relies on a time-reversible version of the Melnikov approach in\cite{wechselberger2002a}, used in \cite{wechselberger_existence_2005} to provethe transcritical bifurcations for all odd values of $\mu$. It is known thatthe secondary canards produced by the transcritical and the pitchforkbifurcations only reach the Fenichel slow manifolds on one side of eachtranscritical bifurcation for all $0<\epsilon\ll 1$. In this paper, we providea new geometric explanation for this fact, relying on the symmetry of thenormal form and a separate blowup of the fold lines. We also show that ourapproach for evaluating the Melnikov integrals of the folded node -- based uponlocal characterization of the invariant manifolds by higher order variationalequations and reducing these to an inhomogeneous Weber equation -- applies togeneral, quadratic, time-reversible, unbounded connection problems in $\mathbbR^3$. We conclude the paper by using our approach to present a new proof of thebifurcation of periodic orbits from infinity in the Falkner-Skan equation andthe Nos\'e equations.

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