On the Pitchfork Bifurcation of the Folded Node and Other Unbounded Time-Reversible Connection Problems in $\mathbb R^3$

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Abstract

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 [M. Wechselberger, Dynam. Syst., 17 (2002), pp. 215--233] used in [M. Wechselberger, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 101--139] 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.
Original languageEnglish
JournalSIAM Journal on Applied Dynamical Systems
Volume19
Issue number3
Pages (from-to)2059-2102
ISSN1536-0040
DOIs
Publication statusPublished - 2020

Keywords

  • Folded node
  • Bifurcations
  • Canards
  • Secondary canards
  • Global bifurcations
  • Geometric singular perturbation theory
  • Blowup
  • Falkner-Skan equation
  • Nosé equations
  • Two-fold

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