2020/03/15 by Kristian Uldall Kristiansen, Kristiansen, Kristian Uldall
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Quantum chaos and dynamical systems #math.DS
paper · pdf · doi:10.48550/arxiv.2003.06817
arxiv created 2020/03/15 · openalex publication_date 2020/03/15 · arxiv updated 2020/03/17 · openalex created_date 2022/07/23 · openalex updated_date 2026/07/28
In this paper, we revisit the folded node and the bifurcations of secondary canards at resonances μ∈ \mathbb 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 \citewechselberger2002a, used in \citewechselbergerexistence2005 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. 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 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.