Some unexpected results on the Brillouin singular equation: Fold bifurcation of periodic solutions

Roberto Castelli, Maurizio Garrione*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

In this paper, we find some new patterns regarding the periodic solvability of the Brillouin electron beam focusing equation x¨+β(1+cos⁡(t))x=[Formula presented]. In particular, we prove that there exists β≈0.248 for which a 2π-periodic solution exists for every β∈(0,β], and the bifurcation diagram with respect to β displays a fold for β=β. This result significantly contributes to the discussion about the well-known conjecture asserting that the Brillouin equation admits a periodic solution for every β∈(0,1/4), leading to doubt about its truthfulness. For the first time, moreover, we prove multiplicity of periodic solutions for a range of values of β near β. The technique used relies on rigorous computation and can be extended to some generalizations of the Brillouin equation, with right-hand side equal to 1/xp; we briefly discuss the cases p=2 and p=3.

Original languageEnglish
Pages (from-to)2502-2543
Number of pages42
JournalJournal of Differential Equations
Volume265
Issue number6
DOIs
Publication statusPublished - 15 Sep 2018

Keywords

  • Brillouin focusing beam equation
  • Computer-assisted proof
  • Fold bifurcation
  • Non-autonomous singular ODEs
  • Periodic solutions

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