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

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=1/x. 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.