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) over dot + beta(1+cos(t))x = 1/x. In particular, we prove that there exists beta* approximate to 0.248 for which a 2 pi-periodic solution exists for every beta is an element of (0, beta*], and the bifurcation diagram with respect to beta displays a fold for beta = beta*. This result significantly contributes to the discussion about the well-known conjecture asserting that the Brillouin equation admits a periodic solution for every beta is an element of (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 beta near beta*. 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/x(P); we briefly discuss the cases p = 2 and p = 3. (C) 2018 Elsevier Inc. All rights reserved.