TY - JOUR

T1 - Some unexpected results on the Brillouin singular equation

T2 - Fold bifurcation of periodic solutions

AU - Castelli, Roberto

AU - Garrione, Maurizio

PY - 2018/9/15

Y1 - 2018/9/15

N2 - 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.

AB - 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.

KW - Brillouin focusing beam equation

KW - Computer-assisted proof

KW - Fold bifurcation

KW - Non-autonomous singular ODEs

KW - Periodic solutions

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U2 - 10.1016/j.jde.2018.04.042

DO - 10.1016/j.jde.2018.04.042

M3 - Article

AN - SCOPUS:85046639091

VL - 265

SP - 2502

EP - 2543

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 6

ER -