Some unexpected results on the Brillouin singular equation: fold bifurcation of periodic solutions
type: Item
instance of: scholarly article
title: Some unexpected results on the Brillouin singular equation: fold bifurcation of periodic solutions
zbMATH Open document ID: 1415.34078
author: Roberto Castelli
author: Maurizio Garrione
published in: Journal of Differential Equations
publication date: 2018-05-28T00:00:00Z
review text: In the paper the following singular parametric equation \[ \ddot x + \beta (1-\cos t)x =\frac1 x \eqno{(1)} \] is investigated. Such an equation describes the dynamics of an electron beam under the influence of the Brillouin flow, and is called Brillouin electron beam focusing equation. The authors discuss the existence of a classical \(2 \pi\) periodic positive solution of (1) in terms of the values of the parameter \(\beta\). They prove the existence of a threshold value \(\beta^*\) such that the branch of the solutions, parametrized by \(\beta\), undergoes a fold bifurcation at \(\beta=\beta^*\). As a consequence, they are able to state also a multiplicity result near bifurcation. The clever argument is based on numerical techniques combined with fixed point theorems. First, the authors find numerically an approximate \(2 \pi\) periodic solution, and then by a fixed point argument, they prove that a `real' \(2 \pi\) periodic solution exists near the `approximate' one. Finally they show how this technique can be generalized to the equation \[ \ddot x + \beta (1-\cos t)x =\frac1{x^p} \] for \(p=2,3\).
reviewed by: Alessandro Calamai
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 37M20
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 6875584
zbMATH Keywords: Brillouin focusing beam equation
zbMATH Keywords: computer-assisted proof
zbMATH Keywords: fold bifurcation
zbMATH Keywords: periodic solutions
zbMATH Keywords: non-autonomous singular ODEs
describes a project that uses: Matlab
describes a project that uses: INTLAB
MaRDI profile type: Publication
full work available at URL: j.jde.2018.04.042
OpenAlex ID: W2799322258
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Wikidata QID: Q129867898
DOI: 10.1016/J.JDE.2018.04.042
description: scientific article; zbMATH DE number 6875584