Double degeneracy in the problem on unbounded branches of forced oscillations
type: Item
instance of: scholarly article
title: Double degeneracy in the problem on unbounded branches of forced oscillations
zbMATH Open document ID: 1159.34034
DOI: 10.1134/S1064562408020038
author: Alexander Krasnosel'skij
published in: Doklady Mathematics
publication date: 2008-10-27T00:00:00Z
review text: The paper is concerned with the operator equation \[ \mathcal{L}(p;\lambda) x=f(t,x;\lambda) ,\tag{a} \] where \(p=d/dt\) and \(\lambda\in\Lambda\subset\mathbb{R}.\) It is assumed that \(\mathcal{L}\) is a real polynomial in \(p\) and, for any \(\lambda,\) the nonlinearity \(f\) is continuous in \(t\) and \(x\) and \(2\pi\)-periodic in \(t.\) If the polynomial \(\mathcal{L}\) has roots of the form \(ki,\) the set \(\mathfrak{B}_{\varepsilon}\) of all \(2\pi\)-periodic solutions of equation (a) for all \(\left| \lambda-\lambda_{0}\right| \leq\varepsilon\) may be unbounded for any \(\varepsilon.\) Conditions for the set \(\mathfrak{B} _{\varepsilon}\) to be unbounded are derived, and its structure is studied.
reviewed by: Svitlana P. Rogovchenko
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 47J15
zbMATH DE Number: 5357820
zbMATH Keywords: operator equation
zbMATH Keywords: solution set
zbMATH Keywords: parameter
zbMATH Keywords: branching of solutions
zbMATH Keywords: degeneracy
zbMATH Keywords: asymptotic bifurcation point
zbMATH Keywords: periodic solutions
MaRDI profile type: Publication
cites work: Bounded perturbations of forced harmonic oscillators at resonance
cites work: Critical point theory and Hamiltonian systems
cites work: The index at infinity of some twice degenerate compact vector fields
full work available at URL: s1064562408020038
OpenAlex ID: W2023189619
description: scientific article; zbMATH DE number 5357820