Periodic, quasi-periodic and unbounded solutions of radially symmetric systems with repulsive singularities at resonance
type: Item
instance of: scholarly article
title: Periodic, quasi-periodic and unbounded solutions of radially symmetric systems with repulsive singularities at resonance
zbMATH Open document ID: 1330.34069
author: Qihuai Liu
author: Pedro J. Torres
author: Dingbian Qian
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2015-10-15T00:00:00Z
review text: The authors deal with a radially symmetric system of the type \[ u''(t)=-h(|u(t)|,t)\,\frac{u(t)}{|u(t)|}\,, \] where \(h:(0,+\infty)\times{\mathbb R}\to{\mathbb R}\) is continuous and \(2\pi\)-periodic in \(t\). Writing \[ h(x,t)=\frac{n^2}{4} x +f(x,t)\,, \] where \(n\) is a positive integer, they consider the function \(\sigma_f:{\mathbb R}\to{\mathbb R}\) defined as \[ \sigma_f(\theta):=\int_0^{2\pi}\left|\sin\left(\frac{\theta}{2}+\frac{n}{2}t\right)\right|f(+\infty,t)\,dt\,. \] They prove that, if \(\sigma_f(\theta)\) doesn't change sign, then there are infinitely many periodic solutions rotating around the origin. Otherwise, if there is a simple zero of \(\sigma_f(\theta)\), all solutions starting sufficiently far from the origin will be unbounded, either in the past, or in the future.
reviewed by: Alessandro Fonda
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 6494619
zbMATH Keywords: resonance
zbMATH Keywords: periodic solutions
zbMATH Keywords: isochronous system
MaRDI profile type: Publication
full work available at URL: s00030-015-0316-3
OpenAlex ID: W2012884452
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DOI: 10.1007/S00030-015-0316-3
description: scientific article; zbMATH DE number 6494619