On the boundedness of solutions to a nonlinear singular oscillator
type: Item
instance of: scholarly article
title: On the boundedness of solutions to a nonlinear singular oscillator
zbMATH Open document ID: 1189.34070
author: Anna Capietto
author: Walter Dambrosio
author: Bin Liu
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2010-01-29T00:00:00Z
review text: The authors consider a second order scalar differential equation of the form \[ x''+V'(x)=p(t), \] where \(p\) is periodic and the potential \(V\) has a strong singularity of repulsive type. Sufficient conditions are given for boundedness of all solutions, as well as for existence of Aubry-Mather sets. In the proofs, the Hamiltonian is written in area-angle variables and then careful estimates on the Poincaré map are crafted in order to apply Ortega's variant of Moser's twist theorem and Pei's theorem on the existence of Aubry-Mather sets. If compared with related works on regular potential, the presence of the singularity supposes an additional difficulty.
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 37E40
zbMATH DE Number: 5664540
zbMATH Keywords: singular potential
zbMATH Keywords: boundedness
zbMATH Keywords: Moser's twist theorem
zbMATH Keywords: Aubry-Mather sets
MaRDI profile type: Publication
full work available at URL: s00033-008-8094-y
OpenAlex ID: W2090153044
DOI: 10.1007/S00033-008-8094-Y
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description: scientific article; zbMATH DE number 5664540