Two inverse problems for analytic potential systems
type: Item
instance of: scholarly article
title: Two inverse problems for analytic potential systems
zbMATH Open document ID: 1163.34007
author: Pedro J. Torres
author: Francesc Mañosas
published in: Journal of Differential Equations
publication date: 2008-12-12T00:00:00Z
review text: This paper is a study of two basic inverse problems for analytic potential systems on the plane. Namely, the authors deal with the system \[ \dot x = -y, \quad \dot y = V'(x), \quad x,y \in ({\mathbb R},0), \tag{1} \] where \(V\) is an analytic function defined in a neighborhood of the origin such that \(V(0)=0\), \(V'(0)=0\) and \(V''(0)>0\). Let \(F\) be a prescribed analytic function having a local extremum at \(0\) such that \(F(0)>0\). The first inverse problem is to investigate the existence of some potential \(V\) of system (1) such that \(F\) is the period function associated to \(V\). The main result of the paper is the following: if \(F\) is non-constant then there exists a unique analytic function \(V\) such that the period function of system (1) is \(F\), if \(F\) is constant then there exist infinitely many analytic functions \(V\) satisfying this condition. The second inverse problem concerns the way to complete or extend a potential \(V\) working in the semiplane \(y>0\) to the whole phase plane in order to get an isochronous center at the origin. This problem was raised in the previous work of the authors, where the even case was solved. The result obtained in the present paper solves completely the question.
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5382293
zbMATH Keywords: analytic potential systems
zbMATH Keywords: inverse problems
zbMATH Keywords: center
zbMATH Keywords: focus
zbMATH Keywords: isochronous center
reviewed by: Alexey O. Remizov
MaRDI profile type: Publication
full work available at URL: j.jde.2008.01.010
OpenAlex ID: W2093500777
cites work: Q3699161
cites work: Isochronicity for several classes of Hamiltonian systems
cites work: Period function for a class of Hamiltonian systems
cites work: Isochronicity of a class of piecewise continuous oscillators
cites work: Q3282148
cites work: Q5332361
DOI: 10.1016/J.JDE.2008.01.010
description: scientific article; zbMATH DE number 5382293