On the period function of \(x^{\prime\prime}+f(x)x^{\prime2}+g(x)=0\)
type: Item
instance of: scholarly article
title: On the period function of \(x^{\prime\prime}+f(x)x^{\prime2}+g(x)=0\)
zbMATH Open document ID: 1048.34068
DOI: 10.1016/S0022-0396(03)00067-6
published in: Journal of Differential Equations
publication date: 2004-01-26T00:00:00Z
review text: The paper is devoted to a study of the period function \(T(x, y)\), which associates to every point \((x,y)\) from a neighborhood of the center \(O\) of the equation \(x''+f(x){x'}^2+g(x)=0\) the corresponding period \(T\). The function \(T\) has a strong relationship to the existence and uniqueness of the solutions of some boundary value problem. The author considers some classes of planar systems equivalent to such equation. The article contains a sufficient condition for the monotonicity of \(T\), or for the isochronicity of \(O\), which is also necessary, when \(f\) and \(g\) are odd and analytic.
reviewed by: Alexander Grin
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C07
zbMATH DE Number: 2033005
zbMATH Keywords: center
zbMATH Keywords: period function
zbMATH Keywords: monotonicity
zbMATH Keywords: polynomial systems
author: Marco Sabatini
MaRDI profile type: Publication
cites work: Isochronicity via normal form
cites work: The monotonicity of the period function for planar Hamiltonian vector fields
cites work: Remarks on some monotonicity conditions for the period function
cites work: Q3739490
cites work: Period function for a class of Hamiltonian systems
cites work: First derivative of the period function with applications
cites work: Q4771385
cites work: Commutators and linearizations of isochronous centers
cites work: Remarks on Periods of Planar Hamiltonian Systems
cites work: The centres in the reduced Kukles system
cites work: On period functions of Liénard systems
cites work: A class of Hamiltonian systems with increasing periods.
full work available at URL: s0022-0396(03)00067-6
OpenAlex ID: W2036672938
description: scientific article; zbMATH DE number 2033005