On the periodic oscillations of \(\ddot x=g(x)\)
type: Item
instance of: scholarly article
title: On the periodic oscillations of \(\ddot x=g(x)\)
zbMATH Open document ID: 0701.34049
DOI: 10.1016/0022-0396(89)90076-4
author: Gaetano Zampieri
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: The paper is a continuation of the author's work in [J. Differ. Equations 74, No.2, 254-265 (1988; Zbl 0668.34051)]. For the scalar equation \(x''+xf(x)=0\), \(f(0)>0\), and for a strictly positive function m and for a periodic solution \(x(.,x_ 0,x'_ 0)\) the change in time is performed by the formula \(\tau (t,x_ 0,x'_ 0)=\int^{t}_{0}m(x(\theta,x_ 0,x'_ 0))d\theta.\) It is said that the origin is an m-isochronous center for the equation if all solutions in a neighbourhood of the origin have the same period with respect to the new time associated to m. Conditions for m-isochronism are given and it is proved that the functions f for which the origin is f-isochronous and m-isochronous for \(m(x)=(1+\alpha x)^{-2}\) are defined by \(f(x)=f(0)/(1+\alpha x)^ 3.\) A formula is obtained for functions f for which only m-isochronism with respect to the above m holds. It is proved that the origin is a stable equilibrium for the system \(x''+xf(x)=0\), \(y''+yf(x)=0\) iff \(f(0)>0\) and the origin is f-isochronous for the first equation in the system.
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 4148595
zbMATH Keywords: m-isochronous center
zbMATH Keywords: m-isochronism
MaRDI profile type: Publication
full work available at URL: 0022-0396(89)90076-4
OpenAlex ID: W1994403283
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description: scientific article; zbMATH DE number 4148595