The period function for second-order quadratic ODEs is monotone
type: Item
instance of: scholarly article
title: The period function for second-order quadratic ODEs is monotone
zbMATH Open document ID: 1074.34033
DOI: 10.1007/BF02970864
author: Armengol Gasull
author: Antoni Guillamon
author: J. Villadelprat
published in: Qualitative Theory of Dynamical Systems
publication date: 2005-05-03T00:00:00Z
review text: Very little is known about the period function for large families of centers. In one of the pioneering works on this problem, C. Chicone conjectured that all centers encountered in the family of second-order quadratic differential equations \(\ddot{x}=V(x,\dot{x}),\) being \(V\) a quadratic polynomial, should have a monotone period function. C. Chicone solved some of the cases but some others remained unsolved. In this paper, the authors fill up these gaps by using a new technique based on the existence of Lie symmetries. This technique can be used to reprove all the cases that have been already solved, providing in this way a compact proof for all second-order quadratic differential equations. They also prove that the property on the period function is no longer true when \(V\) is a polynomial of which a nonlinear part is homogeneous of degree \(n>2.\)
reviewed by: Valery A. Gaiko
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 2164197
zbMATH Keywords: second-order quadratic differential equation
zbMATH Keywords: monotone period function
zbMATH Keywords: Lie symmetry
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02970864
OpenAlex ID: W2103135356
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description: scientific article; zbMATH DE number 2164197