On the number of critical periods for planar polynomial systems of arbitrary degree
type: Item
instance of: scholarly article
title: On the number of critical periods for planar polynomial systems of arbitrary degree
zbMATH Open document ID: 1203.34049
author: Armengol Gasull
author: Changjian Liu
author: Jiazhong Yang
published in: Journal of Differential Equations
publication date: 2010-07-19T00:00:00Z
review text: A first approach to the celebrate Hilbert's 16th problem is the construction of examples having as many limit cycles as possible, in other words to find lower bounds for the so-called Hilbert's numbers. If one only considers limit cycles surrounding one critical point, then the best lower bound for the number of limit cycles is \(O(n^2)\), while the best results concerning any configuration of limit cycles provide \(O( n^2 \log n)\) lower bounds. It is worth to notice the parallelism between Hilbert's 16th problem and the problem of the number of critical period of a planar vector field of degree \(n\). In this paper the authors construct a class of planar systems of arbitrary degree \(n\) having a reversible center at the origin and such that the number of critical periods on its period annulus grows quadratically with \(n\). Previous results on this subject generated systems having linear growth.
reviewed by: Jaume Giné
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 37C27
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 5757644
zbMATH Keywords: reversible center
zbMATH Keywords: period function
zbMATH Keywords: critical period
zbMATH Keywords: Hilbert's 16th problem
MaRDI profile type: Publication
full work available at URL: j.jde.2010.01.002
OpenAlex ID: W2031615970
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DOI: 10.1016/J.JDE.2010.01.002
description: scientific article; zbMATH DE number 5757644