Reversibility and quasi-homogeneous normal forms of vector fields
type: Item
instance of: scholarly article
title: Reversibility and quasi-homogeneous normal forms of vector fields
zbMATH Open document ID: 1202.34069
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-06-11T00:00:00Z
review text: The authors use tools of quasi-homogeneous normal form theory to study the reversibility of a vector field. The property of reversibility (time-reversal symmetry) of a planar vector field is a sufficient condition for a monodromic singular point to be a center. This provides a strong motivation to study the reversibility of a vector field since the center problem is one of the main open problem in the qualitative theory of planar dynamical systems. In fact, for any planar system having a non-degenerate (respectively, nilpotent) center, there exists a analytic transformation such that the system becomes reversible (respectively, orbitally reversible). Hence, the reversibility problem can provide new information in the center problem. The main results of the paper are the following: There are given necessary conditions for the reversibility of a vector field using reversible generators, see Theorem 2.1. It provides an algorithm, via Lie triangle, that detects the non-reversibility of vector fields, see Theorem 3.19. These results are applied to several examples of nilpotent and degenerate centers. The paper provides families of nilpotent centers which are non-reversible.
reviewed by: Jaume Giné
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5720180
zbMATH Keywords: reversibility
zbMATH Keywords: quasi-homogeneous
zbMATH Keywords: normal form
zbMATH Keywords: center singular point
author: Antonio Algaba
author: Cristóbal García
author: Marco A. Teixeira
MaRDI profile type: Publication
full work available at URL: j.na.2010.03.046
OpenAlex ID: W2025520707
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DOI: 10.1016/J.NA.2010.03.046
description: scientific article; zbMATH DE number 5720180