Analytic integrability for some degenerate planar systems
type: Item
instance of: scholarly article
title: Analytic integrability for some degenerate planar systems
zbMATH Open document ID: 1278.34001
published in: Communications on Pure and Applied Analysis
publication date: 2013-06-06T00:00:00Z
review text: The authors prove that an analytic system of ordinary differential equations in a neighborhood of the origin of the form \[ \dot x = y^3 + 3Ax^2y + \cdots,\,\dot y = -x^3 - 3Axy^2 + \cdots \] (where the omitted terms vanish through order three) is analytically integrable if and only if it is formally equivalent to its homogeneous part of degree 3. They also characterize analytic integrability in terms of existence of a formal Lie symmetry and in terms of formal orbital reversibility. They demonstrate that the family contains systems that have a center at the origin yet have no analytic first integral.
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 6171074
zbMATH Keywords: nonlinear differential equations
zbMATH Keywords: integrability
zbMATH Keywords: degenerate center
author: Antonio Algaba
author: Cristóbal García
author: Jaume Giné
MaRDI profile type: Publication
full work available at URL: cpaa.2013.12.2797
OpenAlex ID: W2334465368
DOI: 10.3934/CPAA.2013.12.2797
description: scientific article; zbMATH DE number 6171074