Partial integrals and the first focal value in the problem of centre
type: Item
instance of: scholarly article
title: Partial integrals and the first focal value in the problem of centre
zbMATH Open document ID: 0817.34019
DOI: 10.1007/BF01194012
author: Dumitru Cozma
author: Alexandru Şubă
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 1995-08-13T00:00:00Z
review text: The authors prove that for planar polynomial differential systems of degree \(n\geq 3\) if \(O(0,0)\) is weak focus of order \(\geq 2\), and there exist \(N= (n^ 2+ n- 4)/2\) partial algebraical integrals, then \(O\) is a centre. For cubic systems they obtain 14 series of conditions each ensuring that the origin is a centre. In all these cases the partial integrals are all straight lines. They also give examples to show that for cubic systems sometimes the number of integral straight lines can be \(<4\) in order to ensure \(O(0,0)\) to be a centre; but sometimes on increasing the number of focal values equal to zero along with a corresponding decrease of the number of integral straight lines does not always imply the existence of a centre at \(O\). Notice that for \(n= 2\) \((N= 1)\), assuming that the partial integral is a straight line, a Chinese mathematician N. D. Zhu has already proved in 1965 that \(O\) is a centre. On discussing a similar problem, we refer the reader to a recent paper of \textit{R. E. Kooij} [J. Differ. Equations 116, No. 2, 249-264 (1995)].
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 736848
zbMATH Keywords: planar polynomial differential systems
zbMATH Keywords: partial algebraical integrals
zbMATH Keywords: centre
zbMATH Keywords: cubic systems
zbMATH Keywords: integral straight lines
reviewed by: Yanqian Ye
MaRDI profile type: Publication
cites work: Q5801898
cites work: Q3794514
cites work: Q3805047
cites work: Q4383511
description: scientific article; zbMATH DE number 736848