Configurations of fans and nests of limit cycles for polynomial vector fields in the plane
type: Item
instance of: scholarly article
title: Configurations of fans and nests of limit cycles for polynomial vector fields in the plane
zbMATH Open document ID: 0703.34040
DOI: 10.1016/0022-0396(89)90168-X
author: Anna Cima
author: Jaume Llibre
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: Consider the set of limit cycles for a differential system (*) \(\dot x=P(x,y)\), \(\dot y=Q(x,y)\), where P and Q are polynomials of degree n in x and y. Let X be the polynomial vector field associated to the system (*). Two limit cycles \(C_ 1\) and \(C_ 2\) are f-equivalent if and only if \(s(C_ 1)=s(C_ 2)\), where s(C) is the set, which is formed by the critical points of X belonging to Int C. The limit cycles \(C_ 1\) and \(C_ 2\) are n-equivalent if \(s(C_ 1)\subseteq s(C_ 2)\) and there does not exists another limit cycle \(C_ 3\neq C_ 1\) such that \(C_ 3\subset Int C_ 2\setminus Int C_ 1\) [resp. vice versa]. Using the classes of equivalence the authors introduce the concepts of nest and fan of limit cycles. These concepts are helpful in the study of configurations of limit cycles of X.
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 4152882
zbMATH Keywords: limit cycles
zbMATH Keywords: polynomial vector field
zbMATH Keywords: nest
zbMATH Keywords: fan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4152882