Quadratic systems with a 3rd-order (or 2nd-order) weak focus
type: Item
instance of: scholarly article
zbMATH Open document ID: 0995.34023
author: Pingguang Zhang
publication date: 2002-10-15T00:00:00Z
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37C27
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 1691287
zbMATH Keywords: quadratic system
zbMATH Keywords: limit cycle
zbMATH Keywords: weak focus
zbMATH Keywords: Liénard equation
MaRDI profile type: Publication
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title: Quadratic systems with a 3rd-order (or 2nd-order) weak focus
published in: Annals of Differential Equations
review text: It is well known that a limit cycle of quadratic systems (QS) surrounds just one critical point which must be of focus type. Furthermore, only two different nests of limit cycles can coexist for QS. It is also known that if one of these singularities is a weak focus of order three (respectively, two) then it has no (respectively, at most one) limit cycles surrounding it.NEWLINENEWLINENEWLINEThe main result of this paper is that in both cases the other singularity that can also be surrounded by limit cycles, which is a strong focus, has at most one limit cycle around it. This result is coherent with the conjecture that asserts that QS have at most four limit cycles and that when this number is attained three of these limit cycles surround a focus and the fourth one surrounds the other one. The proof reduces the problem to prove the uniqueness of limit cycles for a Liénard differential equation of the form \(x''+f(x)x'+g(x)=0.\) This equation is studied by using Filippov's transformation. Nothing is said about the hyperbolicity of the limit cycle surrounding the strong focus. Similar results are presented in a paper of the same author and \textit{S. Zhao} [Appl. Math., Ser. B (Engl. Ed.) 16, No.~2, 127-132 (2001; Zbl 0994.34019)].
reviewed by: Armengol Gasull
description: scientific article; zbMATH DE number 1691287