Linear estimation and distribution of the limit cycles bifurcated from a kind of degenerate polycycles
type: Item
instance of: scholarly article
title: Linear estimation and distribution of the limit cycles bifurcated from a kind of degenerate polycycles
zbMATH Open document ID: 1123.34025
published in: Science in China. Series A
publication date: 2007-06-14T00:00:00Z
review text: The paper studies the number and distribution of limit cycles bifurcated from a degenerate planar polycycle through three singular points: a saddle-node \(P_0\), a fine saddle \(P_1\) with finite order \(m\in\mathbb N\), and a contractive (attracting) saddle \(P_2\) with the hyperbolicity ratio \(q_2(0)\notin\mathbb Q\). The connection between \(P_0\) and \(P_1\) is of hh-type and the connection between \(P_0\) and \(P_2\) is of hp-type. It is assumed that the connection between \(P_0\) to \(P_2\) and \(P_0\) to \(P_1\) keeps unbroken. It is shown that the cyclicity of this polycycle is not more than \(3m+1\), which is linearly dependent on the order of the resonant saddle \(P_1\). It is also shown that the cyclicity is not more than \(m+3\) if \(q_2(0)>m\), and that the nearer \(q_2(0)\) is close to 1, the more limit cycles are bifurcated.
reviewed by: Eugene Ershov
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 37G25
Mathematics Subject Classification ID: 37C29
zbMATH DE Number: 5164226
zbMATH Keywords: degenerate polycycle
zbMATH Keywords: cyclicity
zbMATH Keywords: finitely-smooth normal form
zbMATH Keywords: transition map
author: Liqin Zhao
MaRDI profile type: Publication
full work available at URL: s11425-007-0015-6
OpenAlex ID: W2023533391
cites work: A pure point spectrum of the stochastic one-dimensional Schrödinger operator
cites work: Q3831770
cites work: Hilbert's 16th problem for quadratic vector fields
cites work: Q4841398
cites work: Cyclicity of elementary polycycles and ensembles with codimension \(3\) degeneration.
cites work: Cyclicity of a kind of degenerate polycycles through three singular points
cites work: Finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type
cites work: Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics
cites work: Degenerate and non-trivial hyperbolic 2-polycycles: appearance of two independent Ecalle-Roussarie compensators and Khovanskii's theory
cites work: Elementary graphics of cyclicity 1 and 2
cites work: Finitely-smooth normal forms of local families of diffeomorphisms and vector fields
cites work: Q5751632
cites work: On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields
DOI: 10.1007/S11425-007-0015-6
description: scientific article; zbMATH DE number 5164226