Cyclicity of homoclinic loops and degenerate cubic Hamiltonians
type: Item
instance of: scholarly article
title: Cyclicity of homoclinic loops and degenerate cubic Hamiltonians
zbMATH Open document ID: 0959.34022
DOI: 10.1007/BF02880079
published in: Science in China. Series A
publication date: 2000-11-16T00:00:00Z
review text: New conditions for a planar system of the form \[ \dot x=H_y+\varepsilon f(x,y,\varepsilon ,\delta),\quad \dot y=-H_x+\varepsilon g(x,y,\varepsilon ,\delta), \] with \(H,f,g\in C^{\infty}\), \(\varepsilon\geq 0\) is small, \(\delta\in\mathbb R^n\) with \(n>3\) to have for \(\varepsilon +|\delta -\delta _0|\) small at most two limit cycles near a homoclinic loop of the unperturbed system (\(\varepsilon =0\)) are obtained. As an application it is proved that a homoclinic loop of a degenerate cubic Hamiltonian has cyclicity two under arbitrary quadratic perturbations.
reviewed by: Eugene Ershov
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 1531239
zbMATH Keywords: perturbations of planar Hamiltonian systems
zbMATH Keywords: homoclinic loop
zbMATH Keywords: bifurcation
zbMATH Keywords: cyclicity
zbMATH Keywords: quadratic systems
zbMATH Keywords: limit cycle
author: Maoan Han
author: Yanqian Ye
author: Deming Zhu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1531239