Perturbations from a kind of quartic Hamiltonians under general cubic polynomials
type: Item
instance of: scholarly article
title: Perturbations from a kind of quartic Hamiltonians under general cubic polynomials
zbMATH Open document ID: 1192.34044
author: Qi Wang
author: Liqin Zhao
published in: Science in China. Series A
publication date: 2009-12-07T00:00:00Z
review text: The authors extend results by Dumortier and Li on perturbations of a class of quartic Hamiltonian planar vector fields \(X\) (or Liénard equations), admitting now arbitrary cubic perturbations. They prove that if \(X\) has more than one equilibrium, the number of limit cycles of the perturbed system which surround only one center of \(X\) does not exceed twelve. If \(X\) has a saddle loop, they show the existence of a perturbation exhibiting at least three limit cycles, and, moreover, they present a subclass of \(X\) whose perturbations produce at most three limit cycles near the original homoclinic loop. The proofs use standard methods involving Abelian integrals, Picard-Fuchs equations, ratios of (derivatives of) abelian integrals and corresponding Riccati equations. For the expansion of the Abelian integrals near the homoclinic loop, the coefficients up to order four are determined by tedious calculations. As for the zeros of Abelian integrals, the authors apply theorems on the number of intersection points of planar curves that are similar to Khovanskii's results.
reviewed by: Josef Hainzl
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C08
zbMATH DE Number: 5643426
zbMATH Keywords: planar polynomial vector field
zbMATH Keywords: perturbation
zbMATH Keywords: limit cycle
zbMATH Keywords: abelian integral
zbMATH Keywords: homoclinic loop
MaRDI profile type: Publication
full work available at URL: s11425-009-0009-7
OpenAlex ID: W2355296867
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DOI: 10.1007/S11425-009-0009-7
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description: scientific article; zbMATH DE number 5643426