Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians
type: Item
instance of: scholarly article
title: Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians
zbMATH Open document ID: 0961.34016
author: Yulin Zhao
author: Zhifen Zhang
published in: Journal of Differential Equations
publication date: 2000-09-15T00:00:00Z
review text: When studying the second part of the 16th Hilbert problem one of the interesting problems is that posed by V. I. Arnold: estimate the number of limit cycles for a system on the plane that is a polynomial perturbation of a Hamiltonian polynomial one. Then, due to the Poincaré-Pontryagin method, the problem of estimating the number of cycles (periodic solutions) to the system is reduced to the estimate of the number of zeros of some polynomial Abelian integral along ovals of the unperturbed Hamiltonian for a 1-form depending on the perturbation. In this paper, the authors obtain a linear estimate \(B(n)\leq 7n+5\) in \(n\) (maximal degree of the perturbation) for the Hamiltonian \(H(x,y)=y^{2}/2+U(x)\), \(\deg U(x)=4\), having at least one center singular point.
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 1309592
zbMATH Keywords: periodic solution
zbMATH Keywords: 16th Hilbert problem
zbMATH Keywords: Hamiltonian
zbMATH Keywords: perturbation
zbMATH Keywords: limit cycles
reviewed by: L. M. Lerman
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3581
OpenAlex ID: W2093030779
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DOI: 10.1006/JDEQ.1998.3581
description: scientific article; zbMATH DE number 1309592