Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle
type: Item
instance of: scholarly article
title: Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle
zbMATH Open document ID: 1345.34041
author: Jihua Yang
author: Liqin Zhao
published in: Nonlinear Analysis. Real World Applications
publication date: 2015-12-22T00:00:00Z
review text: Consider the perturbed polynomial system \[ \frac{dx}{dt} = \frac{\partial H}{\partial y} + \varepsilon f(x,y),\quad \frac{dy}{dt}=-\frac{\partial H}{\partial x} + \varepsilon g(x,y) , \] where \(H(x,y)\) is a polynomial function of degree \(m+1\), \(f(x,y)\) and \(g(x,y)\) are polynomial functions of degree \(n\), and \(\varepsilon\) is a small parameter. Suppose that for \(\varepsilon =0\), the system has a family of closed orbits \(\Gamma_{h}\) define by \(H(x,y)=h\). Let \(Z(m,n)\) be the least upper bound for the number of isolated zeros of the abelian integral \[ I(h) = \oint_{\Gamma_{h}}^{} g(x,y) dx -f(x,y) dy . \] The weakened Hilbert's 16th problem is to find \(Z(m,n)\) in terms of \(m\) and \(n\). There are many results for certain values of \(m\) and \(n\). The authors consider the Hamiltonian \[ H(x,y) = -x^{2} +x^{4} + y^{4} + r\,x^{2}y^{2},\,\,r \geq 0, r\neq 2. \] The unperturbed system has three singular points \[ S(0,0),C_{1} (-\frac{\sqrt{2}}{2}), C_{2}(\frac{\sqrt{2}}{2},0). \] The period annulus around \(C_{1}\) (\(C_{2}\)) is denoted by \(\Gamma_{h}^{L}\) (\(\Gamma_{h}^{R}\)). Let \(B_{1}(n)\) be the value of \(Z\) corresponding to \(\Gamma_{h}^{L}\) or \(\Gamma_{h}^{R}\). Let \(B_{2}\) be the value of \(Z\) corresponding to the unbounded period annulus. Using a known procedure, the authors prove that \[ \begin{aligned} B_{1} (n) &\leq 40n +6 \left[ \frac{n-1}{4} \right] + 6 \left[ \frac{n-2}{4} \right] + 11,\\ B_{2}(n) &\leq 3n + 3 \left[ \frac{n-1}{4} \right] + 13.\end{aligned} \]
reviewed by: M. A. M. Alwash
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6523606
zbMATH Keywords: abelian integral
zbMATH Keywords: weakened Hilbert's 16th problem
zbMATH Keywords: Picards-Fuchs equation
zbMATH Keywords: Chebyshev space
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2015.08.005
OpenAlex ID: W1931587871
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DOI: 10.1016/J.NONRWA.2015.08.005
description: scientific article; zbMATH DE number 6523606