Principal Poincaré-Pontryagin function of polynomial perturbations of the Hamiltonian triangle
type: Item
instance of: scholarly article
title: Principal Poincaré-Pontryagin function of polynomial perturbations of the Hamiltonian triangle
zbMATH Open document ID: 1124.34020
author: Marco Uribe
published in: Journal of Dynamical and Control Systems
publication date: 2007-05-24T00:00:00Z
review text: The author investigates the displacement map \(D(t,\varepsilon)\) for a small polynomial perturbation \[ dF(x,y)+\varepsilon[P_d(x,y)dx+Q_d(x,y)dy]=0, \] where \(F(x,y)=x[y^2-(x-3)^2]\) is the so called Hamiltonian triangle and \(P_d,Q_d\) are polynomials of degree \(n\). The first nonzero coefficient \(M_k(t)\) in the expansion with respect to the small parameter, \[ D(t,\varepsilon)=\varepsilon^k M_k(t)+O(\varepsilon^{k+1}), \] is sometimes called Poincaré-Pontryagin function and its zeroes correspond to the limit cycles that tend as \(\varepsilon\to 0\) to periodic orbits of the unperturbed system. The number \(k\) depends on the perturbation. In the quadratic case \(d=2\), the function \(D(t,\varepsilon)\) was first studied by \textit{H. Żołądek} [J. Differ. Equations 109, 223--273 (1994; Zbl 0797.34044)] and by \textit{I. D. Iliev} [Adv. Differ. Equ. 1, No. 4, 689--708 (1996; Zbl 0851.34042), J. Differ. Equations 128, No. 1, 309--326 (1996; Zbl 0853.58084)]. It was established that \(M_1\) and \(M_2\) are Abelian integrals but \(M_3\) is not; moreover, \(M_1(t)\neq 0\), \(M_2(t)\) has at most one zero, \(M_3\) and \(D(t,\varepsilon)\) have at most three zeroes. In this paper the author proves for any \(d\) that the first nonzero term \(M_k(t)\) belongs to a \({\mathbf C}[t,1/t]\) module of the same three integrals over \(F=t\) as those included in \(M_3(t)\) when \(d=2\). Also, for \(d\geq 5\), the coefficient \(M_2(t)\) is not (generically) an Abelian integral, too.
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C07
zbMATH DE Number: 5156804
zbMATH Keywords: Poincaré-Pontryagin function
zbMATH Keywords: Abelian integral
zbMATH Keywords: polynomial perturbation
zbMATH Keywords: weakened Hilbert 16th problem
zbMATH Keywords: Hamiltonian triangle
reviewed by: Iliya D. Iliev
MaRDI profile type: Publication
full work available at URL: s10450-006-9687-4
OpenAlex ID: W2057401125
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DOI: 10.1007/S10450-006-9687-4
description: scientific article; zbMATH DE number 5156804