Center conditions. II: Parametric and model center problems
type: Item
instance of: scholarly article
title: Center conditions. II: Parametric and model center problems
zbMATH Open document ID: 0989.34021
DOI: 10.1007/BF02803516
published in: Israel Journal of Mathematics
publication date: 2002-07-28T00:00:00Z
review text: The paper is concerned with the following initial value problem for Abel's equation \[ y'=p(x)y^2+ q(x)y^3,\quad y(0)=y_0, \tag{1} \] where \(p\) and \(q\) are polynomials of the degree \(d_1\) and \(d_2\), respectively. A center condition for the IVP (1) is known as \(y_0=y(0) \equiv y(1)\) for any solution \(y(x)\) to (1). Introducing a parameter in the IVP (1) \[ y'=p(x) y^2+ \varepsilon q(x)y^3, \quad y(0)=y_0, \tag{2} \] where \(p\) and \(q\) are as above and \(\varepsilon\) is a complex number, the authors formulate a parametric center condition for the IVP (2) as \(y(\varepsilon,0) \equiv y(\varepsilon,1)\) for any solution \(y(\varepsilon,x)\) to (2). It is shown that the parametric center condition implies that all the moments \(m_k(1)\) vanish, where \[ m_k(x)= \int^x_0 P^k (t)q(t)dt, \quad P(z)=\int^x_0 p(t)dt. \] The paper is a continuation of the papers by the same authors [Nonlinearity 11, No. 3, 431-443 (1998; Zbl 0905.34051)] and Center conditions, compositions of polynomials and moments on algebraic curves, Ergodic Theory Dyn. Syst. 19, No. 5, 1201-1220 (1999; Zbl 0990.34017)], and one of its main purposes is to prove a conjecture stated in the latter paper for some special cases developing moment techniques and an algebraic analysis of the \(m_k(x)\) and their representation in a canonical form. Finally, a combinatorial approach to the problem of stabilization of the moment sequence is briefly discussed.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 1506582
zbMATH Keywords: polynomial Abel equation
zbMATH Keywords: parametric center condition
zbMATH Keywords: center-focus problem
author: M. Briskin
author: Jean-Pierre Francoise
author: Yosef Yomdin
reviewed by: Svitlana P. Rogovchenko
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02803516
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description: scientific article; zbMATH DE number 1506582