Parametric centers for trigonometric Abel equations
type: Item
instance of: scholarly article
title: Parametric centers for trigonometric Abel equations
zbMATH Open document ID: 1163.34017
author: Jean-Pierre Francoise
published in: Journal of Dynamics and Differential Equations
publication date: 2008-12-01T00:00:00Z
review text: The paper is concerned with the one-dimensional version of the center problem for the trigonometric Abel equation \[ \frac{\mathrm{d}y}{\mathrm{d}\theta}=p( \theta) y^{2}+\varepsilon q( \theta) y^{3},\tag{1} \] where \(\varepsilon\) is a small parameter, \(p\) and \(q\) are polynomials in \(\sin\theta\) and \(\cos\theta,\) and \(y( 2\pi,\varepsilon,c) \) denotes the solution of equation (1) satisfying the initial condition \(y( 0,\varepsilon) =c.\) The author is interested in dependence of solutions on the parameter \(\varepsilon\) and, in particular, in obtaining the formula for the first different from zero derivative of the solution with respect to the parameter \(\varepsilon.\) Using the expansion \[ -\frac{1}{y}=-\frac{1}{c}+\varepsilon Y_{1}( c) +\cdots +\varepsilon^{k}Y_{k}( c) +\cdots, \] one can compute the successive derivatives of \(-1/y( 2\pi) \) easier than those of \(y( 2\pi) .\) A recursive formula for the derivatives is obtained through an algorithm that runs for as long as derivatives vanish. In particular, one has \[ Y_{1}( c) =\int_{0}^{2\pi}\frac{cq( \theta) }{1-cP( \theta) }\mathrm{d}\theta, \] where \(q( \theta) \) and \(P( \theta) \) are given trigonometric polynomials, \(P( \theta) =\int_{0}^{\theta}p( \psi) \mathrm{d}\psi.\) Necessary conditions for the derivative \(Y_{1}( c) \) to vanish are derived by studying iterated integrals over the unit complex disc using the tools of complex analysis. This new promising direction of research relates the discussed problem to many interesting research topics like composition of Laurent polynomials or Cauchy integrals of algebraic functions.
reviewed by: Svitlana P. Rogovchenko
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34E10
zbMATH DE Number: 5375865
zbMATH Keywords: center
zbMATH Keywords: trigonometric Abel differential equation
zbMATH Keywords: perturbation
zbMATH Keywords: iterated integral
MaRDI profile type: Publication
full work available at URL: s10884-008-9107-4
OpenAlex ID: W1969702651
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DOI: 10.1007/S10884-008-9107-4
description: scientific article; zbMATH DE number 5375865