Center conditions. III: Parametric and model center problems
type: Item
instance of: scholarly article
title: Center conditions. III: Parametric and model center problems
zbMATH Open document ID: 0989.34022
DOI: 10.1007/BF02803517
published in: Israel Journal of Mathematics
publication date: 2002-07-28T00:00:00Z
review text: The authors consider the Abel equations \[ (*)\;y'=p(x)y^2+ q(x)y^3\quad\text{and}\quad (**)\;y'=p(x)y^2+ \varepsilon q(x)y^3 \] with \(p(x)\), \(q(x)\) polynomials in \(x\) of degrees \(d_1,d_2\), respectively, and \(\varepsilon\) a complex parameter. A center condition for (*) is that \(y(0)=y(1)\) for any solution \(y(x)\) to (*). The equation (**) is said to have a parameter center if, for any \(\varepsilon\) and for any solution \(y(x,\varepsilon)\) to (**), \(y(0, \varepsilon)= y(1,\varepsilon)\). The authors first give another proof, shown before by the same authors, that the parametric center condition implies vanishing of all the moments \(m_k(1)\), where \(m_k(x)= \int^x_0 P^k(t) q(t)dt\), \(P(x)=\int^x_0 p(t)dt\), and then investigate the structure of zeros of \(m_k(x)\) and generalize a canonical representation of \(m_k(x)\). On this basis, the authors study some parametric center problems. For part II see the review above.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1506583
zbMATH Keywords: Abel equations
author: M. Briskin
author: Jean-Pierre Francoise
author: Yosef Yomdin
reviewed by: Maoan Han
MaRDI profile type: Publication
cites work: On a condition for a centre of cubic non-autonomous equations
cites work: Non-autonomous equations related to polynomial two-dimensional systems
cites work: Periodic solutions of a quartic nonautonomous equation
cites work: Q4435496
cites work: The Bautin ideal of the Abel equation
cites work: Center conditions, compositions of polynomials and moments on algebraic curves
cites work: Center conditions. II: Parametric and model center problems
cites work: Small-amplitude limit cycles in polynomial Liénard systems
cites work: Word problems related to periodic solutions of a non-autonomous system
cites work: Word problems related to derivatives of the displacement map
cites work: On the number of solutions of the equation \(\sum^n_{j=0}a_j(t)x^j,0\leq t\leq 1\), for which \(x(0)=x(1)\)
full work available at URL: https://doi.org/10.1007/bf02803517
OpenAlex ID: W2078486315
description: scientific article; zbMATH DE number 1506583