The composition conjecture for Abel equation
type: Item
instance of: scholarly article
title: The composition conjecture for Abel equation
zbMATH Open document ID: 1190.34048
author: M. A. M. Alwash
published in: Expositiones Mathematicae
publication date: 2009-06-30T00:00:00Z
review text: The paper deals with the composition conjecture for the Abel differential equation \[ \dot{z}=A(t)z^3+B(t)z^2, \] where \(z\) is real and \(A,B\) are continuous functions. It states that if all solutions in a neighborhood of the origin are periodic then the indefinite integrals of coefficients from the equation are compositions of a periodic functions. By means of the Cherkas transformation the author explains the relations with the classical center problem. At present the conjecture is verified for a particular cases, when the coefficients are polynomial functions of small degrees or have small number of terms. Therefore, the author presents a widely accessible exposition of the conjecture and verifies it for certain classes of coefficients.
reviewed by: Alexander Grin
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 5572050
zbMATH Keywords: Abel differential equation
zbMATH Keywords: two-dimensional polynomial system
zbMATH Keywords: periodic solution
zbMATH Keywords: center condition
zbMATH Keywords: composition conjecture
zbMATH Keywords: moment
MaRDI profile type: Publication
full work available at URL: j.exmath.2009.02.002
OpenAlex ID: W1967089765
Wikidata QID: Q123238329
cites work: On a condition for a centre of cubic non-autonomous equations
cites work: On the center conditions of certain cubic systems
cites work: Q4444803
cites work: Non-autonomous equations related to polynomial two-dimensional systems
cites work: Generalized centre conditions and multiplicities for polynomial Abel equations of small degrees
cites work: Center conditions at infinity for Abel differential equations
cites work: Q3371979
cites work: Q4090690
cites work: Qualitative theory of planar differential systems
cites work: Cubic systems and Abel equations
cites work: Solution of the polynomial moment problem
cites work: Q3264374
cites work: A counterexample to the &#x201C;composition conjecture&quot;
cites work: On polynomials orthogonal to all powers of a given polynomial on a segment
cites work: Q4817701
DOI: 10.1016/J.EXMATH.2009.02.002
description: scientific article; zbMATH DE number 5572050