Centers with degenerate infinity and their commutators.
type: Item
instance of: scholarly article
title: Centers with degenerate infinity and their commutators.
zbMATH Open document ID: 1037.34024
DOI: 10.1016/S0022-247X(02)00625-X
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-04T00:00:00Z
review text: This paper is devoted to study the isochronicity of the origin of planar polynomial systems of the form \[ \dot x= P(x,y)=-y+P_2(x,y)+\cdots+ P_n(x,y), \] \[ \dot y= Q(x,y)=x+Q_2(x,y)+\cdots+ Q_n(x,y), \] where \(P_i\) and \(Q_i\) are homogeneous polynomials of degree \(i\) and \(xQ_n(x,y)\equiv yP_n(x,y).\) This last identity forces the infinity of the Poincaré compactification of the vector field to be full of critical points and, for this reason, the above systems are called systems with degenerate infinity. The authors derive necessary conditions for a system with degenerate infinity \(X=(P,Q)\) to admit a transversal polynomial commutator \(Y,\) i.e., a polynomial vector field \(Y\) such that is transversal to \(X\) in a punctured neighborhood of the origin and verifies \([X,Y]=0.\) Recall that if such an \(Y\) exists then the origin of \(X\) is an isochronous center. The results obtained are applied to constant angular speed systems (also called rigid systems), which are systems \(X\) such that \(xQ_i(x,y)\equiv yP_i(x,y)\) for \(i=2,\ldots,n,\) giving that \(Y\) has to be of the form \(Y(x,y)= (xK(x,y),yK(x,y))\) for some polynomial \(K\) of degree at most \(n-1.\) Finally, the case \(X=(-y+P_s(x,y)+xH(x,y), x+Q_s(x,y)+yH(x,y))\) where \(H\) is a polynomial of degree at most \(n-1\) and having no monomials of degree lower than \(s,\) is studied. The paper ends with several comments, examples and open problems. For instance, the system \(\dot x=-y+xy+xy^3,\) \(\dot y=x+y^2+y^4,\) that does not admit polynomial commutators, has an isochronous center. Is it true that all centers for systems with constant angular speed are either reversible or admit a polynomial commutator?
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34A26
zbMATH DE Number: 1903475
zbMATH Keywords: polynomial vector field
zbMATH Keywords: center problem
zbMATH Keywords: isochronous center
zbMATH Keywords: commutator
zbMATH Keywords: reversibility
author: Antonio Algaba
author: M. Reyes
MaRDI profile type: Publication
cites work: Isochronicity via normal form
cites work: Q4524908
cites work: Non-autonomous equations related to polynomial two-dimensional systems
cites work: Periodic solutions of a quartic nonautonomous equation
cites work: Integrability of cubic systems with degenerate infinity
cites work: Isochronous Centers in Planar Polynomial Systems
cites work: Small-amplitude limit cycles in polynomial Liénard systems
cites work: Q4283276
cites work: Conditions for a centre in a simple class of cubic systems
cites work: Coexisting Isochronous and Nonisochronous Centres
cites work: Q4394773
cites work: The Number of Periodic Solutions of the Equation Ż=z N +p 1 (t )z
cites work: Q3308384
cites work: Quadratic-like cubic systems
cites work: Q5513883
cites work: Darboux linearization and isochronous centers with a rational first integral
cites work: Linearization of isochronous centers
cites work: Q3803878
cites work: Q5665494
cites work: Q4102153
cites work: Quadratic Isochronous centers commute
cites work: Characterizing isochronous centres by Lie brackets
cites work: On period functions of Liénard systems
cites work: Q3282148
cites work: Isochronicity and commutation of polynomial vector fields
description: scientific article; zbMATH DE number 1903475