Strong isochronism of a center and a focus for systems with homogeneous nonlinearities
type: Item
instance of: scholarly article
title: Strong isochronism of a center and a focus for systems with homogeneous nonlinearities
zbMATH Open document ID: 1198.34047
DOI: 10.1134/S0012266109020025
published in: Differential Equations
publication date: 2009-10-08T00:00:00Z
review text: Given a real polynomial planar differential equation \[ x'=-y+p_n(x,y),~y'=x+q_n(x,y), \] where \(p_n\) and \(q_n\) are homogeneous polynomials of degree \(n\geq2,\) the author proposes a new method to detect isochronism. It is based on the existence of a transformation, \(X=x+f(x,y)\), \(Y=y,\) that converts it into a system of the form \[ X'=-Y+X\,S(X,Y),~Y'=X+Y\,S(X,Y). \] Notice that in polar coordinates this last system writes as \(\theta'=1\) and so it has at the origin either an isochronous center, or an isochronous focus. The author applies this method when \(2\leq n\leq 5\) and gives a list of isochronous centers and foci. As fas as the reviewer knows, some of the given systems exhibiting isochronism are new. It is also noticed in the paper that this method always provides so called strong isochronous systems because the transformation leaves the \(x\)-axis invariant and the systems present isochronism with respect to it.
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 5611131
zbMATH Keywords: isochronous center
zbMATH Keywords: isochronous focus
zbMATH Keywords: normal form
zbMATH Keywords: homogeneous nonlinearities
author: A. E. Rudenok
MaRDI profile type: Publication
cites work: Isochronous and strongly isochronous oscillations of two-dimensional monodromic holomorphic dynamical systems
cites work: Linearization of isochronous centers
cites work: Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial
cites work: Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials
cites work: Q3772536
description: scientific article; zbMATH DE number 5611131