Strong isochronism of polynomial differential systems with a center
type: Item
instance of: scholarly article
zbMATH Open document ID: 0978.34023
author: V. V. Amelkin
author: Kasim Mukhamed Al'-Khaider
publication date: 2001-03-19T00:00:00Z
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34C07
zbMATH DE Number: 1577610
zbMATH Keywords: strong isochronicity
zbMATH Keywords: center
zbMATH Keywords: polynomial system
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: symmetry
zbMATH Keywords: orbits
MaRDI profile type: Publication
title: Strong isochronism of polynomial differential systems with a center
published in: Differential Equations
review text: Let \(\dot x= P(x,y)\), \(\dot y= Q(x,y)\) be a plane differential system \((S)\). A center \(O\) of \((S)\) is said to be strongly isochronous of order \(n\) when, in a neighbourhood \(U\) of \(O\), all trajectories starting at time \(t_0\) from the ray with polar angle \(\phi_0\) pass from the ray \(\phi= \phi_0+{2k\pi\over n}\) to the ray \(\phi= \phi_0+ {2(k+ 1)\pi\over n}\), \(k= 0,\dots, n-1\), in the same time \(T= {2\pi\over n}\).NEWLINENEWLINENEWLINEHere, some necessary and/or sufficient conditions for the strong isochronicity of a center of a polynomial system are given. Hamiltonian systems of degree 4 are considered, as well as special classes of non-Hamiltonian polynomial systems of degree 3, 4, 5. Most of the paper is concerned with strong isochronicity of order 2, which is often related to symmetry properties of the orbits to \((S)\).
reviewed by: Marco Sabatini
description: scientific article