Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree
type: Item
instance of: scholarly article
title: Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree
zbMATH Open document ID: 0854.34033
author: C. B. Collins
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-01-06T00:00:00Z
review text: The author considers a vector field in the plane given by an ordinary differential equation \(dx/dt= X(x)\), where the components of \(X\) are homogeneous polynomials of degree \(n- 1\) and \(n+ 1\). Algebraic conditions for the origin to be a centre and a stable/unstable focus are given. These necessary and sufficient conditions are applied to simple examples (linear, cubic and quintic cases). For some classes of systems with homogeneous nonlinearities criteria for the origin to be a (uniformly isochronous) centre are also given. Finally, some errors in previously published phase portraits of homogeneous cubic systems are provided in the appendix.
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 861793
zbMATH Keywords: homogeneous polynomial planar vector field
zbMATH Keywords: centre
zbMATH Keywords: focus
reviewed by: Yuriy V. Rogovchenko
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1385
OpenAlex ID: W2081704054
DOI: 10.1006/JMAA.1995.1385
Recommended article: Conditions for a centre in a simple class of cubic systems
Recommended article: Sufficient conditions for the existence of a center in polynomial systems of arbitrary degree
Recommended article: On the center conditions of certain cubic systems
Recommended article: Centers and foci for a class of cubic systems
Recommended article: The centre conditions for a certain cubic system
Recommended article: Q4796637
Recommended article: Q3473414
Recommended article: Invariant algebraic curves and conditions for a centre
Recommended article: Q4927903
Recommended article: Q4568510
description: scientific article; zbMATH DE number 861793