Bifurcation of limit cycles in a particular class of quadratic systems with two centres
type: Item
instance of: scholarly article
title: Bifurcation of limit cycles in a particular class of quadratic systems with two centres
zbMATH Open document ID: 0814.34024
author: Robert E. Kooij
author: W. T. Van Horssen
published in: Journal of Differential Equations
publication date: 1994-12-19T00:00:00Z
review text: Suppose a differential equation in the plane has an invariant annulus of periodic solutions. A classic problem is to determine if any of these periodic solutions continue to exist after adding a small perturbation. The problem is solved by studying the fixed points of the Poincaré return map, parametrized by the perturbation, and defined on a curve transverse to the period annulus. In particular, the derivatives required to find the fixed points of the resulting parametrized family of maps are computable in terms of certain integrals around the unperturbed periodic orbits. However, in an application of the theory to a specific system, the analysis of the integrals so obtained is usually very difficult. In the paper under review, an analysis of this type is carried out for the perturbations, within the class of quadratic systems, for the period annuli surrounding the centers at the origin and at the point with coordinates \((0,1/n)\) of the quadratic system \[ \dot x= -y+ ny^ 2,\qquad \dot y= x- xy,\qquad 0< n< 1. \] Using clever analytic arguments and some numerical experiments, the authors are able to determine the number of periodic orbits that survive a quadratic perturbation in most cases. Their precise results are too numerous to formulate here. However, they find perturbations with at least two limit cycles. At present, within the class of quadratic systems, it seems that there is no known example of a system with a period annulus such that three of its periodic orbits continue to limit cycles after a perturbation.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 703251
zbMATH Keywords: bifurcation
zbMATH Keywords: quadratic systems
zbMATH Keywords: number of periodic orbits
zbMATH Keywords: quadratic perturbation
zbMATH Keywords: limit cycles
zbMATH Keywords: period annulus
reviewed by: C. Chicone
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1160
OpenAlex ID: W2016007647
DOI: 10.1006/JDEQ.1994.1160
description: scientific article; zbMATH DE number 703251