Separatrix cycles and mutiple limit cycles in a class of quadratic systems
type: Item
instance of: scholarly article
title: Separatrix cycles and mutiple limit cycles in a class of quadratic systems
zbMATH Open document ID: 0809.34045
author: André Zegeling
published in: Journal of Differential Equations
publication date: 1994-10-24T00:00:00Z
review text: The three parameter family \(C_{ll}\) of quadratic systems given by \(\dot x= -y+ \delta x+ \ell x^ 2+ m xy\), \(\dot y= x+ x^ 2\) defines a subclass of the type \(ll\) quadratic systems with limit cycles in the ``Chinese classification''. The number and position of the limit cycles of \(C_{ll}\) is not known, although there is reason to believe that the system has at most two limit cycles. Some delicate properties of the bifurcation structure of the system in \((\ell,\delta,m)\) parameter space are considered in the paper under review. In particular, the author determines some of the surfaces given by \(\delta:= \delta(l,m)\) on which \(C_{ll}\) has either a multiple limit cycle or a separatrix cycle. For example, for the asymptotics given by \(m\) fixed and \(\ell\to\infty\), it is shown that, for \(\epsilon:= 1/\ell\), the curves in the \((\epsilon,\delta)\) space corresponding to systems with a multiple limit cycle or with a separatrix cycle have the same asymptote near \(\epsilon= 0\), in fact, in both cases, \(\delta(\ell,m)= (2- m)/4\ell+ o(1/\ell)\). This and several other results are obtained from a nontrivial analysis using the properties of quadratic systems and singular perturbation theory. As a corollary, it follows that at least two limit cycles can be created in the class of quadratic systems from an invariant parabola, from a cyclic formed by the equator of the Poincaré sphere and an invariant line, or from a separatrix cycle formed by two critical points at infinity, one of them an elementary saddle, the other a degenerate critical point.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 665705
zbMATH Keywords: quadratic systems
zbMATH Keywords: multiple limit cycle
zbMATH Keywords: separatrix cycle
zbMATH Keywords: singular perturbation theory
reviewed by: C. Chicone
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1128
OpenAlex ID: W2088361072
DOI: 10.1006/JDEQ.1994.1128
description: scientific article; zbMATH DE number 665705