Limit cycles in Kukles systems of arbitrary degree with invariant ellipse
type: Item
instance of: scholarly article
title: Limit cycles in Kukles systems of arbitrary degree with invariant ellipse
zbMATH Open document ID: 1124.34017
author: Javier Chavarriga
author: Isaac A. García
author: Eduardo Sáez
author: Ivan Szanto
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-06-06T00:00:00Z
review text: This paper is concerned with planar polynomial differential systems of Kukles type, that is, systems of the form \[ \dot{x} = -y, \quad \dot{y} = Q(x,y), \] where \(Q(x,y)\) is a real polynomial, which is assumed to be not constant and not divisible by \(y\). The degree of the system is defined as the degree of the polynomial \(Q(x,y)\) and will be denoted by \(n\). This work deals with the existence, uniqueness and location of limit cycles in a Kukles system with an invariant ellipse. The existence of invariant algebraic curves and its distribution on the real plane implies properties on the phase portrait of the system and, in particular, on the number of limit cycles. A result on the relation between invariant algebraic curves and existence limit cycles is, for instance, that a quadratic system with an invariant algebraic curve of degree \(2\) has at most one limit cycle, see \textit{C. Christopher} [Proc. Roy. Soc. Edinburgh Sect. A 112, 113--134 (1989; Zbl 0677.34034)]. In particular, C. Christopher shows that if a quadratic system has an invariant ellipse and a limit cycle, this limit cycle is the ellipse, it is unique and it is hyperbolic. Other related results appear in [Nonlinear Anal. 59, 673--693 (2004; Zbl 1076.34029)], see also the references therein, where an example of a cubic Kukles system with an invariant hyperbola and two small-amplitude limit cycles is given. The first main result of this work is a normal form for the Kukles systems of degree \(n\) with an invariant ellipse. Any system with these characteristics is shown to be affine-equivalent to the system: \[ \dot{x} = -y, \quad \dot{y} = -ac + bx + q_{n-2}(x,y) f(x,y), \tag{1} \] where \(f(x,y) = a-2acx+bx^2+y^2\), \(a\), \(b\) and \(c\) are real numbers such that \(b>0\), \(a<0\) and \(q_{n-2}(x,y)\) is a real polynomial of degree \(n-2\) such that \(q_{n-2}(0,0) = c\). We observe that the origin \((0,0)\) is a singular point of system (1), the algebraic curve \(f(x,y)=0\) is the invariant e…
reviewed by: Maite Grau
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5161903
zbMATH Keywords: polynomial differential equations
zbMATH Keywords: invariant algebraic curves
MaRDI profile type: Publication
full work available at URL: j.na.2006.06.035
OpenAlex ID: W1980096996
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DOI: 10.1016/J.NA.2006.06.035
description: scientific article; zbMATH DE number 5161903