Alien limit cycles in Liénard equations
type: Item
instance of: scholarly article
title: Alien limit cycles in Liénard equations
zbMATH Open document ID: 1262.34036
author: Freddy Dumortier
author: B. Coll
author: Rafel Prohens
published in: Journal of Differential Equations
publication date: 2013-01-18T00:00:00Z
review text: The second part of Hilbert's 16th problem deals about the existence of a finite upper bound, only depending on the degree of the vector field, of the number of limit cycles of a planar real polynomial differential system. An approach to this problem is by using perturbation methods, i.e., to ask for the maximum number of limit cycles when perturbing from polynomial Hamiltonian systems (the infinitesimal Hilbert's 16th problem). In perturbing from a polynomial Hamiltonian system, large part of the cycliclity can be reduced to studying the zeros of the so-called abelian integrals. However, this study does not finish the investigation perturbing Hamiltonian systems and some attention has to be paid to the singular cycles located at the boundary of the annuli of periodic orbits. The limit cycles that can be perturbed from the boundary cannot always been detected by the abelian integrals. Such limit cycles are called alien limit cycles. To prove the existence of alien limit cycles for concrete families of vector fields is far from being trivial and in some cases impossible. In a previous work [the second author and \textit{R. Roussarie}, J. Differ. Equations 227, No. 1, 116--165 (2006; Zbl 1111.34028)], it was shown that the simplest possibility to get alien limit cycles is the hyperbolic two-saddle cycle, and a first example was provided by \textit{S. Luca} et al. [Discrete Contin. Dyn. Syst. 25, No. 4, 1081--1108 (2009; Zbl 1194.34084)]. In the present paper, an example of a family of polynomial Liénard equations exhibiting an alien limit cycle is given. This limit cycle is perturbed from a two-saddle cycle in the boundary of an annulus of periodic orbits given by a Hamiltonian vector field. The Hamiltonian represents a truncated pendulum of degree four. In comparison to the former polynomial example, not only the equations are simpler, but a lot of tedious calculations can be avoided, making this example also interesting with respect to simplicity in treatment.
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6126900
zbMATH Keywords: planar vector fields
zbMATH Keywords: Liénard equation
zbMATH Keywords: Hamiltonian perturbation
zbMATH Keywords: limit cycle
zbMATH Keywords: abelian integral
zbMATH Keywords: two-saddle cycle
reviewed by: Jaume Giné
MaRDI profile type: Publication
full work available at URL: j.jde.2012.11.005
OpenAlex ID: W1967680978
DOI: 10.1016/J.JDE.2012.11.005
description: scientific article; zbMATH DE number 6126900