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Title:  More limit cycles than expected in Lienard equations 
Authors:  DUMORTIER, Freddy Panazzolo, Daniel Roussarie, Robert 
Issue Date:  2007 
Publisher:  AMER MATHEMATICAL SOC 
Citation:  PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 135(6). p. 18951904 
Abstract:  The paper deals with classical polynomial Lienard equations, i.e. planar vector fields associated to scalar second order differential equations x" + f(x)x' + x = 0 where f is a polynomial. We prove that for a wellchosen polynomial f of degree 6, the equation exhibits 4 limit cycles. It induces that for n >= 3 there exist polynomials f of degree 2n such that the related equations exhibit more than n limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Lienard equations as above, with f of degree 2n, the maximum number of limit cycles is n. The limit cycles that we found are relaxation oscillations which appear in slowfast systems at the boundary of classical polynomial Lienard equations. More precisely we find our example inside a family of second order differential equations ex" + f mu(x)x' + x = 0. Here, f mu is a wellchosen family of polynomials of degree 6 with parameter mu is an element of R4 and e is a small positive parameter tending to 0. We use bifurcations from canard cycles which occur when two extrema of the critical curve of the layer equation are crossing (the layer equation corresponds to epsilon = 0). As was proved by DUMORTIER and Roussarie (2005) these bifurcations are controlled by a rational integral computed along the critical curve of the layer equation, called the slow divergence integral. Our result is deduced from the study of this integral. 
Notes:  Univ Hasselt, B3500 Diepenbeek, Belgium. Univ Sao Paulo, Inst Matemat 7 Estatist, BR05508090 Sao Paulo, Brazil. Univ Bourgogne, CNRS, UMR 5584, Inst Math, F21078 Dijon, France.DUMORTIER, F, Univ Hasselt, Campus Diepenbeek,Agoralaan Gebouw D, B3500 Diepenbeek, Belgium.freddy.dumortier@uhasselt.be dpanazzo@ime.usp.br roussari@ubourgogne.fr 
URI:  http://hdl.handle.net/1942/4042 
DOI:  10.1090/S0002993907086881 
ISI #:  000244445600036 
ISSN:  00029939 
Category:  A1 
Type:  Journal Contribution 
Validation:  ecoom, 2008

Appears in Collections:  Research publications

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