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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

More limit cycles than expected in Liénard equations


Authors: Freddy Dumortier, Daniel Panazzolo and Robert Roussarie
Journal: Proc. Amer. Math. Soc. 135 (2007), 1895-1904
MSC (2000): Primary 34C05, 34C26
Published electronically: January 12, 2007
MathSciNet review: 2286102
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Abstract: The paper deals with classical polynomial Liénard 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 well-chosen polynomial $ f$ of degree $ 6,$ the equation exhibits $ 4$ limit cycles. It induces that for $ n\geq 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 Liénard 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 slow-fast systems at the boundary of classical polynomial Liénard equations. More precisely we find our example inside a family of second order differential equations $ \varepsilon x''+f_\mu(x)x'+x=0.$ Here, $ f_\mu$ is a well-chosen family of polynomials of degree $ 6$ with parameter $ \mu\in \mathbb{R}^4$ and $ \varepsilon$ 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 $ \varepsilon =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.


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Additional Information

Freddy Dumortier
Affiliation: Universiteit Hasselt, Campus Diepenbeek, Agoralaan - Gebouw D, B-3590 Diepenbeek, Belgium
Email: freddy.dumortier@uhasselt.be

Daniel Panazzolo
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010 - São Paulo, SP, 05508-090, Brazil
Email: dpanazzo@ime.usp.br

Robert Roussarie
Affiliation: Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S., Université de Bourgogne, B.P. 47 870, 21078 Dijon Cedex, France
Email: roussari@u-bourgogne.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08688-1
PII: S 0002-9939(07)08688-1
Keywords: Limit cycles, Li\'enard equation, slow-fast equation.
Received by editor(s): June 29, 2005
Received by editor(s) in revised form: February 27, 2006
Published electronically: January 12, 2007
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.