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
Abstract: The paper deals with classical polynomial Liénard equations, i.e. planar vector fields associated to scalar second order differential equations where is a polynomial. We prove that for a well-chosen polynomial of degree the equation exhibits limit cycles. It induces that for there exist polynomials of degree such that the related equations exhibit more than limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Liénard equations as above, with of degree the maximum number of limit cycles is 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 Here, is a well-chosen family of polynomials of degree with parameter and is a small positive parameter tending to 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 . 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.
- [DR1] Freddy Dumortier and Robert Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc. 121 (1996), no. 577, x+100. With an appendix by Cheng Zhi Li. MR 1327208, https://doi.org/10.1090/memo/0577
- [DR2] Freddy Dumortier and Robert Roussarie, Multiple canard cycles in generalized Liénard equations, J. Differential Equations 174 (2001), no. 1, 1–29. MR 1844521, https://doi.org/10.1006/jdeq.2000.3947
- [DR3] F. Dumortier, R. Roussarie, Bifurcation of relaxation oscillations in dimension , preprint I.M.B. (2005).
- [LMP] A. Lins, W. de Melo, and C. C. Pugh, On Liénard’s equation, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Springer, Berlin, 1977, pp. 335–357. Lecture Notes in Math., Vol. 597. MR 0448423
- [R] R. Roussarie, Putting a boundary to the space of Liénard equations, to appear in Discr. and Cont. Dyn. Sys. (2005).
- [S] Steve Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15. MR 1631413, https://doi.org/10.1007/BF03025291
- F. Dumortier, R. Roussarie, Canard Cycles and Center Manifolds, Memoirs of Amer. Math. Soc. Vol. 121, n 577, 1-100 (1996).MR 1327208 (96k:34113)
- F. Dumortier, R. Roussarie, Multiple Canard Cycles in Generalized Liénard Equations, Journ. Diff. Equa. vol. 174, 1-29 (2001). MR 1844521 (2002k:34076)
- F. Dumortier, R. Roussarie, Bifurcation of relaxation oscillations in dimension , preprint I.M.B. (2005).
- A. Lins Neto, W. de Melo, C.C. Pugh, On Liénard Equations, Proc. Symp. Geom. and Topol., Springer Lectures Notes in Math. n 597, 335-357 (1977).MR 0448423 (56:6730)
- R. Roussarie, Putting a boundary to the space of Liénard equations, to appear in Discr. and Cont. Dyn. Sys. (2005).
- S. Smale, Mathematical Problems for the Next Century, Springer-Verlag, New York, Vol. 20, n 2, 7-15 (1998). MR 1631413 (99h:01033)
Affiliation: Universiteit Hasselt, Campus Diepenbeek, Agoralaan - Gebouw D, B-3590 Diepenbeek, Belgium
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010 - São Paulo, SP, 05508-090, Brazil
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
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.