Bifurcations of multiple relaxation oscillations in polynomial Liénard equations
HTML articles powered by AMS MathViewer
- by P. De Maesschalck and F. Dumortier PDF
- Proc. Amer. Math. Soc. 139 (2011), 2073-2085 Request permission
Abstract:
In this paper, we prove the presence of limit cycles of given multiplicity, together with a complete unfolding, in families of (singularly perturbed) polynomial Liénard equations. The obtained limit cycles are relaxation oscillations. Both classical Liénard equations and generalized Liénard equations are treated.References
- P. De Maesschalck and F. Dumortier, Canard cycles in the presence of slow dynamics with singularities, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), no. 2, 265–299. MR 2406691, DOI 10.1017/S0308210506000199
- Freddy Dumortier, Daniel Panazzolo, and Robert Roussarie, More limit cycles than expected in Liénard equations, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1895–1904. MR 2286102, DOI 10.1090/S0002-9939-07-08688-1
- Freddy Dumortier and Robert Roussarie, Multiple canard cycles in generalized Liénard equations, J. Differential Equations 174 (2001), no. 1, 1–29. MR 1844521, DOI 10.1006/jdeq.2000.3947
- Freddy Dumortier and Robert Roussarie, Bifurcation of relaxation oscillations in dimension two, Discrete Contin. Dyn. Syst. 19 (2007), no. 4, 631–674. MR 2342266, DOI 10.3934/dcds.2007.19.631
- F. Dumortier, Slow divergence integral and balanced canard solutions, Qualitative Theory of Dynamical Systems, to appear.
- M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518
- 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) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 335–357. MR 0448423
- Robert Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst. 17 (2007), no. 2, 441–448. MR 2257444, DOI 10.3934/dcds.2007.17.441
Additional Information
- P. De Maesschalck
- Affiliation: Hasselt University, Campus Diepenbeek, Agoralaan gebouw D, B-3590 Diepenbeek, Belgium
- Email: peter.demaesschalck@uhasselt.be
- F. Dumortier
- Affiliation: Hasselt University, Campus Diepenbeek, Agoralaan gebouw D, B-3590 Diepenbeek, Belgium
- Email: freddy.dumortier@uhasselt.be
- Received by editor(s): March 24, 2010
- Received by editor(s) in revised form: May 31, 2010
- Published electronically: November 3, 2010
- Additional Notes: The first author was supported by the Research Foundation Flanders.
- Communicated by: Yingfei Yi
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2073-2085
- MSC (2010): Primary 37G15, 34E17; Secondary 34C07, 34C26
- DOI: https://doi.org/10.1090/S0002-9939-2010-10610-X
- MathSciNet review: 2775385