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http://hdl.handle.net/1942/11743
Title: | Classical Lienard equations of degree n >= 6 can have [n-1/2]+2 limit cycles | Authors: | DE MAESSCHALCK, Peter DUMORTIER, Freddy |
Issue Date: | 2011 | Publisher: | ACADEMIC PRESS INC ELSEVIER SCIENCE | Source: | JOURNAL OF DIFFERENTIAL EQUATIONS, 250(4). p. 2162-2176 | Abstract: | Based on geometric singular perturbation theory we prove the existence of classical Lienard equations of degree 6 having 4 limit cycles. It implies the existence of classical Lienard equations of degree n >= 6, having at least [n-1/2] + 2 limit cycles. This contradicts the conjecture from Lins, de Melo and Pugh formulated in 1976, where an upperbound of [n-1/2] limit cycles was predicted. This paper improves the counterexample from Dumortier, Panazzolo and Roussarie (2007) by supplying one additional limit cycle from degree 7 on, and by finding a counterexample of degree 6. We also give a precise system of degree 6 for which we provide strong numerical evidence that it has at least 3 limit cycles. (c) 2010 Elsevier Inc. All rights reserved. | Notes: | [De Maesschalck, P.; Dumortier, F.] Hasselt Univ, B-3590 Diepenbeek, Belgium. | Keywords: | Slow–fast system; Singular perturbations; Limit cycles; Relaxation oscillation; Classical Liénard equations;Slow-fast system; Singular perturbations; Limit cycles; Relaxation oscillation; Classical Lienard equations | Document URI: | http://hdl.handle.net/1942/11743 | ISSN: | 0022-0396 | e-ISSN: | 1090-2732 | DOI: | 10.1016/j.jde.2010.12.003 | ISI #: | 000286447000014 | Rights: | Elsevier Science | Category: | A1 | Type: | Journal Contribution | Validations: | ecoom 2012 |
Appears in Collections: | Research publications |
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morelc-jde-final.pdf | Peer-reviewed author version | 354.6 kB | Adobe PDF | View/Open |
peter 1.pdf Restricted Access | 224.28 kB | Adobe PDF | View/Open Request a copy |
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