Classical Lienard equations of degree n ≥ 6 can have [n-1/2]+2 limit cycles

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:250

Article

De Maesschalck, P.¹  Dumortier, F.¹ 

[1] Hasselt Univ, B-3590 Diepenbeek, Belgium

关键词: Slow-fast system;    Singular perturbations;    Limit cycles;    Relaxation oscillation;    Classical Lienard equations;   

DOI  :  10.1016/j.jde.2010.12.003

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

Free