Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 250, Issue 4, Pages 2162-2176Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2010.12.003
Keywords
Slow-fast system; Singular perturbations; Limit cycles; Relaxation oscillation; Classical Lienard equations
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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.
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