Abstract

In this talk, we prove the presence of limit cycles of given multiplicity, together with a complete unfolding, in families of (singularly perturbed) polynomial Lienard equations. The obtained limit cycles are relaxation oscillations. Both classical Lienard equations and generalized Lienard equations are treated. In a second part, we prove the existence of classical Lienard equations of degree 6 having 4 limit cycles. This contradicts the conjecture from Lins, de Melo and Pugh formulated in 1976, where an upperbound of 2 limit cycles was predicted for degree 6. This result 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.

The talk is based on 2 submitted papers, joint work with F. Dumortier.