Classical Liénard equations of degree n⩾6 can have limit cycles

Based on geometric singular perturbation theory we prove the existence of classical Liénard equations of degree 6 having 4 limit cycles. It implies the existence of classical Liénard equations of degree n⩾6, having at least limit cycles. This contradicts the conjecture from Lins, de Melo and Pugh formulated in 1976, where an upperbound of 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.