Limit cycles of the classical Liénard differential systems: A survey on the Lins Neto, de Melo and Pugh's conjecture

In 1977 Lins Neto et al. (1977) conjectured that the classical Liénard system ẋ=y−F(x),ẏ=−x, with F(x) a real polynomial of degree n, has at most [(n−1)/2] limit cycles, where [⋅] denotes the integer part function. In this paper we summarize what is known and what is still open on this conjecture. For the known results on this conjecture we present a complete proof.