Abstract

The work deal with Liénard equations of the form $x\prime =y$, $y\prime \; =\; P(x)\; +\; \delta \; yQ(x)$ with $\delta$ small and $P$ and $Q$ polynomials of degree respectively 3 and 2, and attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree four, exhibiting respectively a saddle loop, a cuspidal loop, a global center and a figure eight-loop. It is proved that the least upper bound of the number of zeros of the related elliptic integral is respectively 2, 4, 4 , and 5, and this is a sharp bound, multiplicity taken into account. This is a joint work with Freddy Dumortier, consists of 4 papers, the first two of them were published in JDE (2001, 175 and 176), and the last two just appear in the same journal (2003, 188, no.2).