The paper deals with Lienard equations of the form x = y, y = P(x) + yQ(x)
with P and Q polynomials of degree respectively 3 and 2. Attention goes to
perturbations of the Hamiltonian vector field with an elliptic Hamiltonian
of degree 4, exhibiting a cuspidal loop. It is proven that the least upper
bound for the number of zeros of the related elliptic integral is four, and
this upper bound is a sharp one.
This permits to prove the existence of Lienard equations of type (3, 2) wit
h at least four limit cycles. The paper also contains a complete result on
the respective number of "small" and "large" limit cycles. (C) 2001 Academi
c Press.