Perturbations from an Elliptic Hamiltonian of Degree Four: II. Cuspidal Loop

The paper deals with Liénard 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 Liénard equations of type (3, 2) with at least four limit cycles. The paper also contains a complete result on the respective number of “small” and “large” limit cycles.