Perturbations from an Elliptic Hamiltonian of Degree Four: I. Saddle Loop and Two Saddle Cycle

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 fields with an elliptic Hamiltonian of degree 4 and especially to the study of the related elliptic integrals. Besides some general results the paper contains a complete treatment of the Saddle Loop case and the Two Saddle Cycle case. It is proven that the related elliptic integrals have at most two zeros, respectively one zero, the multiplicity taken into account. The bifurcation diagram of the zeros is also obtained.