Perturbations from an elliptic Hamiltonian of degree four I. Saddle loop and two saddle cycle

The paper deals with Lienard equations of the form (x) over dot = y, (y) ov er dot = 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 a n elliptic Hamiltonian of degree 4 and especially to the study of the relat ed elliptic integrals. Besides some general results the paper contains a co mplete 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, respe ctively one zero. the multiplicity taken into account. The bifurcation diag ram of the zeros is also obtained. (C) 2001 Academic Press.