Perturbations from an elliptic Hamiltonian of degree four. II: Cuspidal loop (Q5949595)

The authors study Liénard equations of the form \(\dot{x}=y\), \(\dot{y}=P(x)+y Q(x)\), with \(P\) and \(Q\) polynomials of degree 3 and 2, respectivily. 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 corresponding number of ``small'' and ``large'' limit cycles.NEWLINENEWLINEFor part I see ibid. 176, 114--157 (2001; Zbl 1004.34018).