A rigorous study of possible configurations of limit cycles bifurcating from a hyper-elliptic Hamiltonian of degree five

We consider a hyper-elliptic Hamiltonian of degree five, chosen from a generic set of parameters, and study what configurations of limit cycles can bifurcate from the corresponding differential system under quartic perturbations. Perturbations of Lienard type are considered separately. Several different configurations with seven (four) limit cycles, bifurcating from the given system for general (Lienard type) quartic perturbations, are constructed. We also discuss how to construct perturbations yielding a given configuration, and how to validate the correctness of such a candidate perturbation.