The cyclicity of period annuli for a class of cubic Hamiltonian systems with nilpotent singular points
In the paper, the authors investigate the maximal number of limit cycles that might occur in a class of symmetric cubic Hamiltonian systems with a nilpotent singular point under polynomial perturbations. It is well-known that the number of zeros of the abelian integral provides an upper bound for the number of limit cycles of such systems. They first obtain the algebraic structure of the abelian integral in terms of nine generators. Then, three Picard-Fuchs equations satisfied by the generators are derived. From the existence of a second-order differential operator that acts on the abelian integral, the number of zeros of the abelian integral can be estimated. The number of limit cycles follows from the Poincaré-Pontryagin Theorem.
scientific article; zbMATH DE number 6946020
scientific article