Generalized Normal Forms of Two-Dimensional Autonomous Systems with a Hamiltonian Unperturbed Part

Abstract: Generalized pseudo-Hamiltonian normal forms (GPHNF) and an effective method of obtaining them are introduced for two-dimensional systems of autonomous ODEs with a Hamiltonian quasi-homogeneous unperturbed part of an arbitrary degree. The terms that can be additionally eliminated in a GPHNF are constructively distinguished, and it is shown that after removing them GPHNF becomes a generalized normal form (GNF). By using the introduced method, all the GNFs are obtained in cases where the unperturbed part of the system is Hamiltonian and has monomial components, which allowed to generalize some results by Takens, Baider and Sanders, as well as Basov et al.