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

TL;DR

This paper introduces generalized pseudo-Hamiltonian normal forms and an effective method for two-dimensional autonomous systems with Hamiltonian unperturbed parts, extending previous results and providing a systematic approach to simplify such systems.

Contribution

It develops a new method to obtain generalized pseudo-Hamiltonian normal forms for 2D autonomous systems with Hamiltonian quasi-homogeneous parts, generalizing prior work.

Findings

All generalized normal forms are obtained for systems with Hamiltonian monomial unperturbed parts.

The method distinguishes terms that can be eliminated in the normal form.

The approach extends and generalizes results by Takens, Baider and Sanders, and Basov et al.

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.