On a computer-aided approach to the computation of Abelian integrals

TL;DR

This paper introduces a precise computational method for Abelian integrals, enabling detailed analysis of phase portraits and bifurcations in perturbed Hamiltonian systems, exemplified through cubic perturbations of elliptic Hamiltonians.

Contribution

It presents a novel accurate enclosure method for Abelian integrals, facilitating the study of bifurcations in polynomial Hamiltonian systems.

Findings

Accurate enclosures of Abelian integrals achieved

Enhanced understanding of limit cycle bifurcations

Application to cubic perturbations of elliptic Hamiltonians

Abstract

An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four.