On the cyclicity of the period annulus of quadratic Hamiltonian triangle vector field

TL;DR

This paper investigates the cyclicity of the period annulus in quadratic Hamiltonian triangle vector fields under quadratic perturbations, correcting previous displacement function formulas and establishing that the cyclicity is exactly three.

Contribution

It provides a corrected form of the displacement function and proves the cyclicity of the period annulus is three, advancing understanding of quadratic Hamiltonian systems.

Findings

Cyclicity of the period annulus is exactly three.

Corrected the displacement function for quadratic Hamiltonian triangle vector fields.

Extended previous studies with more accurate mathematical tools.

Abstract

This paper is concerned with the cyclicity of the period annulus of quadratic Hamiltonian triangle vector field under quadratic perturbations. This problem has been studied by Iliev (J. Differential Equations {\bf 128}(1996)), based on the displacement function obtained by \.{Z}o{\l}adek (J. Differential Equations {\bf 109}(1994)). Recently, P. Marde\v{s}i\'{c} etc. (J. Dynamical and Control Systems {\bf 17}(2011)) studied unfoldings of the Hamiltonian triangle within quadratic vector fields. It turned out that the displacement function is not precise of the form given by \.{Z}o{\l}adek. Using the corrected form of the displacement function obtained by P. Marde\v{s}i\'{c} etc, it is proved in this paper that the cyclicity of the period annulus under quadratic perturbations is equal to three.