Isochronous centers of polynomial Hamiltonian systems and a conjecture of Jarque and Villadelprat

TL;DR

This paper proves the conjecture that centers of planar polynomial Hamiltonian systems of even degree are nonisochronous for quadratic and quartic cases, and characterizes nonisochronous systems of arbitrary large degree.

Contribution

It confirms the conjecture for quadratic and quartic systems and introduces a method to describe nonisochronous Hamiltonian systems of any even degree.

Findings

Proved the conjecture for quadratic and quartic systems.

Developed a characterization method for isochronicity using vector field correction.

Described a large class of nonisochronous systems of arbitrary large even degree.

Abstract

We study the conjecture of Jarque and Villadelprat stating that every center of a planar polynomial Hamiltonian system of even degree is nonisochronous. This conjecture is proved for quadratic and quartic systems. Using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields, wa are able to describe a very large class of nonisochronous Hamiltonian system of even degree of degree arbitrary large.