The local period function for Hamiltonian systems with applications

TL;DR

This paper develops a method to compute the Taylor expansion of the period function for planar Hamiltonian systems, aiding the analysis of limit cycles and bifurcations with applications to physical models.

Contribution

It introduces a constructive procedure for the Taylor expansion of the period function, applicable to various Hamiltonian systems and useful for studying bifurcations.

Findings

Derived Taylor expansions for specific Hamiltonian systems.

Applied Chebyshev systems to analyze zeroes of Abelian integrals.

Demonstrated the method on examples like the whirling pendulum.

Abstract

In the first part of the paper we develop a constructive procedure to obtain the Taylor expansion, in terms of the energy, of the period function for a non-degenerated center of any planar analytic Hamiltonian system. We apply it to several examples, including the whirling pendulum and a cubic Hamiltonian system. The knowledge of this Taylor expansion of the period function for this system is one of the key points to study the number of zeroes of an Abelian integral that controls the number of limit cycles bifurcating from the periodic orbits of a planar Hamiltonian system that is inspired by a physical model on capillarity. Several other classical tools, like for instance Chebyshev systems are applied to study this number of zeroes. The approach introduced can also be applied in other situations.