Zeros of complete elliptic integrals and its application to Melnikov functions

TL;DR

This paper investigates the zeros of linear combinations of complete elliptic integrals and applies the results to analyze the Melnikov functions in a Hamiltonian system with invariant lines.

Contribution

It establishes bounds on the zeros of elliptic integral combinations and applies these findings to the study of Melnikov functions in perturbed Hamiltonian systems.

Findings

Bound on the number of zeros of elliptic integral combinations

Linear independence results for elliptic integrals

Application to Hamiltonian systems with invariant lines

Abstract

In this paper, we first discuss the linear independence of the complete elliptic integrals of the first, second and third kinds $K(k)$, $E(k)$ and $\Pi(\mu(k),k)$, and then obtain an upper bound for the number of zeros of a function of the form \begin{eqnarray*} p(k)K(k)+q(k)E(k)+r(k)\Pi(\mu(k),k),\ k\in(-1,1), \end{eqnarray*} where $p(k)$, $q(k)$ and $r(k)$ are real polynomials, $\mu(k)$ is a real polynomial or rational function. Finally, we apply it to a Hamiltonian triangle with three invariant straight lines under small real polynomials piecewise smooth perturbation.