Cubic Perturbations of Symmetric elliptic Hamiltonians of degree four in a Complex domain

TL;DR

This paper investigates the effects of cubic perturbations on symmetric elliptic Hamiltonians of degree four, focusing on the computation and analysis of Melnikov functions to understand bifurcations in complex domains.

Contribution

It provides explicit formulas for the second Melnikov function in cases where the first vanishes, advancing the understanding of perturbations in elliptic Hamiltonian systems.

Findings

Computed the general form of M_2 when M_1 vanishes.

Analyzed zeros of M_2 in complex domains.

Enhanced understanding of bifurcations in perturbed Hamiltonian systems.

Abstract

We consider arbitrary one-parameter cubic deformations of the Duffing oscillator $x"=x-x^3$. In the case when the first Melnikov function $M_1$ vanishes, but $M_2\neq 0$ we compute the general form of $M_2$ and study its zeros in a suitable complex domain.