A parametrization algorithm to compute lower dimensional elliptic tori in Hamiltonian systems

TL;DR

This paper introduces an algorithm for constructing lower-dimensional elliptic tori in Hamiltonian systems using parametrization, with an application to coupled pendula demonstrating its practical implementation.

Contribution

It develops a parametrization algorithm that prescribes tangent and normal frequencies, adjusting parameters to compute elliptic tori in Hamiltonian systems.

Findings

Successfully computed 2-dimensional elliptic tori in a 4-degree-of-freedom system

Algorithm adjusts parameters to match prescribed frequencies

Implementation demonstrated on coupled pendula system

Abstract

We present an algorithm for the construction of lower dimensional elliptic tori in parametric Hamiltonian systems by means of the parametrization method with the tangent and normal frequencies being prescribed. This requires that the Hamiltonian system has as many parameters as the dimension of the normal dynamics, and the algorithm must adjust these parameters. We illustrate the methodology with an implementation of the algorithm computing $2$--dimensional elliptic tori in a system of $4$ coupled pendula (4 degrees of freedom).