Hamiltonian Control of Magnetic Field Lines: Computer Assisted Results Proving the Existence of KAM Barriers

TL;DR

This paper demonstrates how a control term can induce invariant tori in magnetic field line models, acting as transport barriers, using a combination of frequency analysis and computer-assisted KAM algorithms, with publicly available code.

Contribution

It introduces a novel computer-assisted method to prove the existence of KAM barriers in magnetic fields, including accessible software for broader applications.

Findings

Invariant tori appear due to control, acting as transport barriers

The method is general and applicable to quasi-integrable Hamiltonian systems

All codes used are publicly available for reproducibility

Abstract

We reconsider a control theory for Hamiltonian systems, that was introduced on the basis of KAM theory and applied to a model of magnetic field in previous articles. By a combination of Frequency Analysis and of a rigorous (Computer Assisted) KAM algorithm we prove that in the phase space of the magnetic field, due to the control term, a set of invariant tori appear, and it acts as a transport barrier. Our analysis, which is common (but often also limited) to Celestial Mechanics, is based on a normal form approach; it is also quite general and can be applied to quasi-integrable Hamiltonian systems satisfying a few additional mild assumptions. As a novelty with respect to the works that in the last two decades applied Computer Assisted Proofs into the framework of KAM theory, we provide all the codes allowing to produce our results. They are collected in a software package that is publicly available from the {\it Mendeley Data} repository. All these codes are designed in such a way to be easy-to-use, also for what concerns eventual adaptations for applications to similar problems.