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 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.

Valvo, L., Locatelli, U. (2022). Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers. JOURNAL OF COMPUTATIONAL DYNAMICS, 9(4), 505-527 [10.3934/jcd.2022002].

Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers

Locatelli U.
2022-01-01

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 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.
2022
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/07 - FISICA MATEMATICA
English
Con Impact Factor ISI
Computer-assisted proofs
Controlled magnetic fields
Hamiltonian systems
Kam theory
Normal form methods
Valvo, L., Locatelli, U. (2022). Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers. JOURNAL OF COMPUTATIONAL DYNAMICS, 9(4), 505-527 [10.3934/jcd.2022002].
Valvo, L; Locatelli, U
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/326803
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