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

Lorenzo Valvo; Ugo Locatelli

doi:10.3934/jcd.2022002

Article Contents

2022, Volume 9, Issue 4: 505-527. Doi: 10.3934/jcd.2022002

This issue Previous Article Next Article High-order symplectic Lie group methods on

$ SO(n) $

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

Lorenzo Valvo and
Ugo Locatelli^,

Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", via della Ricerca Scientifica 1, 00133 — Rome, Italy

^* Corresponding author: Ugo Locatelli
^* Corresponding author: Ugo Locatelli

Received: May 2021

Revised: January 2022

Early access: February 2022

Published: October 2022

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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.

Keywords:

Mathematics Subject Classification: Primary: 37J40; Secondary: 37N05, 70–08, 70F07, 70H08.

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Figure 1. Phase portraits given by the time-$ 2\pi $ mappings for the Hamiltonian $ H+v+f $ (see equations (2), (3), (10)). The equations of motion were numerically integrated by using a leap-frog method in the case with $ \varepsilon = 0.003 $

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Figure 2. Two FAMs for the Hamiltonian system corresponding to the Hamiltonians $ H+v $ (on the left) and $ H+v+f $ (see equations (2), (3) and (10)), for $ \varepsilon = 0.0012 $. In both cases we chose 200 equidistributed values of $ \psi_0 $, and for each of them we run a simulation of $ (2^{15}+1) $ perturbation periods. The initial values of the variables $ \theta $ and $ \varphi $ were always set to $ 0 $. The equations of motion were solved by a symmetric splitting method of order two

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Figure 3. The action-frequency map for the system $ H+v+f $ at different values of $ \varepsilon $, in the region where the KAM tori appeared as a consequence of the control term. For each value of $ \varepsilon $ we numerically computed $ 150 $ trajectories of $ (2^{15}+1) $ perturbation periods

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Figure 4. Action-frequency map analysis for a region of phase space of the system $ H+f $ filled with invariant tori, for $ \varepsilon = 0.004 $. The graphics is made of $ 200 $ points. For each point we run a simulation of $ (2^{16}+1) $ perturbation periods

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Figure 5. Decay of the norms of the terms appearing in the expansion (61) of $ H^{(0)} $, that is defined in formulæ (62)–(63)

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