American Institute of Mathematical Sciences

doi: 10.3934/jcd.2022002
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Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers

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

* Corresponding author: Ugo Locatelli

Received  May 2021 Revised  January 2022 Early access February 2022

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.

Citation: Lorenzo Valvo, Ugo Locatelli. Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers. Journal of Computational Dynamics, doi: 10.3934/jcd.2022002
References:
 [1] S. S. Abdullaev, Construction of Mappings for Hamiltonian Systems and Their Applications, Lecture Notes in Physics. Spinger, Berlin Heidelberg, 2006. [2] P. M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press, July 2008. [3] G. Benettin, L. Galgani, A. Giorgilli and J.-M. Strelcyn, A proof of Kolmogorov's theorem on invariant tori using canonical transformations defined by the Lie method, Nuovo Cimento B, 79 (1984), 201-223.  doi: 10.1007/BF02748972. [4] C. Caracciolo and U. Locatelli, Computer-assisted estimates for Birkhoff normal forms, J. Comput. Dyn., 7 (2020), 425-460.  doi: 10.3934/jcd.2020017. [5] A. Carati, M. Zuin, A. Maiocchi, M. Marino, E. Martines and L. Galgani, Transition from order to chaos, and density limit, in magnetized plasmas, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 033124, 7 pp. doi: 10.1063/1.4745851. [6] A. Celletti, A. Giorgilli and U. Locatelli, Improved estimates on the existence of invariant tori for Hamiltonian systems, Nonlinearity, 13 (2000), 397-412.  doi: 10.1088/0951-7715/13/2/304. [7] C. Chandre, M. Vittot, G. Ciraolo, P. Ghendrih and R. Lima, Control of stochasticity in magnetic field lines, Nuclear Fusion, 46 (2006), 33-45. [8] G. Ciraolo, F. Briolle, C. Chandre, E. Floriani, R. Lima, M. Vittot, M. Pettini, C. Figarella and P. Ghendrih, Control of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas, Physical Review E, 69 (2004), 056213. [9] G. Ciraolo, C. Chandre, R. Lima, M. Vittot and M. Pettini, Control of chaos in Hamiltonian systems, Celestial Mechanics and Dynamical Astronomy, 90 (2004), 3-12.  doi: 10.1007/s10569-004-6445-3. [10] C. Di Troia, From charge motion in general magnetic fields to the non perturbative gyrokinetic equation, Physics of Plasmas, 22 (2015), 042103. [11] D. F. Escande, Contributions of plasma physics to chaos and nonlinear dynamics, Plasma Physics and Controlled Fusion, 58 (2016), 113001. [12] J.-Ll. Figueras, A. Haro and A. Luque, Rigorous computer-assisted application of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193.  doi: 10.1007/s10208-016-9339-3. [13] F. Gabern, A. Jorba and U. Locatelli, On the construction of the Kolmogorov normal form for the Trojan asteroids, Nonlinearity, 18 (2005), 1705-1734.  doi: 10.1088/0951-7715/18/4/017. [14] A. Giorgilli and U. Locatelli, Kolmogorov theorem and classical perturbation theory, Z. Angew. Math. Phys., 48 (1997), 220-261.  doi: 10.1007/PL00001475. [15] A. Giorgilli and U. Locatelli, On classical series expansions for quasi-periodic motions, MPEJ, 3 (1997), Paper 5, 25 pp. [16] A. Giorgilli, U. Locatelli and M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies, Celestial Mech. Dynam. Astronom., 104 (2009), 159-173.  doi: 10.1007/s10569-009-9192-7. [17] A. Giorgilli, U. Locatelli and M. Sansottera, Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories, Regul. Chaotic Dyn., 22 (2017), 54-77.  doi: 10.1134/S156035471701004X. [18] A. Giorgilli and M. Sansottera, Methods of algebraic manipulation in perturbation theory, Workshop Series of the Asociacion Argentina de Astronomia, (2011), 147–183. [19] R. D. Hazeltine and J. D. Meiss, Plasma Confinement, Addison-Wesley Publishing Company, 2003. [20] S. Kim and S. Östlund, Simultaneous rational approximations in the study of dynamical systems, Physical Review A, 34 (1986), 3426-3434.  doi: 10.1103/PhysRevA.34.3426. [21] J. Laskar, Introduction to frequency map analysis, In Hamiltonian Systems with Three or More Degrees of Freedom (editor C. Simó), NATO ASI Series, 134–150. Springer Netherlands, Dordrecht, 1999. [22] E. Lega and C. Froeschlé, Numerical investigations of the structure around an invariant KAM torus using the frequency map analysis, Physica D: Nonlinear Phenomena, 95 (1996), 97-106.  doi: 10.1016/0167-2789(96)00046-2. [23] U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems, Celestial Mechanics and Dynamical Astronomy, 78 (2000), 47-74.  doi: 10.1023/A:1011139523256. [24] R. S. MacKay and J. Stark, Locally most robust circles and boundary circles for area-preserving maps, Nonlinearity, 5 (1992), 867-888.  doi: 10.1088/0951-7715/5/4/002. [25] A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys., 78 (1995), 1607-1617.  doi: 10.1007/BF02180145. [26] T. Northrop, The Adiabatic Motion of Charged Particles, Interscience Publishers, Inc, 1963. [27] L. Stefanelli and U. Locatelli, Kolmogorov's normal form for equations of motion with dissipative effects, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2561-2593.  doi: 10.3934/dcdsb.2012.17.2561. [28] M. Vittot, Perturbation theory and control in classical or quantum mechanics by an inversion formula, Journal of Physics A: Mathematical and General, 37 (2004), 6337-6357.  doi: 10.1088/0305-4470/37/24/011. [29] M. Vittot, C. Chandre, G. Ciraolo and R. Lima, Localised control for non-resonant Hamiltonian systems, Nonlinearity, 18 (2005), 423-440.  doi: 10.1088/0951-7715/18/1/021.

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References:
 [1] S. S. Abdullaev, Construction of Mappings for Hamiltonian Systems and Their Applications, Lecture Notes in Physics. Spinger, Berlin Heidelberg, 2006. [2] P. M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press, July 2008. [3] G. Benettin, L. Galgani, A. Giorgilli and J.-M. Strelcyn, A proof of Kolmogorov's theorem on invariant tori using canonical transformations defined by the Lie method, Nuovo Cimento B, 79 (1984), 201-223.  doi: 10.1007/BF02748972. [4] C. Caracciolo and U. Locatelli, Computer-assisted estimates for Birkhoff normal forms, J. Comput. Dyn., 7 (2020), 425-460.  doi: 10.3934/jcd.2020017. [5] A. Carati, M. Zuin, A. Maiocchi, M. Marino, E. Martines and L. Galgani, Transition from order to chaos, and density limit, in magnetized plasmas, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 033124, 7 pp. doi: 10.1063/1.4745851. [6] A. Celletti, A. Giorgilli and U. Locatelli, Improved estimates on the existence of invariant tori for Hamiltonian systems, Nonlinearity, 13 (2000), 397-412.  doi: 10.1088/0951-7715/13/2/304. [7] C. Chandre, M. Vittot, G. Ciraolo, P. Ghendrih and R. Lima, Control of stochasticity in magnetic field lines, Nuclear Fusion, 46 (2006), 33-45. [8] G. Ciraolo, F. Briolle, C. Chandre, E. Floriani, R. Lima, M. Vittot, M. Pettini, C. Figarella and P. Ghendrih, Control of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas, Physical Review E, 69 (2004), 056213. [9] G. Ciraolo, C. Chandre, R. Lima, M. Vittot and M. Pettini, Control of chaos in Hamiltonian systems, Celestial Mechanics and Dynamical Astronomy, 90 (2004), 3-12.  doi: 10.1007/s10569-004-6445-3. [10] C. Di Troia, From charge motion in general magnetic fields to the non perturbative gyrokinetic equation, Physics of Plasmas, 22 (2015), 042103. [11] D. F. Escande, Contributions of plasma physics to chaos and nonlinear dynamics, Plasma Physics and Controlled Fusion, 58 (2016), 113001. [12] J.-Ll. Figueras, A. Haro and A. Luque, Rigorous computer-assisted application of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193.  doi: 10.1007/s10208-016-9339-3. [13] F. Gabern, A. Jorba and U. Locatelli, On the construction of the Kolmogorov normal form for the Trojan asteroids, Nonlinearity, 18 (2005), 1705-1734.  doi: 10.1088/0951-7715/18/4/017. [14] A. Giorgilli and U. Locatelli, Kolmogorov theorem and classical perturbation theory, Z. Angew. Math. Phys., 48 (1997), 220-261.  doi: 10.1007/PL00001475. [15] A. Giorgilli and U. Locatelli, On classical series expansions for quasi-periodic motions, MPEJ, 3 (1997), Paper 5, 25 pp. [16] A. Giorgilli, U. Locatelli and M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies, Celestial Mech. Dynam. Astronom., 104 (2009), 159-173.  doi: 10.1007/s10569-009-9192-7. [17] A. Giorgilli, U. Locatelli and M. Sansottera, Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories, Regul. Chaotic Dyn., 22 (2017), 54-77.  doi: 10.1134/S156035471701004X. [18] A. Giorgilli and M. Sansottera, Methods of algebraic manipulation in perturbation theory, Workshop Series of the Asociacion Argentina de Astronomia, (2011), 147–183. [19] R. D. Hazeltine and J. D. Meiss, Plasma Confinement, Addison-Wesley Publishing Company, 2003. [20] S. Kim and S. Östlund, Simultaneous rational approximations in the study of dynamical systems, Physical Review A, 34 (1986), 3426-3434.  doi: 10.1103/PhysRevA.34.3426. [21] J. Laskar, Introduction to frequency map analysis, In Hamiltonian Systems with Three or More Degrees of Freedom (editor C. Simó), NATO ASI Series, 134–150. Springer Netherlands, Dordrecht, 1999. [22] E. Lega and C. Froeschlé, Numerical investigations of the structure around an invariant KAM torus using the frequency map analysis, Physica D: Nonlinear Phenomena, 95 (1996), 97-106.  doi: 10.1016/0167-2789(96)00046-2. [23] U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems, Celestial Mechanics and Dynamical Astronomy, 78 (2000), 47-74.  doi: 10.1023/A:1011139523256. [24] R. S. MacKay and J. Stark, Locally most robust circles and boundary circles for area-preserving maps, Nonlinearity, 5 (1992), 867-888.  doi: 10.1088/0951-7715/5/4/002. [25] A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Statist. Phys., 78 (1995), 1607-1617.  doi: 10.1007/BF02180145. [26] T. Northrop, The Adiabatic Motion of Charged Particles, Interscience Publishers, Inc, 1963. [27] L. Stefanelli and U. Locatelli, Kolmogorov's normal form for equations of motion with dissipative effects, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2561-2593.  doi: 10.3934/dcdsb.2012.17.2561. [28] M. Vittot, Perturbation theory and control in classical or quantum mechanics by an inversion formula, Journal of Physics A: Mathematical and General, 37 (2004), 6337-6357.  doi: 10.1088/0305-4470/37/24/011. [29] M. Vittot, C. Chandre, G. Ciraolo and R. Lima, Localised control for non-resonant Hamiltonian systems, Nonlinearity, 18 (2005), 423-440.  doi: 10.1088/0951-7715/18/1/021.
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 $\varepsilon = 0.003$
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
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
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
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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