Dynamical Systems and Topology of Magnetic Fields in Conducting Medium


We discuss application of contemporary methods of the theory of dynamical systems with regular and chaotic hyperbolic dynamics to investigation of topological structure of magnetic fields in conducting media. For substantial classes of magnetic fields, we consider well-known physical models allowing us to reduce investigation of such fields to study of vector fields and Morse–Smale diffeomorphisms as well as diffeomorphisms with nontrivial basic sets satisfying the A axiom introduced by Smale. For the point–charge magnetic field model, we consider the problem of the separator playing an important role in the reconnection processes and investigate relations between its singularities. We consider the class of magnetic fields in the solar corona and solve the problem of topological equivalency of fields in this class. We develop a topological modification of the Zeldovich funicular model of the nondissipative cinematic dynamo, constructing a hyperbolic diffeomorphism with chaotic dynamics that is conservative in the neighborhood of its transitive invariant set.

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  1. 1.

    H. Alfven, “On sunspots and the solar cycle,” Arc. F. Mat. Ast. Fys., 29A, 1–17 (1943).

  2. 2.

    H. Alfven, “Electric currents in cosmic plasmas,” Rev. Geophys. Space Phys., 15, 271 (1977).

    Google Scholar 

  3. 3.

    H. Alfven and C.-G. Fälthammar, Cosmical Electrodynamics: Fundamental principles, Clarendon, Oxford (1963).

    MATH  Google Scholar 

  4. 4.

    D. V. Anosov and V. V. Solodov, “Hyperbolic sets,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fundam. Napravl., 66, 12–99 (1991).

    Google Scholar 

  5. 5.

    V. I. Arnol’d and B. A. Khesin, Topological Methods in Hydrodynamics [in Russian], MTsNMO, Moscow (2007).

  6. 6.

    P. Baum and A. Bratenahl, “Flux linkages of bipolar sunspot groups: a computer study,” Solar Phys., 67, 245–258 (1980).

    Google Scholar 

  7. 7.

    C. Beveridge, E. R. Priest, and D. S. Brown, “Magnetic topologies due to two bipolar regions,” Solar Phys., 209, No. 2, 333–347 (2002).

    Google Scholar 

  8. 8.

    C. Beveridge, E. R. Priest, and D. S. Brown, “Magnetic topologies in the solar corona due to four discrete photospheric flux regions,” Geophys. Astrophys. Fluid Dyn., 98, No. 5, 429–445 (2004).

    MathSciNet  MATH  Google Scholar 

  9. 9.

    C. Bonatti, V. Grines, V. Medvedev, and E. Pecou, “Three-dimensional manifolds admitting Morse-Smale diffeomorphisms without heteroclinic curves,” Topol. Appl., 117, 335–344 (2002).

    MATH  Google Scholar 

  10. 10.

    H. Bothe, “The ambient structure of expanding attractors, II. Solenoids in 3-manifolds,” Math. Nachr., 112, 69–102 (1983).

    MathSciNet  MATH  Google Scholar 

  11. 11.

    D. S. Brown and E. R. Priest, “The topological behaviour of 3D null points in the Sun’s corona,” Astron. Astrophys., 367, 339 (2001).

    Google Scholar 

  12. 12.

    S. Childress and A. D. Gilbert, Stretch, Twist, Fold: The Fast Dynamo, Springer, Berlin–Heidelberg–N.Y. (1995).

  13. 13.

    R. M. Close, C. E. Parnell, and E. R. Priest, “Domain structures in complex 3D magnetic fields,” Geophys. Astrophys. Fluid Dyn., 99, No. 6, 513–534 (2005).

    MathSciNet  MATH  Google Scholar 

  14. 14.

    T. G. Cowling, Magnetohydrodynamics, Interscience, New York (1956).

  15. 15.

    G. Duvaut and J. L. Lions, “Inéquations en thermoélasticité et magnétohydrodynamique,” Arch. Ration. Mech. Anal., 46, 241–279 (1972).

    MATH  Google Scholar 

  16. 16.

    W. M. Elsässer, “Magnetohydrodynamics,” Am. J. Phys., 23, 590 (1955).

  17. 17.

    W. M. Elsässer, “Magnetohydrodynamics,” Usp. Fiz. Nauk, 64, No. 3, 529–588 (1958).

  18. 18.

    A. T. Fomenko, Differential Geometry and Topology, Plenum Press, N.Y.–London (1987).

    MATH  Google Scholar 

  19. 19.

    V. S. Gorbachev, S. R. Kel’ner, B. V. Somov, and A. S. Shvarts, “New topological approach to the problem of trigger for solar flares,” Astron. Zh., 65, 601–612 (1988).

    Google Scholar 

  20. 20.

    V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, and S. Kh. Zinina, “Heteroclinic curves of Morse–Smale diffeomorphisms and separators in the plasma magnetic field,” Nelin. Dinam., 10, 427–438 (2014).

    MATH  Google Scholar 

  21. 21.

    V. Grines, T. Medvedev, and O. Pochinka, Dynamical Systems on 2- and 3-Manifolds, Springer, Berlin (2016).

    MATH  Google Scholar 

  22. 22.

    V. Grines, T. Medvedev, O. Pochinka, and E. Zhuzhoma, “On heteroclinic separators of magnetic fields in electrically conducting fluids,” Phys. D. Nonlin. Phenom., 294, 1–5 (2015).

    MathSciNet  MATH  Google Scholar 

  23. 23.

    V. Z. Grines and O. V. Pochinka, Introduction to Topological Classification of Cascades on Manifolds of Dimension Two and Three [in Russian], Moscow–Izhevsk (2011).

  24. 24.

    V. Z. Grines and O. V. Pochinka, “Morse–Smale cascades on 3-manifolds,” Russ. Math. Surv., 68, No. 1, 117–173 (2013).

    MathSciNet  MATH  Google Scholar 

  25. 25.

    V. Z. Grines and O. V. Pochinka, “Morse–Smale cascades on 3-manifolds,” Usp. Mat. Nauk, 68, No. 1, 129–188 (2013).

    MathSciNet  MATH  Google Scholar 

  26. 26.

    V. Grines and O. Pochinka, “Topological classification of global magnetic fields in the solar corona,” Dyn. Syst., 33, No. 3, 536–546 (2018).

    MathSciNet  MATH  Google Scholar 

  27. 27.

    V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, “New relations for flows and Morse–Smale diffeomorphisms,” Dokl. RAN, 382, No. 6, 730–733 (2002).

    MATH  Google Scholar 

  28. 28.

    V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, and O. V. Pochinka, “Global attractor and repeller of Morse–Smale diffeomorphisms,” Tr. MIAN, 271, 111–133 (2010).

    MathSciNet  MATH  Google Scholar 

  29. 29.

    A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge–N.Y. (1995).

    MATH  Google Scholar 

  30. 30.

    A. Katok and B. Hasselblatt, Introduction to the Theory of Dynamical Systems [in Russian], Faktorial, Moscow (1999).

  31. 31.

    I. Klapper and L.-S. Young, “Rigorous bounds of the fast dynamo growth rate involving topological entropy,” Commun. Math. Phys., 173, 623–646 (1995).

    MathSciNet  MATH  Google Scholar 

  32. 32.

    L. D. Landau and E. M. Lifshits, Theoretical Physics in 10 Volumes. Vol. VIII. Continuum Electrodynamics [in Russian], Fizmatlit, Moscow (2005).

  33. 33.

    D. W. Longcope, “Topological and current ribbons: a model for current, reconnection anf flaring in a complex, evolving corona,” Solar Phys., 169, 91–121 (1996).

    Google Scholar 

  34. 34.

    R. C. Maclean, C. Beveridge, G. Hornig, and E. R. Priest, “Coronal magnetic topologies in a spherical geometry, I. Two bipolar flux sources,” Solar Phys., 235, No. 1-2, 259–280 (2006).

    Google Scholar 

  35. 35.

    R. Maclean, C. Beveridge, D. Longcope, D. Brown, and E. Priest, “A topological analysis of the magnetic breakout model for an eruptive solar flare,” Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci., 461, 2099 (2005).

    MathSciNet  MATH  Google Scholar 

  36. 36.

    R. Maclean, C. Beveridge, and E. Priest, “Coronal magnetic topologies in a spherical geometry, II. Four balanced flux sources,” Solar Phys., 238, 13–27 (2006).

    Google Scholar 

  37. 37.

    R. C. Maclean and E. R. Priest, “Topological aspects of global magnetic field behaviour in the solar corona,” Solar Phys., 243, No. 2, 171–191 (2007).

    Google Scholar 

  38. 38.

    H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fields, Cambridge University Press, Cambridge (1978).

    Google Scholar 

  39. 39.

    H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids [Russian translation], Mir, Moscow (1980).

  40. 40.

    S. A. Molchanov, A. A. Ruzmaykin, and D. D. Sokolov, “Kinematic dynamo in random flow,” Usp. Fiz. Nauk, 145, 593–628 (1985).

    Google Scholar 

  41. 41.

    M. M. Molodenskiy and S. I. Syrovatskiy, “Magnetic fields of active areas and their null points,” Astron. Zh., 54, 1293–1304 (1977).

    Google Scholar 

  42. 42.

    Z. Nitecki, Differential Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, M.I.T. Press, Cambridge–London (1971).

  43. 43.

    A. V. Oreshina, I. V. Oreshina, and B. V. Somov, “Magnetic-topology evolution in NOAA AR 10501 on 2003 November 18,” Astron. Astrophys., 538, 138 (2012).

  44. 44.

    E. N. Parker, “Hydromagnetic dynamo models,” Astrophys. J., 122, 293–314 (1955).

    MathSciNet  Google Scholar 

  45. 45.

    H. Poincaré, “Sur les courbes définies par une équation différentielle, III,” J. Math. Pures Appl., 4, No. 1, 167–244 (1882).

  46. 46.

    E. R. Priest, Solar Magnetohydrodynamics, Springer, Dordrecht (1982).

    Google Scholar 

  47. 47.

    E. Priest, T. Bungey, and V. Titov, “The 3D topology and interaction of complex magnetic flux systems,” Geophys. Astrophys. Fluid Dyn., 84, 127–163 (1997).

    MathSciNet  Google Scholar 

  48. 48.

    E. Priest and T. Forbes, Magnetic Reconnection: MHD Theory and Applications, Cambridge Univ. Press, New York (2000).

    MATH  Google Scholar 

  49. 49.

    E. Priest and T. Forbes, Magnetic Reconnection: MHD Theory and Applications, FML, Moscow (2005).

    MATH  Google Scholar 

  50. 50.

    E. Priest and C. Schriver, “Aspects of three-dimensional magnetic reconnection,” Solar Phys., 190, 1–24 (1999).

    Google Scholar 

  51. 51.

    E. R. Priest and V. S. Titov, “Magnetic reconnection at three-dimensional null points,” Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 354, 2951–2992 (1996).

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Shao Shu-Guang, Wang Shu, Xu Wen-Qing, and Ge. Yu-Li, “On the local C1 solution of ideal magnetohydrodynamical equations,” Discrete Contin. Dyn. Syst., 37, No. 4, 2103–2118 (2007).

  53. 53.

    B. V. Somov, Plasma Astrophysics, Part II: Reconnection and Flares, Springer, N.Y. (2013).

  54. 54.

    S. Smale, “Diffeomorphisms with many periodic points,” Mathematica, 11, No. 4, 88–106 (1967).

    Google Scholar 

  55. 55.

    S. Smale, “Differentiable dynamical systems,” Bull. Am. Math. Soc., 73, 741–817 (1967).

    MathSciNet  Google Scholar 

  56. 56.

    D. D. Sokolov, “Problems of magnetic dynamo,” Usp. Fiz. Nauk, 185, 643–648 (2015).

    Google Scholar 

  57. 57.

    D. D. Sokolov, R. A. Stepanov, and P. G. Frik, “Dynamo: from astrophysic models to laboratory experiment,” Usp. Fiz. Nauk, 184, 313–335 (2014).

    Google Scholar 

  58. 58.

    P. A. Sweet, “The production of high energy particles in solar flares,” Nuovo Cimento Suppl., 8, Ser. X, 188–196 (1958).

  59. 59.

    S. I. Syrovatskiy, “Magnetohydrodynamics,” Usp. Fiz. Nauk, 62, No. 7, 247–303 (1957).

    Google Scholar 

  60. 60.

    S. I. Vaynshteyn and Ya. B. Zel’dovich, “On genesis of magnetic fields in astrophysics (Turbulent mechanisms “dynamo”),” Usp. Fiz. Nauk, 106, 431–457 (1972).

    Google Scholar 

  61. 61.

    Ya. B. Zel’dovich and A. A. Ruzmaykin, “Hydromagnetic dynamo as a source of planetary, solar, and galactic magnetism,” Usp. Fiz. Nauk, 152, 263–284 (1987).

    Google Scholar 

  62. 62.

    E. V. Zhuzhoma and N. V. Isaenkova, “On zero-measure solenoidal basic sets,” Mat. Sb., 202, No. 3, 47–68 (2011).

    MathSciNet  MATH  Google Scholar 

  63. 63.

    E. V. Zhuzhoma, N. V. Isaenkova, and V. S. Medvedev, “On topological structure of magnetic field of regions of the photosphere,” Nelin. Dinam., 13, No. 3, 399–412 (2017).

    MathSciNet  MATH  Google Scholar 

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Correspondence to V. Z. Grines.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 3, Differential and Functional Differential Equations, 2017.

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Grines, V.Z., Zhuzhoma, E.V. & Pochinka, O.V. Dynamical Systems and Topology of Magnetic Fields in Conducting Medium. J Math Sci 253, 676–691 (2021). https://doi.org/10.1007/s10958-021-05261-1

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