Dynamics of Field Line Mappings in Magnetic Flux Tubes - Publication - Bridge of Knowledge

Search

Dynamics of Field Line Mappings in Magnetic Flux Tubes

Abstract

We study the topological constraints on the dynamics of magnetic field lines in flux tubes. Our approach is based on the application of the topological invariant: fixed point index. We consider periodic flux tubes and find various restrictions on the field lines that come from the sequence of fixed point indices of iterations. We also analyze the case of a tube with a cylindrical obstacle, deducing some special dynamical properties of the field line mapping from geometrical properties of the tube and the additional assumptions put on the field.

Citations

  • 1

    CrossRef

  • 0

    Web of Science

  • 1

    Scopus

Cite as

Full text

download paper
downloaded 20 times
Publication version
Accepted or Published Version
License
Creative Commons: CC-BY open in new tab

Keywords

Details

Category:
Articles
Type:
artykuł w czasopiśmie wyróżnionym w JCR
Published in:
MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY no. 21, pages 1 - 18,
ISSN: 1385-0172
Language:
English
Publication year:
2018
Bibliographic description:
Graff G., Signerska-Rynkowska J.: Dynamics of Field Line Mappings in Magnetic Flux Tubes// MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY. -Vol. 21, (2018), s.1-18
DOI:
Digital Object Identifier (open in new tab) 10.1007/s11040-018-9284-2
Bibliography: test
  1. Arango, J., Gómez, A.: Flows and diffeomorphisms. Rev. Colombiana Mat. 32, 13-27 (1998) open in new tab
  2. Babenko, I.K., Bogatyi, S.A.: The behavior of the index of periodic points under iterations of a mapping. Math. USSR Izv. 38, 1-26 (1992) open in new tab
  3. Berger, M.A., Field, G.B.: The topological properties of magnetic helicity. J. Fluid. Mech. 147, 133- 148 (1984) open in new tab
  4. Chow, S.N., Mallet-Parret, J., Yorke, J.A.: A periodic point index which is a bifurcation invariant, Geometric dynamics (Rio de Janeiro, 1981), pp. 109-131. Springer Lecture Notes in Math, Berlin (1983). 1007 open in new tab
  5. Fetter, A.L.: Rotating trapped Bose-Einstein condensates. Rev. Mod. Phys. 81, 647-691 (2009) open in new tab
  6. Franks, J.: Rotation numbers and instability sets. Bull. Amer. Math. Soc. (N.S.) 40, 263-279 (2003) open in new tab
  7. Franks, J.: Geodesics on S 2 and periodic points of annulus homeomorphisms. Invent. Math. 108, 403-418 (1992) open in new tab
  8. Freund, I.: Critical point explosions in two-dimensional wave fields. Opt. Commun. 159, 99-117 (1999) open in new tab
  9. Graff, G.: Minimal number of periodic points for smooth self-maps of two-holed 3-dimensional closed ball. Topol. Method Nonl. Ann. 33(1), 121-130 (2009) open in new tab
  10. Graff, G.: Minimal periods of maps of rational exterior spaces. Fund. Math. 163(2), 99-115 (2009) open in new tab
  11. Graff, G., Jezierski, J.: Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds. Geom. Dedicata 187(1), 241-258 (2017) open in new tab
  12. Graff, G., Jezierski, J.: Minimal number of periodic points for C 1 self-maps of compact simply- connected manifolds. Forum Math. 21(3), 491-509 (2009) open in new tab
  13. Graff, G., Nowak-Przygodzki, P.: General form of fixed point indices of an iterated C 1 map and infiniteness of minimal periods. Dyn. Syst. 23(4), 491-504 (2008) open in new tab
  14. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press, Cambridge (1995)
  15. Handel, M.: The rotation set of a homeomorphism of the annulus is closed. Comm. Math. Phys. 127, 339-349 (1990) open in new tab
  16. Jezierski, J., Marzantowicz, W.: Homotopy Methods in Topological Fixed and Periodic Points Theory Topological Fixed Point Theory and its Applications, vol. 3. Springer, Dordrecht (2006) open in new tab
  17. Matsuoka, T., Shiraki, H.: Smooth maps with finitely many periodic points. Mem. Fac. Sci., Kochi Univ. (Math) 11, 1-6 (1990)
  18. Pontin, D.I., Wilmot-Smith, A.L., Hornig, G., Galsgaard, K.: Dynamics of braided coronal loops. II. Cascade to multiple small-scale reconnection events. Astron. Astrophys. 525, A57 (2011) open in new tab
  19. Rubin, J.E., Signerska-Rynkowska, J., Touboul, J.D., Vidal, A.: Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations. Discrete Contin. Dyn. Syst. Ser. B 22(10), 4003- 4039 (2017)
  20. Taylor, J.B.: Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 1139 (1974) open in new tab
  21. Class for Physics of the Royal Swedish Academy of Sciences, Topological phase transitions and topological phases of matter. Scientifc Background on the Nobel Prize in Physics (2016) open in new tab
  22. Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Springer, Berlin (1996) open in new tab
  23. Yeates, A.R., Hornig, G.: Dynamical constraints from field line topology in magnetic flux tubes. J. Phys. A: Math. Theor. 44, 265501 (2011) open in new tab
  24. Yeates, A.R., Hornig, G., Wilmot-Smith, A.L.: Topological constraints on magnetic relaxation. Phys. Rev. Lett. 105, 085002 (2010) open in new tab
Verified by:
Gdańsk University of Technology

seen 106 times

Recommended for you

Meta Tags