Advertisement

Regular and Chaotic Dynamics

, Volume 22, Issue 2, pp 122–135 | Cite as

On the number of heteroclinic curves of diffeomorphisms with surface dynamics

  • Vyacheslav Z. GrinesEmail author
  • Elena Ya. Gurevich
  • Olga V. Pochinka
Article
  • 30 Downloads

Abstract

Separators are fundamental plasma physics objects that play an important role in many astrophysical phenomena. Looking for separators and their number is one of the first steps in studying the topology of the magnetic field in the solar corona. In the language of dynamical systems, separators are noncompact heteroclinic curves. In this paper we give an exact lower estimation of the number of noncompact heteroclinic curves for a 3-diffeomorphism with the so-called “surface dynamics”. Also, we prove that ambient manifolds for such diffeomorphisms are mapping tori.

Keywords

separator in a magnetic field heteroclinic curves mapping torus gradient-like diffeomorphisms 

MSC2010 numbers

37D20 37D15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andronov, A.A. and Pontryagin, L. S., Rough Systems, Dokl. Akad. Nauk SSSR, 1937, vol. 14, no. 5, pp. 247–250 (Russian).Google Scholar
  2. 2.
    Baer, R., Isotopie von Kurven auf orientierbaren, geschlossenen Flächen und ihr Zusammenhang mit der topologischen Deformation der Flächen, J. Reine Angew. Math., 1928, vol. 159, pp. 101–116.zbMATHGoogle Scholar
  3. 3.
    Brown, M., Locally Flat Imbeddings of Topological Manifolds, Ann. of Math. (2), 1962, vol. 75, pp. 331–341.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonatti, C., Grines, V., Medvedev, V., and Pécou, E., Three-Manifolds Admitting Morse–Smale Diffeomorphisms without Heteroclinic Curves, Topology Appl., 2002, vol. 117, no. 3, pp. 335–344.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonatti, C., Grines, V., Medvedev, V., and Pécou, E., Topological Classification of Gradient-Like Diffeomorphisms on 3-Manifolds, Topology, 2004, vol. 43, no. 2, pp. 369–391.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grines, V., Gurevich, E., Zhuzhoma, E., and Zinina, S., Heteroclinic Curves of Morse–Smale Diffeomorphisms and Separators in the Magnetic Field of the Plasma, Nelin. Dinam., 2014, vol. 10, no. 4, pp. 427–438 (Russian).CrossRefzbMATHGoogle Scholar
  7. 7.
    Grines, V., Medvedev, T., Pochinka, O., and Zhuzhoma, E., On Heteroclinic Separators of Magnetic Fields in Electrically Conducting Fluids, Phys. D, 2015, vol. 294, pp. 1–5.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Grines, V. Z., Medvedev, V. S., and Zhuzhoma, E.V., On Surface Attractors and Repellers in 3-Manifolds, Math. Notes, 2005, vol. 78, nos. 5–6, pp. 757–767; see also: Mat. Zametki, 2005, vol. 78, no. 6, pp. 813–826.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grines, V. Z., Zhuzhoma, E.V., and Medvedev, V. S., New Relations for Morse–Smale Systems with Trivially Embedded One-Dimensional Separatrices, Sb. Math., 2003, vol. 194, nos. 7–8, pp. 979–1007; see also: Mat. Sb., 2003, vol. 194, nos. 7–8, pp. 979–1007; see also: Mat. Sb., 2003, vol. 194, no. 7, pp. 25–56MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grines, V., Pochinka, O., Medvedev, V., and Levchenko, Yu., The Topological Classification of Structural Stable 3-Diffeomorphisms with Two-Dimensional Basic Sets, Nonlinearity, 2015, vol. 28, no. 11, pp. 4081–4102.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ghys, E. and Sergiescu, V., Stabilité et conjugaison différentiable pour certains feuilletages, Topology, 1980, vol. 19, no. 2, pp. 179–197.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mañé, R., A Proof of the C1 Stability Conjecture, Publ. Math. Inst. Hautes Études Sci., 1987, vol. 66, pp. 161–210.CrossRefzbMATHGoogle Scholar
  13. 13.
    Moise, E. E., Geometric Topology in Dimensions 2 and 3, Grad. Texts in Math., vol. 47, New York: Springer, 1977.CrossRefzbMATHGoogle Scholar
  14. 14.
    Nielsen, J., Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math., 1927, vol. 50, pp. 189–358.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Palis, J., On Morse–Smale Dynamical Systems, Topology, 1969, vol. 8, no. 4, pp. 385–404.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Palis, J. and de Melo, W., Geometric Theory of Dynamical Systems: An Introduction, New York: Springer, 1982.CrossRefzbMATHGoogle Scholar
  17. 17.
    Peixoto, M. M., Structural Stability on Two-Dimensional Manifolds, Topology, 1962, vol. 1, no. 2, pp. 101–120.Google Scholar
  18. 17a.
    Peixoto, M. M., Structural Stability on Two-Dimensional Manifolds: A Further Remark, Topology, 1963, vol. 2, nos. 1–2, pp. 179–180.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 18.
    Grines, V. Z., Levchenko, Yu.A., Medvedev, V. S., and Pochinka, O. V., On the Dynamical Coherence of Structurally Stable 3-Diffeomorphisms, Regul. Chaotic Dyn., 2014, vol. 19, no. 4, pp. 506–512.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 19.
    Priest, E., Solar Magnetohydrodynamics, Dordrecht: Reidel, 1982.CrossRefGoogle Scholar
  21. 20.
    Priest, E.R. and Forbes, T.G., Magnetic Reconnection: MHD Theory and Applications, Cambridge: Cambridge Univ. Press, 2007.zbMATHGoogle Scholar
  22. 21.
    Robbin, J. W., A Structural Stability Theorem, Ann. of Math. (2), 1971, vol. 94, pp. 447–493.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 22.
    Robinson, C., Structural Stability of C1 Diffeomorphisms, J. Differential Equations, 1976, vol. 22, no. 1, pp. 28–73.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 23.
    Robinson, C., Dynamical Systems: Stability, Symbolic Dynamics,Chaos, 2nd ed., Stud. Adv. Math., vol. 28, Boca Raton, Fla.: CRC, 1998.Google Scholar
  25. 24.
    Smale, S., On Gradient Dynamical Systems, Ann. of Math. (2), 1961, vol. 74, pp. 199–206.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 25.
    Smale, S., Differentiable Dynamical Systems, Bull. Amer. Math. Soc., 1967, vol. 73, no. 6, pp. 747–817.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 26.
    Smale, S., The O-Stability Theorem, in Global Analysis: Proc. Sympos. Pure Math. (Berkeley, Calif., 1968): Vol. 14, Providence, R.I.: AMS, 1970, pp. 289–297.CrossRefGoogle Scholar
  28. 27.
    Zieschang, H., Vogt, E., and Coldewey, H.-D., Surfaces and Planar Discontinuous Groups, Lecture Notes in Math., vol. 835, Berlin: Springer, 1980.CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • Vyacheslav Z. Grines
    • 1
    Email author
  • Elena Ya. Gurevich
    • 1
  • Olga V. Pochinka
    • 1
  1. 1.National Research University Higher School of EconomicsNizhny NovgorodRussia

Personalised recommendations