ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Pochinka O. V., Galkina S. Y., Shubin D. D. Modeling of gradient-like flows on n-sphere. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 6, pp. 63-72. DOI: 10.18500/0869-6632-2019-27-6-63-72

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 233)
Article type: 

Modeling of gradient-like flows on n-sphere

Pochinka Olga Vitalievna, National Research University "Higher School of Economics"
Galkina Svetlana Yurievna, National Research University "Higher School of Economics"
Shubin Danila Denisovich, National Research University "Higher School of Economics"

A general idea of the qualitative study of dynamical systems, going back to the works by A. Andronov, E. Leontovich, A. Mayer, is a possibility to describe dynamics of a system using combinatorial invariants. So M. Peixoto proved that the structurally stable flows on surfaces are uniquely determined, up to topological equivalence, by the isomorphic class of a directed graph. Multidimensional structurally stable flows does not allow entering their classification into the framework of a general combinatorial invariant. However, for some subclasses of such systems it is possible to achieve the complet combinatorial description of their dynamics.

In the present paper, based on classification results by S. Pilyugin, A. Prishlyak, V. Grines, E. Gurevich, O. Pochinka, any connected bi-color tree implemented as gradient-like flow of n-sphere, n > 2 without heteroclinic intersections. This problem is solved using the appropriate gluing operations of the so-called Cherry boxes to the flow-shift. This result not only completes the topological classification for such flows, but also allows to model systems with a regular behavior. For such flows, the implementation is especially important because they model, for example, the reconnection processes in the solar corona.

  1. Andronov A.A. and Pontryagin L.S. Rough systems. Doklady Akademii nauk SSSR, 1937, vol. 14, no. 5, pp. 247–250 (in Russian).
  2. Leontovich E.A., Mayer A.G. About trajectories determining qualitative structure of sphere partition into trajectories. Doklady Akademii nauk SSSR, 1937, vol. 14, no. 5, pp. 251–257 (in Rissian).
  3. Leontovich E.A., Mayer A.G. About scheme determining topological structure of partition into trajectories. Doklady Akademii nauk SSSR, 1955, vol. 103, no. 4, pp. 557–560 (in Rissian).
  4. Peixoto M. On the Classification of Flows on Two Manifolds. Dynamical systems Proc., 1971.
  5. Pilyugin S.Yu. Phase diagrams determining Morse–Smale systems without periodic trajectories on spheres. Differencial’nyje uravneniya, 1978, vol. 14, no. 2, pp. 245–254 (in Russian).
  6. Prishlyak A.O. Morse–Smale vector-fields without closed trajectories on three-dimensional manifolds. Matematicheskie zametki, 2002, vol. 71, no. 2, pp. 254–260 (in Russian).
  7. Grines V.Z., Gurevich E.Ya., Pochinka O.V. A combinatorial invariant of Morse–Smale diffeomorphisms without heteroclinic intersections on the sphere Sn, n ≥ 4 // Math. Notes. 2019. Vol. 105, no. 1. P. 132–136.
  8. Pesin Ya.B., Yurhcenko A.A. Some physical models described by the reaction-diffusion equation and chains of connected maps. UMN, 2004, vol. 59, no. 3(357), pp. 81–114 (in Russian).
  9. Browns D.S., Priest E.R. The topological behaviour of stable magnetic separators // Sol. Phys. 1999. Vol. 190. P. 25–33.
  10. Grines V., Zhuzhoma E.V., Pochinka O., Medvedev T.V. On heteroclinic separators of magnetic fields in electrically conducting fluids // Physica D: Nonlinear Phenomena. 2015. Vol. 294. P. 1–5.
  11. Priest E., Forbes T. Magnetic Reconnection: MHD Theory and Applications. Cambridge univ. Prees, Cambridge, 2000.
  12. Smale S. Differentiable dynamical systems // Bull. Amer. Soc. 1967. Vol. 73. P. 747–817.
  13. Grines V.Z., Medvedev T.V., Pochinka O.V. Dynamical Systems on 2- and 3- Manifolds // Dev. Math., 46, Springer, Cham, 2016, xxvi+295 pp.
  14. Cantrell J.C. Almost locally flat sphere S n−1 in S n // Proceeding of the American Mathematical society. 1964. Vol. 15. P. 574–578.
  15. Brown M. Locally Flat Imbeddings of Topological Manifolds // Annals of Mathematics Second Series. 1962. Vol. 75. P. 331–341.
  16. Grines V.Z., Gurevich E.Ya., Medvedev V.S. Classification of Morse–Smale diffeomorphisms with one-dimensional set of unstable separatrices // Proc. Steklov Inst. Math. 2010. Vol. 270. P. 57–79.
  17. Harary F. Graph Theory. Addison-Wesley, Reading, MA, 1969. 
Short text (in English):
(downloads: 120)