**MSC2010** 37C15

### Morse-Smale flows and the model of the topology of magnetic fields in plasma

#### A. N. Sakharov^{1}, A. A. Shilovskaya^{2}

Annotation | This article is a continuation of the work \cite {KST2016} and is devoted to the presentation of results related to the construction of models of magnetic fields in an electrically conductive medium (plasma) in terms of dynamic systems. Research in this direction has been intensively pursued over the past 20 years. Since the solution of equations magnetic hydrodynamics is associated with certain difficulties, then approximate models are used in this approach magnetic field. A class of models of vector fields is constructed, united by the common name {\em model of the topology of magnetic charges}. Fields from this class generate continuous dynamical systems (flows) on three-dimensional manifolds with a sufficiently simple structure. First, the non-wandering set is finite and consists of from hyperbolic equilibrium states, and secondly, these flows allow the existence of a so-called self-indexing energy function that allows them to be complete topological classification. In addition, fields from of this class can be arbitrarily closely approximated by vector fields generating structurally stable flows. Particular attention is paid to the fields in the corona of the Sun, which is associated with the actual task estimates of energy release in solar flares. |
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Keywords | singular points of the field, magnetic field lines, sources, sinks, separatrix, separators, heteroclinic curves |

^{1}**Alexandr N. Sakharov**, Assistant professor of department of higher mathematic and theoretical mechanics, FGBOU VO "Nizhegorodskaya State Agricultural Academy" (603107, Russia, Nizhny Novgorod, Gagarin Avenue, 97), ORCID:http://orcid.org/0000-0002-4520-8062, ansakharov2008@yandex.ru^{2}**Anna A. Shilovskaia**, postgraduate student of the Department of differential equations, mathematical and numerical analysis, Institute of Informational Technology, Mathematics and Mechanics, N.I. Lobachevsky State University (23 Gagarin Av., Nizny Novgorod 603950, Russia), ORCID:http://orcid.org/0000-0002-3328-8157, a.shilovskaia@gmail.com

**Citation**: A. N. Sakharov, A. A. Shilovskaya, "[Morse-Smale flows and the model of the topology of magnetic fields in plasma]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:1 (2017) 88–101 (In Russian)

**DOI** 10.15507/2079-6900.19.2017.01.88-101