**MSC2010** 37D15

### On the simplest Morse-Smale flows with heteroclinical intersections on the sphere $S^n$

#### E. Gurevich^{1}, D. Pavlova^{2}

Annotation | This paper is the first step in stydying a structure of decomposition of phase space of dimension $n\geq 4$ on the trajectories of Morse-Smale flows (structurally stable flows whose non-wandering set consisits of finite number of equilibria and closed trajectories) admitting heteroclinic intersections. More precisely, the class of Morse-Smale flows on the sphere $ S ^ n $, the non-wandering set of which consists of two nodal and two saddle equilibrium states is considered. It is proved that for any flow from the class under consideration the intersection of invariant manifolds of two different saddle equilibrium states is nonempty and consists of a finite number of connected components. Heteroclinic intersections are a mathematical model of magnetic field separators, and the study of their structure, as well as the question of their existence, is one of the principal problems of magnetic hydrodynamics. |
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Keywords | Morse-Smale flows, heteroclinic intersection |

^{1}**Elena Gurevich**, Associate Professor, Department of Fundamental Mathematics, National Research University Higher School of Economics (25 Bolshaya Pechyorskaya Str., Nizhnii Novgorod 603155, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0003-1815-3120, egurevich@hse.ru^{2}**Daria Pavlova**, student, National Research University Higher School of Economics (25 Bolshaya Pechyorskaya Str., Nizhnii Novgorod 603155, Russia), ORCID: http://orcid.org/0000-0001-8634-4143, dapavlova$_1$@mail.ru

**Citation**: E. Gurevich, D. Pavlova, "[On the simplest Morse-Smale flows with heteroclinical intersections on the sphere $S^n$]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:2 (2017) 25–30 (In Russian)

**DOI** 10.15507/2079-6900.19.201701.025-030