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

This paper is the first step in stydying structure of decomposition of phase space with dimension $n\geq 4$ on the trajectories of Morse-Smale flows (structurally stable flows with non-wandering set consisting of finite number of equilibria and closed trajectories) allowing heteroclinic intersections. More precisely, special class of Morse-Smale flows on the sphere $S^n$ is studied. The non-wandering set of the flow of interest consists of two nodal and two saddle equilibrium states. It is proved that for every 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 connectivity components. Heteroclinic intersections are mathematical models for magnetic field separators. Study of their structure, as well as the question of their existence, is one of the principal problems of magnetic hydrodynamics.