Maxim Gorkiy str., 162,18, Nizhniy Novgorod,Russia
Lobachevsky State University of Nizhny Novgorod
Grines V. Z., Kruglov E. V., Pochinka O. V.
The Topological Classification of Diffeomorphisms of the Two-Dimensional Torus with an Orientable Attractor
2020, Vol. 16, no. 4, pp. 595-606
This paper is devoted to the topological classification of structurally stable diffeomorphisms of the two-dimensional torus whose nonwandering set consists of an orientable one-dimensional attractor and finitely many isolated source and saddle periodic points, under the assumption that the closure of the union of the stable manifolds of isolated periodic points consists of simple pairwise nonintersecting arcs. The classification of one-dimensional basis sets on surfaces has been exhaustively obtained in papers by V. Grines. He also obtained a classification of some classes of structurally stable diffeomorphisms of surfaces using combined algebra-geometric invariants. In this paper, we distinguish a class of diffeomorphisms that admit purely algebraic differentiating invariants.
Pochinka O. V., Kruglov E. V., Dolgonosova A.
Scenario of reconnection in the solar corona with a simple discretization
2017, Vol. 13, No. 4, pp. 573–578
In this paper, one of the possible scenarios for the creation of heteroclinic separators in the solar corona is described and realized. This reconnection scenario connects the magnetic field with two zero points of different signs, the fan surfaces of which do not intersect, with a magnetic field with two zero points which are connected by two heteroclinic separators. The method of proof is to create a model of the magnetic field produced by the plasma in the solar corona and to study it using the methods of dynamical systems theory. Namely, in the space of vector fields on the sphere $S^3$ with two sources, two sinks and two saddles, we construct a simple arc with two saddle-node bifurcation points that connects the system without heteroclinic curves to a system with two heteroclinic curves. The discretization of this arc is also a simple arc in the space of diffeomorphisms. The results are new.