Dmitry Malyshev
ul. Gagarina 23, Nizhny Novgorod, 603950 Russia
Nizhnii Novgorod State University
Publications:
Kruglov V., Malyshev D. S., Pochinka O. V., Shubin D.
On Topological Classification of Gradientlike Flows on an $n$sphere in the Sense of Topological Conjugacy
2020, vol. 25, no. 6, pp. 716728
Abstract
In this paper, we study gradientlike flows without heteroclinic intersections on an
$n$sphere up to topological conjugacy. We prove that such a flow is completely defined by a
bicolor tree corresponding to a skeleton formed by codimension one separatrices. Moreover, we
show that such a tree is a complete invariant for these flows with respect to the topological
equivalence also. This result implies that for these flows with the same (up to a change
of coordinates) partitions into trajectories, the partitions for elements, composing isotopies
connecting timeone shifts of these flows with the identity map, also coincide. This phenomenon
strongly contrasts with the situation for flows with periodic orbits and connections, where
one class of equivalence contains continuum classes of conjugacy. In addition, we realize every
connected bicolor tree by a gradientlike flow without heteroclinic intersections on the $n$sphere.
In addition, we present a lineartime algorithm on the number of vertices for distinguishing these
trees.

Grines V. Z., Malyshev D. S., Pochinka O. V., Zinina S. K.
Efficient Algorithms for the Recognition of Topologically Conjugate Gradientlike Diffeomorhisms
2016, vol. 21, no. 2, pp. 189203
Abstract
It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradientlike systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration algorithm. However, an efficient algorithm (i. e., polynomial in the number of vertices) has not yet been developed for it, and the problem has not been proved to be intractable (i. e., NPcomplete). We give polynomialtime algorithms for recognition of the corresponding graphs for two gradientlike systems. Moreover, we present efficient algorithms for determining the orientability and the genus of the ambient surface. This result, in particular, sheds light on the classification of configurations that arise from simple, pointsource potentialfield models in efforts to determine the nature of the quietSun magnetic field.
