Дифференциальные уравнения и их приложения в математическом моделировании: материалы XIII Международной научной конференции (Саранск, 12–16 июля 2017 г.)
A lot of many sorts of graphs (directed, multicolored, bipartite, etc.) were repeatedly used to describe and realize systems with regular dynamics on surfaces. For example, Morse-Smale flows are completely described by a directed graph equipped with a subgraph system. In addition, their dynamics can be described by three-color graphs. Four-color graphs describe the dynamics of some non structurally unstable vector fields, and a directed bipartite graph, equipped with additional information, is a complete topological invariant for Ω-stable flows. In this paper, for each oriented equipped bipartite graph, we construct a standard Ω-stable flow on a closed surface.
Complete affine foliations (i.e., foliations admitting the affine geometry as the transversal structure) are investigated. The strong transversal equivalence of complete affine foliations is considered, which is a more refined notion than the transverse equivalence of foliations in the sense of Molino. The classification of complete affine foliations with respect to the strong transversal equivalence is reduced to the classification up to conjugacy of countable subgroups of the affine group $Aff(A^q)$. It is shown that each equivalence class contains a two-dimensional suspended foliation on the manifold, which is an Elenberg--MacLane space of type $K(\pi,1)$.