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О ТОПОЛОГИИ ПЛОСКИХ ВЕЩЕСТВЕННЫХ РАСПАДАЮЩИХСЯ КРИВЫХ СТЕПЕНЕЙ 7 И 8
С. 29–30.
We consider the problem of isotopic classification of plane real algebraic curves that decompose into
several irreducible factors, which is related to the first part of Hilbert’s 16th problem. A review of the
results obtained by the authors in the last two years about curves of degree 7 that decompose into three
factors, and about curves of degree 8, that decompose into two factors, is given.
Keywords: Hilbert’s 16th problem, decomposable plane real algebraic curves, topological classification.
In book
Каз.: Издательство Академии наук Республики Татарстан, 2021.
Gurevich E., Imaev R., Russian Journal of Nonlinear Dynamics 2026 Vol. 22 No. 2 P. 469–483
We show that, for Morse – Smale diffemorphism f: M^n→M^n, n⩾4, the closure of one- and (n−1)-dimensional separatrices in a basin of a sink point ω forms a trivial frame that contrasts with the case n=3. This result is a first step in the solution of problems of topological classification, embedding in a flow and existence of energy functions for such ...
Added: July 14, 2026
Gurevich E., Saraev I., Известия РАН. Серия математическая 2026 Т. 90 № 3 С. 19–57
In this paper, we consider a class of gradient-like ows without heteroclinic
intersections, dened on closed manifolds of dimension four. We show that for
such ows, the problem of complete topological classication can be reduced to
the combinatorial problem of distinguishing special framed graphs describing
the mutual arrangement of invariant manifolds and the action of the ow on a
wandering ...
Added: May 18, 2026
Galkin V., Pochinka O., Успехи математических наук 2026 Т. 81 № 2(422) С. 71–144
Настоящая работа посвящена изучению динамики регулярных четырехмерных потоков, их топологической классификации и взаимосвязи с топологией несущего многообразия. Регулярные потоки являются топологическими аналогами потоков Морса–Смейла. Их появление мотивировано двумя фактами: 1) существованием топологических многообразий размерности 4 и выше, не имеющих гладкой структуры; 2) развитием методов топологической классификации гладких систем, использующих чисто топологические свойства этих систем и ...
Added: April 1, 2026
Pochinka O., Chilina E., Математический сборник 2026 Т. 217 № 3 С. 112–134
Работа посвящена исследованию сохраняющих ориентацию гомеоморфизмов трехмерных многообразий с неблуждающим множеством, состоящим из конечного числа двумерных аттракторов и репеллеров, каждый из которых является дизъюнктным объединением цилиндрически вложенных замкнутых поверхностей, ограничение некоторой степени на каждую из которых топологически сопряжено сохраняющему ориентацию псевдоаносовскому гомеоморфизму. Получена топологическая классификация модельных гомеоморфизмов, реализованных на каждом многообразии, допускающем гомеоморфизмы исследуемого класса. Доказано, ...
Added: March 3, 2026
Мартынов Т. Д., Pochinka O., Chilina E., Математика и теоретические компьютерные науки 2025 Т. 3 № 3 С. 87–109
According to J. Nielsen and H. Hang, each class of topological conjugacy of periodic homeomorphisms of orientable compact surfaces is completely described by a finite set of data called the characteristic. For a two-dimensional sphere, exhaustive classification results with the construction of linear representatives in each conjugacy class were obtained by B. Kerekyarto. For a ...
Added: October 18, 2025
Gurevich E., Математика и теоретические компьютерные науки 2025 Т. 3 № 3 С. 20–42
A topological classification of smooth, structurally stable flows on four-dimensional closed manifolds is obtained, the wandering set of which contains isolated trajectories connecting saddle equilibria (heteroclinic curves). For dimensional reasons, the heteroclinic curves of such flows belong to the intersection of invariant manifolds of saddles of adjacent Morse indices. We assume that the non-wandering set of ...
Added: October 18, 2025
Gurevich E., Saraev I., Communications in Nonlinear Science and Numerical Simulation 2026 Vol. 152 No. D Article 109301
We consider homeomorphisms with a finite and topologically hyperbolic chain-recurrent set and explore the interrelation between the structure of the chain-recurrent set and the topology of carrying manifolds of dimension n > 3. We demonstrate that the Lefschetz fixed-point theorem can be refined for this class of homeomorphisms as follows: if the (n-1)-dimensional invariant manifold ...
Added: October 4, 2025
Galkin V., Pochinka O., Математические заметки 2025 Т. 117 № 6 С. 861–878
Под регулярным топологическим потоком на замкнутом nn-многообразии понимается поток, цепно рекуррентное множество которого состоит из конечного числа топологически гиперболических неподвижных точек и периодических орбит. Такой поток называется неособым, если его цепно рекуррентное множество не содержит неподвижных точек. Топологической эквивалентности маломерных неособых потоков в предположениях различной общности посвящен целый ряд работ. Начиная с размерности 4 имеется пока незначительное число ...
Added: June 2, 2025
Elena Ya. Gurevich, Ilya A. Saraev, Regular and Chaotic Dynamics 2025 Vol. 30 No. 2 P. 254–278
S.~Smale has shown that any closed smooth manifold admits a gradient-like flow, which is a structurally stable flow with a finite non-wandering set. Polar flows are a specific type of gradient-like flows characterized by the simplest non-wandering set for the given manifold, consisting of exactly one source, one sink, and a finite number of saddle ...
Added: November 21, 2024
Gurevich E., Saraev I., Математические заметки 2024 Т. 116 № 1 С. 45–66
In the paper, the problem of topological classification of polar flows on closed four-dimensional manifolds whose set of saddle equilibrium states consists only of points having two-dimensional stable and unstable manifolds. It is shown that the complete topological invariant for such flows is the Kirby diagram, which is a framed link on a sphere intersecting ...
Added: July 1, 2024
E.Y. Gurevich, I.A. Saraev, Partial Differential Equations in Applied Mathematics 2024 Vol. 11 Article 100759
We study relations between a structure of non-wandering set of a Morse–Smale diffeomorphism f and its carrying closed manifold M^n. We prove that if f has no any saddle periodic points with one-dimensional unstable manifolds, and for any periodic point p of Morse index (n−1) its unstable manifolds do not intersect stable invariant manifolds of saddle periodic points different from ...
Added: June 22, 2024
Shubin D., Pochinka O., / Series math "arxiv.org". 2024.
In this paper we consider non-singular Morse-Smale flows on closed orientable 3-manifolds, under the assumption that among the periodic orbits of the flow there is only one saddle orbit and it is twisted. It is found that any manifold admitting such flows is either a lens space, or a connected sum of a lens space ...
Added: May 14, 2024
Горская В. А., В кн.: Математика в современном мире: Материалы II Всероссийской конференции, посвящённой 100-летию со дня рождения видного российского математика Д.А. Граве.: Вологда: Вологодский государственный университет, 2023. С. 62–64.
В настоящей работе продолжается исследование вещественных алгебраических кривых, распадающихся в произведение двух неособых кривых степени 2 и неособой кривой степени 3, начатое в работах [4], [5]. Автор благодарит Г.М. Полотовского за предложенную задачу и полезные обсуждения. ...
Added: May 10, 2024
Morozov A., Pochinka O., Moscow Mathematical Journal 2023 Vol. 23 No. 4 P. 571–590
In this paper, we consider orientation-preserving Morse-Smale diffeomorphisms on orientable closed surfaces. Such diffeomorphisms can have infinitely many heteroclinic orbits, which makes their topological classification very difficult. In fact, even in the case of a finite number of heteroclinic orbits, there are no exhaustive classification results. The main problem is that for all currently known ...
Added: November 29, 2023