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О включении диффеоморфизмов Морса–Смейла на 3-многообразии в топологический поток
Математический сборник. 2012. Т. 203. № 12. С. 81-104.
Grines V., Gurevich E., Pochinka O., Математические заметки 2012 Т. 91 № 5 С. 791-794
В работе получены необходимые и достаточные условия включения в топологический поток диффеоморфизмов Морса-Смейла без гетероклинически пересечений инвариантных многообразий седловых периодических точек, заданных на многообразии размерности три и выше. ...
Added: September 27, 2014
Gurevich E., Pochinka O., Grines V., Современная математика. Фундаментальные направления 2014
В работе изучается класс $G(M^n)$ сохраняющих ориентацию диффеоморфизмов Морса-Смейла на связном замкнутом гладком многообразии $M^n$ размерности $n\geq 4$, выделенный следующими условиями: для любого $f\in G(M^n)$ инвариантные многообразия седловых периодических точек имеют размерность $1$ и $(n-1)$; инвариантные многообразия различных седловых точек не пересекаются. Доказано, что, в зависимости от соотношения между числом седловых и узловых периодических ...
Added: October 24, 2014
Grines V., Gurevich E., Pochinka O., Современная математика. Фундаментальные направления 2020 Т. 66 № 2 С. 160-181
This review presents the results of recent years on solving of the J. Palis's problem on finding necessary and sufficient conditions for the embedding of Morse – Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomophisms given on manifolds of dimension two. The result for the circle ...
Added: June 4, 2020
Pochinka O., Shubin D., / Cornell University. Series math "arxiv.org". 2022.
In the present paper, 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 one and it is twisted are considered. An exhaustive description of the topology of such manifolds is obtained. Namely, it has been established that any manifold admitting such ...
Added: January 30, 2023
Pochinka O., Shubin D., Applied Mathematics and Nonlinear Sciences 2020 Vol. 5 No. 2 P. 261-266
In the present paper we construct an example of 4-dimensional flows on $S^3\times S^1$ whose saddle periodic orbit has a wildly embedded 2-dimensional unstable manifold. We prove that such a property has every suspension under a non-trivial Pixton's diffeomorphism. Moreover we give a complete topological classification of these suspensions. ...
Added: October 14, 2019
Gurevich E., Смирнова А. С., Динамические системы 2018 Т. 2 № 15 С. 159-172
We consider a class $G$ of Morse-Smale diffeomorphisms on the sphere $S^n$ of dimension $n\geq 4$ such that invariant manifolds of different saddle periodic points of any diffeomorphisms from $G$ have no intersection. Dynamics of an arbitrary diffeomorphism $f\in G$ can be represented as ``sink-source'' dynamics where the ``sink'' $A_f$ (the ``source'' $R_f$) is the ...
Added: November 2, 2018
Grines V., Gurevich E., Kurenkov E., Математические заметки 2020 Т. 107 № 1 С. 145-148
In the paper the topological classification of gradient-like flows on mapping tori is obtained. Such flows naturally appear in the modelling of processes with at least on cyclic coordinate. ...
Added: October 17, 2019
Kruglov V., Malyshev D., Pochinka O. et al., Regular and Chaotic Dynamics 2020 Vol. 25 No. 6 P. 716-728
In this paper, we study gradient-like flows without heteroclinic intersections on n-sphere up to topological conjugacy. We prove that such a flow is completely defined by a bi-colour tree corresponding to a skeleton formed by co-dimension one separatrices. Moreover, we show that such a tree is a complete invariant for these
flows with respect to the ...
Added: November 15, 2020
Grines V., Gurevich E., Pochinka O., Математические заметки 2019 Т. 105 № 1 С. 136-141
We provide a definition of a two-colored graph of a Morse-Smale diffeomorphism without heteroclinical intersection defined on the sphere $S^n$, $n\geq 4$ and prove that this graph is the complete topological invariant for such diffeomorphisms. ...
Added: October 13, 2018
Gurevich E., Павлова Д. А., Журнал Средневолжского математического общества 2018 Т. 20 № 4 С. 378-383
We study a structure of four-dimensional phase space decomposition on trajectories of Morse-Smale flows admitting heteroclinical intersections. We consider a class $G(S^4)$ of Morse-Smale flows on the sphere $S^4$ such that for any flow $f\in G(S^4)$ its non-wandering set consists of exactly four equilibria: source, sink and two saddles. Wandering set of such flows ...
Added: November 11, 2018
Kruglov V., Malyshev D., Pochinka O. et al., Discrete and Continuous Dynamical Systems 2020
In this paper, we study gradient-like flows without heteroclinic intersections on n-sphere up to topological conjugacy. We prove that such a flow is completely defined by a bi-colour tree corresponding to a skeleton formed by co-dimension one separatrices. Moreover, we show that such a tree is a complete invariant for these flows with respect to ...
Added: October 17, 2019
Pochinka O., Гринес В. З., Успехи математических наук 2013 Т. 68 № 1 (409) С. 129-188
Исследования связаны с каскадами Морса-Смейла на ориентируемых 3-многообразиях и включают в себя их полную топологическую классификацию, установление взаимосвязи их динамики с топологией объемлющего многообразия, критерий включения в топологический поток, а также необходимые и достаточные условия существования для таких каскадов энергетической функции. ...
Added: March 25, 2014
Gurevich E., Сяинова Д. Т., Журнал Средневолжского математического общества 2014 Т. 16 № 2 С. 46-56
We specify S. Batterson's results of [7] where classes of isotopic maps on torus contained Morse-Smale diffeomorphisms are described. Following to ideas of paper [7], we describe isotopic classes, contained gradient-like diffeomorphisms, present all admitted types of periodic data of such diffeomorphisms and provide an algorithm of realization of each type of periodic data. ...
Added: October 14, 2014
Pochinka O., Grines V., Gurevich E. et al., Математический сборник 2012 Т. 203 № 12 С. 81-104
В настоящей работе для разнообразий размерности 3 решена проблема Дж. Палиса о нахождении необходимых и достаточных условий включения каскада Иорса-Смейла в топологический поток. Кроме того, в работе выделен класс диффеоморфизмов, включающихся в топологический поток, для которых полным топологическим вариантом является граф, аналогичный схеме Е.А. Андроновой, А.Г. Майера и графу М. Пейкшото ...
Added: March 25, 2014
Левченко Ю. А., Grines V., Pochinka O., Математические заметки 2015 Т. 97 № 2 С. 318-320
В статье получена топологическая классификация структурно устойчивых диффеоморфизмов трехмерных многообразий, неблуждающее множество которых состоит из двумерных базисных множеств. ...
Added: September 2, 2015
Grines V., Левченко Ю. А., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2012 Т. 447 № 2 С. 127-129
The paper is devoted to topological classifiication of cascades on 3-manifolds whose nonwandering set consists of surface 2-dimensional basic sets. ...
Added: February 25, 2015
Grines V., Gurevich E., Medvedev V., Труды Математического института им. В.А. Стеклова РАН 2020 Т. 310 С. 119-134
В работе рассматривается класс G(S^n) сохраняющих ориентацию диффеоморфизмов Морса-Смейла, заданных на сфере S^n размерности n≥4 в предположении, что инвариантные многообразия различных седловых периодических точек не пересекаются. Для диффеоморфизмов из этого класса описан алгоритм реализации всех классов топологической сопряженности. ...
Added: June 4, 2020
Gurevich E., Malyshev D., Журнал Средневолжского математического общества 2016 Т. 18 № 4 С. 30-33
We consider a class $G$ of orientation preserving Morse-Smale diffeomorphisms without heteroclinical intersection defined on the sphere $S^{n}$ of dimension $n>3$. We put a colored graph $\Gamma_f$, endowed by a automorphism $P_f$ into the correspondence for every diffeomorphism $f\in G$ and give a definition of an isomorphism of such graphs. There is stated that there ...
Added: November 16, 2016
Pochinka O., Митрякова Т. М., Труды Средневолжского математического общества 2015 Т. 17 № 4 С. 37-40
Получены необходимые и достаточные условия топологической сопряженности диффеоморфизмов на три-моногобразиях с конечным числом орбит гетероклинического касания. ...
Added: December 7, 2015
Горская В. А., Polotovskiy G., Журнал Средневолжского математического общества 2020 Т. 22 № 1 С. 24-37
In the first part of the 16th Hilbert problem the question about the topology of
nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of
algebraic manifolds with singularities belong to this subject too. The particular case of this problem
is the study of curves that are decompozable into the product of curves in a ...
Added: April 16, 2020
Pochinka O., Shubin D., / Cornell University. Серия math "arxiv.org". 2021.
In the present paper the exhaustive topological classification of nonsingular Morse-Smale flows of n-manifolds with two limit cycles is presented. Hyperbolicity of periodic orbits implies that among them one is attracting and another is repelling. Due to Poincare-Hopf theorem Euler characteristic of closed manifold Mn which admits the considered flows is equal to zero. Only torus and Klein ...
Added: December 3, 2021
Polotovskiy G., Борисов И. М., Итоги науки и техники. Современная математика и ее приложения. Тематические обзоры 2020 Т. 176 С. 3-18
The problem of topological classification of locations in the real projective plane of the union of nonsingular curves of degrees 2 and 6 is considered under some conditions of maximality and general position. After listing the permissible topological models of such locations to be investigated using the Orevkov method, based on the theory of braides ...
Added: October 25, 2019
Gurevich E., Труды Средневолжского математического общества 2015 Т. 17 № 3 С. 120-126
We define a class of gradient-like diffeomorphisms that can be presented as local products of diffeomorphisms on the circle and on a surface, provide their topological classification and specify topology of the ambient manifold. ...
Added: December 4, 2015
Grines V., Zhuzhoma E. V., Medvedev V. et al., Siberian Advances in Mathematics 2018 Т. 21 № 2 С. 163-180
In this paper, we study the relationship between the structure of the set of equilibrium states of a gradient-like flow and the topology of a carrier manifold of dimension 4 and higher. We introduce a class of manifolds admitting a generalized Heegaard decomposition. It is established that if a non-wandering set of a gradient-like flow ...
Added: May 27, 2018