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О топологической классификации градиентно-подобных потоков без гетероклинических пересечений посредством энергетической функции
Журнал Средневолжского математического общества. 2013. Т. 15. № 4. С. 91-100.
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
Kurenkov E., Gurevich E., Труды Средневолжского математического общества 2015 Т. 17 № 2 С. 15-26
We introduce the definition of consistent equivalence of energy Morse-Bott functions for Morse-Smale flows on surfaces and state that consistent equivalence of that functions is necessary and sufficient condition for such flows. ...
Added: October 12, 2015
Grines V., Gurevich E., Pochinka O., Математические заметки 2014 Т. 96 № 6 С. 856-863
For gradient-like flows without heteroclinic intersections of the stable and unstable manifolds of saddle periodic points all
of whose saddle equilibrium states have Morse index 1 or n − 1, the notion of consistent equivalence of energy functions
is introduced. It is shown that the consistent equivalence of energy functions is necessary and sufficient for topological
equivalence. ...
Added: September 29, 2014
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., 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., Gurevich E., Pochinka O., Математический сборник 2012 Т. 203 № 12 С. 81-104
В настоящей работе для многообразий размерности 3 решена проблема Дж. Палиса о нахождении необходимых и достаточных условий включения каскада Морса–Смейла в топологический поток. Множество таких потоков открыто в пространстве всех диффеоморфизмов, в то время как множество произвольных диффеоморфизмов, включающихся в гладкий поток, является нигде не плотным. Кроме того, в работе выделен класс диффеоморфизмов, включающихся в топологический ...
Added: September 27, 2014
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
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
Kolobianina A., Kruglov V., Журнал Средневолжского математического общества 2019 Т. 21 № 4 С. 460-468
The paper is devoted to the study of the class of Ω-stable flows without limit cycles on
surfaces, i.e. flows on surfaces with non-wandering set consisting of a finite number of hyperbolic
fixed points. This class is a generalization of the class of gradient-like flows, differing by forbiddance
of saddle points connected by separatrices. The results of the ...
Added: October 22, 2019
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
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
Gurevich E., Kurenkov E., Журнал Средневолжского математического общества 2014 Т. 16 № 3 С. 36-40
We introduce the denition of consistent equivalence of Meyer ξ -functions for Morse- Smale ows on surfaces (that are Lyapunov funñtions) and state that consistent equivalence of ξ -functions is necessary and sucient condition for such ows. ...
Added: December 16, 2014
Kruglov V., Pochinka O., Известия высших учебных заведений. Прикладная нелинейная динамика 2021 Т. 29 № 6 С. 835-850
Purpose. The purpose of this study is to consider the class of Morse-Smale flows on surfaces, to
characterize its subclass consisting of flows with a finite number of moduli of stability, and to obtain a topological
classification of such flows up to topological conjugacy, that is, to find an invariant that shows that there exists
a homeomorphism that ...
Added: October 5, 2021
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
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
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., 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
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
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
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
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
Босова А. А., Kruglov V., Pochinka O., Таврический вестник информатики и математики 2017 № 4(37) С. 51-58
In this paper the class of simplest not rough Ω-stable flows on a sphere is considered. We call simplest not rough Ω-stable flow an Ω-stable flow with least number of fixed points, a single separatrix connecting saddle points and without limit cycles. For such flows we design the Morse energy function. ...
Added: March 9, 2018