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О топологической классификации градиентно-подобных потоков с поверхностной динамикой на 3-многообразиях
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.
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
В. З. Гринес, Е. Я. Гуревич, Успехи математических наук 2022 Т. 77 № 4(466) С. 201-202
A result on the possibility of a complete topological classification of gradient-like flows without heteroclinic intersections, given on a manifold of dimension $n\geq 3$, homeomorphic to the connected sum $\S^{n-1}\times S^1$ is provided. This result significantly extends the class of structurally stable flows for which a topological classification has been obtained. ...
Added: June 24, 2022
Grines V., Gurevich E., Математический сборник 2023 Т. 214 № 5 С. 97-127
We obtain necessary and sufficient conditions for the topological equivalence of gradient-like flows without heteroclinic intersections, defined on a connected sum of a finite number of manifolds homeomorphic to $\mathbb{S}^{n-1}\times \mathbb{S}^1$, $ n\geq 3$. For the case $n>3$, this result essentially extends the class of manifolds for which the topological classification of structurally stable systems ...
Added: December 11, 2022
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., 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
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
Grines V., Gurevich E., Yakovlev E., Журнал Средневолжского математического общества 2021 Т. 23 № 4 С. 379-393
We consider a class GSD(M3) of gradient-like diffeomorphisms with surface dynamics
given on closed oriented manifold M3 of dimension three. In [3] it was proved that manifods,
admitting such diffemorohpsims, are mapping tori under oriented surface of genus g, and the number
of heteroclinic curves no less that 12g. In this paper we determine a subset of GSD(M3) ...
Added: October 24, 2022
Gurevich E., Chernov A., Ivanov A., Динамические системы 2020 Vol. 10 No. 2 P. 129-138
Manifolds admitting a Morse function with three critical points are called projective-like, by analogy with the projective plane. Eells and Kuiper showed that the dimension n of such manifolds takes on the values 2, 4, 8, and 16, and the critical points of the Morse function have indices 0, n / 2, and n. Zhuzhoma ...
Added: November 16, 2020
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
Grines V., Gurevich E., Kevlia S. S., Lobachevskii Journal of Mathematics 2021 Vol. 42 No. 5 P. 901-910
We consider a class of gradient-like flows on three-dimensional closedmanifolds whose
attractors and repellers belongs to a finite union of embedded surfaces and find conditions when the
ambient manifold is Seifert. ...
Added: April 28, 2021
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
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
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
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
Zhuzhoma E. V., Medvedev V., Математический сборник 2016 Т. 207 № 5 С. 69-92
Рассматриваются непрерывные потоки Морса–Смейла, неблуждающее множество которых состоит из трех состояний равновесия, на замкнутых многообразиях. Получены необходимые и достаточные условия топологи- ческой эквивалентности таких потоков и описана топологическая струк- тура несущих многообразий. ...
Added: June 8, 2016
О. В. Починка, Д. Д. Шубин, Математические заметки 2022 Т. 112 № 3 С. 426-443
The topological equivalence of nonsingular Morse–Smale flows under assumptions of various generality has been considered in many works (see, e.g., [1]–[4]). However, in the case of a small number of periodic orbits, it is possible to significantly simplify the known invariants and, most importantly, bring the classification problem to implementation by describing the admissibility of the ...
Added: August 28, 2022
Grines V., Gurevich E., Zhuzhoma E. V. et al., Siberian Advances in Mathematics 2019 Vol. 29 No. 2 P. 116-127
We study relations between the structure of the set of equilibrium points of a gradient-like flows
and the topology of the support manifold of dimension 4 and higher. We introduce a class
of manifolds that admit a generalized Heegaard splitting. We consider gradient-like
flows such that the non-wandering set consists of exactly μ node and ν
saddle equilibrium points of indices equal to either 1 or n − 1. We show ...
Added: May 29, 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., 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
Gurevich E., Журнал Средневолжского математического общества 2017 Т. 19 № 2 С. 25-30
This paper is the first step in stydying structure of decomposition of phase space with dimension n≥4" role="presentation" style="position: relative;">n≥4n≥4n\geq 4 on the trajectories of Morse-Smale flows (structurally stable flows with non-wandering set consisting of finite number of equilibria and closed trajectories) allowing heteroclinic intersections. More precisely, special class of Morse-Smale flows on the sphere ...
Added: October 10, 2017
Zhuzhoma E. V., Medvedev V., Sbornik Mathematics 2016 Vol. 207 No. 5 P. 702-723
Continuous Morse-Smale flows on closed manifolds whose nonwandering set consists of three equilibrium positions are considered. Necessary and sufficient conditions for topological equivalence of such flows are obtained and the topological structure of the underlying manifolds is described. Bibliography: 36 titles. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd. ...
Added: September 13, 2016
Saraev I., Журнал Средневолжского математического общества 2023 Т. 25 № 2 С. 62-75
В статье рассматривается класс G градиентно-подобных потоков на связных замкнутых многообразиях размерности n≥4, такой что для любого потока f^t∈G устойчивые и неустойчивые многообразия седловых состояний равновесия размерности (n−1) не пересекаются с инвариантными многообразиями других седловых состояний равновесия. Известно, что несущее многообразие любого потока f^t из класса G раскладывается в связную сумму сферы S^n, g≥0 копий ...
Added: July 11, 2023