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Nonsingular Morse-Smale flows of n-manifolds with attractor-repeller dynamics
Cornell University
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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 bottle can be ambient manifolds for such flows in case of n=2. Authors established that there exist exactly two classes of topological equivalence of such flows of torus and three of the Klein bottle. There are no constraints for odd-dimensional manifolds which follow from the fact that Euler characteristic is zero. However, it is known that orientable 3-manifold admits a flow of considered class if and only if it is a lens space. In this paper, it is proved that up to topological equivalence each of S3 and RP3 admit one such flow and other lens spaces two flows each. Also, it is shown that the only non-orientable n-manifold (for n>2), which admits considered flows is the twisted I-bundle over (n−1)-sphere. Moreover, there are exactly two classes of topological equivalence of such flows. Among orientable n-manifolds only the product of (n−1)-sphere and the circle can be ambient manifold of a considered flow and the flows are split into two classes of topological equivalence.
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
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
V. Kruglov, O. Pochinka, G. Talanova, Proceedings of the International Geometry Center 2020 Vol. 13 No. 1 P. 49-60
Currently, a complete topological classification has been obtained with respect to the topological equivalence of Morse-Smale flows, [9,7], as well as their generalizations of Ω-stable flows on closed surfaces, [4]. Some results on topological conjugacy classification for such systems are also known. In particular, the coincidence of the classes of topological equivalence and conjugacy of ...
Added: June 28, 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
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., Gurevich E., Zhuzhoma E. V. et al., Успехи математических наук 2019 Т. 74 № 1 С. 41-116
The review is devoted to the presentation of results, including recently obtained by the authors, on the topological classification of Morse-Smale systems and the topology of ambient manifolds. ...
Added: November 20, 2018
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
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., 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
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
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
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
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
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
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
А.Л. Добролюбова, В.Е. Круглов, Журнал Средневолжского математического общества 2022 Т. 24 № 1 С. 40-53
В настоящей работе рассмотрены неособые потоки с двумя предельными
циклами на многообразии S2 × S1. Для таких потоков получена классификация с точностью до топологической сопряжённости, показано, что они имеют функциональный
модуль устойчивости. Поскольку для каждого фиксированного аргумента функциональный модуль устойчивости принимает своё значение, из наличия функционального
модуля следует наличие бесконечного числа числовых модулей устойчивости. Для получения данного результата ...
Added: May 4, 2022
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
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
Pochinka O., Shubin D., Nonlinearity 2022 Vol. 35 No. 3 P. 1485-1499
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 1, 2021
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., 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
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
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., 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