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## Morse-Smale Systems and Topological Structure of Carries Manifolds

Journal of Mathematical Sciences. 2019. Vol. 239. No. 5. P. 549-581.

We revier the results describing the connection between the global dynamics of Morse-Smale systems on closed manifolds and the topology of carrier manifolds. Also we consider the rezults related to topological classification of Morse-Smale systems.

Grines V., Gurevich E., Pochinka O., Moscow Mathematical Journal 2019 Vol. 19 No. 4 P. 739-760

J.~Palis found necessary conditions for a Morse-Smale diffeomorphism on a closed $n$-dimensional manifold $M^n$ to embed into a topological flow and proved that these conditions are also sufficient for $n=2$. For the case $n=3$ a possibility of wild embedding of closures of separatrices of saddles is an additional obstacle for Morse-Smale cascades to embed into ...

Added: October 17, 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

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., 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., 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

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

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

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., 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

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

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

Grines V., Zhuzhoma E. V., Medvedev V., / Cornell University. Series math "arxiv.org". 2018.

We study a topological structure of a closed $n$-manifold $M^n$ ($n\geq 3$) which admits a Morse-Smale diffeomorphism such that codimension one separatrices of saddles periodic points have no heteroclinic intersections different from heteroclinic points. Also we consider gradient like flow on $M^n$ such that codimension one separatices of saddle singularities have no intersection at all. ...

Added: October 22, 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., 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., 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

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

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

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

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., Malyshev D., Pochinka O. et al., Regular and Chaotic Dynamics 2016 Vol. 21 No. 2 P. 189-203

It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration
algorithm. However, an efficient ...

Added: April 5, 2016

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

Grines V., Gurevich E., Pochinka O., / Cornell University Library. 2018.

A numerical solution of differential equation is adequate if, in particular, the obtained discrete model is topologically conjugate to the time one map of the original flow. B.~Garay showed that the Runge-Kutta discretization of a gradient-like flow $(n > 2)$ on the $n$-disk is topologically conjugate to the time one map for a sufficiently ...

Added: October 13, 2018

E. V. Zhuzhoma, V. S. Medvedev, Mathematical notes 2022 Vol. 111 No. 5-6 P. 870-878

We introduce high-dimensional Morse-Smale systems with king-saddles. Every saddle of such system has a separatrix with heteroclinic intersections. We get the necessary and sufficient conditions of conjugacy of Morse-Smale diffeomorphisms with king-saddles. For every polar Morse-Smale system with two saddles on the $n$-sphere $\mathbb{S}^n$, one proves the existence of a king-saddle. This allows to get ...

Added: October 30, 2022