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## On Embedding of Multidimensional Morse-Smale Diffeomorphisms into Topological Flows

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 topological flows. In this paper we show that there are no such obstructions for Morse-Smale diffeomorphisms without heteroclinic intersection given on the sphere $S^n, \,n\geq 4$, and Palis's conditions again are sufficient for such diffeomorphisms.

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., Zhuzhoma E. V., Pochinka O., 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. ...

Added: May 18, 2019

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., 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

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., 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

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

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

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., 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., 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

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., 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

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., 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., 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

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

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., Pochinka O., Russian Math. Surveys 2013 Vol. 68 No. 1 P. 117-173

This is a survey article on Morse-Smale cascades on orientable 3-manifolds. It contains results of the authors and their collaborators on topological classication, the relation of dynamics with the topology of the ambient manifold, criteria for embeddability in a topological ow and a sucient and necessary condition for the existence of an energy function for ...

Added: October 23, 2014

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