?
О топологической классификации диффеоморфизмов на 3-многообразиях с поверхностными двумерными аттракторами и репеллерами
Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика). 2012. Т. 447. № 2. С. 127-129.
Grines V., Левченко Ю. А.
The paper is devoted to topological classifiication of cascades on 3-manifolds whose nonwandering set consists of surface 2-dimensional basic sets.
Ilyashenko Y., Kudryashov Y., Schurov I., / Cornell University. Series math "arxiv.org". 2015. No. 1506.06797.
This is the first part of a two parts paper dedicated to global bifurcations in the plane. In this part we construct an open set of three parameter families whose topological classification has a numerical invariant that may take an arbitrary positive value. In the second part we construct an open set of six parameter ...
Added: June 24, 2015
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., 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
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
Yu. Ilyashenko, Kudryashov Y., I. Schurov, Inventiones Mathematicae 2018 Vol. 213 No. 2 P. 461-506
We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below “families” are “families of vector fields in the two-sphere”. This result disproves an Arnold’s conjecture of 1985. Then we construct an ...
Added: February 6, 2018
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., 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
Grines V., Kurenkov E., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2019 Т. 485 № 2 С. 135-138
In the present paper axiom $A$ diffeomorphisms of closed 2-manifolds of genus $p \geq 2$ whose nonwandering set contains perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of the paper is the following. ...
Added: October 20, 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
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
Ilyashenko Y., Solodovnikov N., Goncharuk N. B., Ergodic Theory and Dynamical Systems 2018
We prove that generic one-parameter families of vector fields on S2 with a parabolic cycle (families of class PC) are structurally stable. The bifurcations in these families are classified. ...
Added: December 15, 2017
Е.В. Жужома, Исаенкова Н. В., В.С. Медведев, Журнал Средневолжского математического общества 2018 Т. 20 № 1 С. 23-29
In the paper we construct some example of smooth dieomorphism of closed manifold. This dieomorphism has one-dimensional (in topological sense) basic set with stable invariant manifold of arbitrary nonzero dimension and the unstable invariant manifold of arbitrary dimension not less than two. The basic set has a saddle type, i.e. is neither attractor nor repeller. ...
Added: May 25, 2018
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., 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., 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
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
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., Kurenkov E., Труды Средневолжского математического общества 2016 Т. 18 № 2 С. 16-24
This paper deals with the study of the dynamics in the neighborhood of one-dimensional basic sets of Ck , k ≥ 1 , endomorphism satisfying axiom of A and given on surfaces. It is established that if one-dimensional basic set of endomorphism f has the type (1; 1) and is a onedimensional submanifold without boundary, ...
Added: June 7, 2016
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
Pochinka O., Левченко Ю. А., Grines V., Нелинейная динамика 2014 Т. 10 № 1 С. 17-33
Consider the class of diffeomorphisms of three-dimensional manifolds and satisfying aksiomA by Smale on the assumption that the non-wandering set of each diffeomorphism consists of surface two-dimensional basic sets. We find interrelations between the dynamics of such a diffeomorphism and the topology of the ambient manifold. Also found that each such diffeomorphism is Ω-conjugate to ...
Added: August 16, 2014