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Построение энергетической функции для трёхмерных каскадов с двумерным растягивающимся аттрактором
Труды Московского математического общества. 2015. Т. 76. № 2. С. 271-286.
Keywords: энергетическая функцияenergy functionрастягивающийся аттракторDA-diffeomorphismexpanding attractorDA-диффеомрфизм
Publication based on the results of:
Barinova M., Grines V., Pochinka O., Journal of difference equations and applications 2023 Vol. 29 No. 9-12 P. 1275-1286
In this paper, we consider a class of A-diffeomorphisms given on a 3-manifold, assuming that all the basic sets of the diffeomorphisms are two dimensional. It is known that such basic sets are either attractors or repellers and they are two types only, surface or expanding (contracting). One of the results of the paper is ...
Added: July 13, 2022
Grines V., Pochinka O., Van Strien S., Moscow Mathematical Journal 2016 Vol. 16 No. 4 P. 727-749
This paper is a step towards the complete topological classication of omega-stable diffeomorphisms on an orientable closed surface, aiming to give necessary and suffcient conditions for two such dieomorphisms to be topologically conjugate without assuming that the dieomorphisms are necessarily close to each other. In this paper we will establish such a classication within a ...
Added: June 6, 2016
Босова А. А., Kruglov V., Pochinka O., Таврический вестник информатики и математики 2017 № 4(37) С. 51-58
In this paper the class of simplest not rough Ω-stable flows on a sphere is considered. We call simplest not rough Ω-stable flow an Ω-stable flow with least number of fixed points, a single separatrix connecting saddle points and without limit cycles. For such flows we design the Morse energy function. ...
Added: March 9, 2018
Pochinka O., Zinina S., Regular and Chaotic Dynamics 2021 Vol. 26 No. 4 P. 350-369
In this paper, we consider regular topological flows on closed n-manifolds. Such
flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number
of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale
flows, which are closely related to the topology of the supporting manifold. This ...
Added: July 16, 2021
Kurenkov E., Gurevich E., Труды Средневолжского математического общества 2015 Т. 17 № 2 С. 15-26
We introduce the definition of consistent equivalence of energy Morse-Bott functions for Morse-Smale flows on surfaces and state that consistent equivalence of that functions is necessary and sufficient condition for such flows. ...
Added: October 12, 2015
Grines V., Pochinka O., Современная математика. Фундаментальные направления 2017 Т. 63 № 2 С. 191-222
This review focuses on the presentation of results related to the energy function of discrete dynamical systems, as well as with the technique of constructing such functions for certain classes of Ω-stable and structurally stable diffeomorphisms on manifolds of dimension 2 and 3. ...
Added: September 5, 2017
Gurevich E., Сахаров А. Н., Трегубова Е. В., Журнал Средневолжского математического общества 2013 Т. 15 № 4 С. 91-100
Работа является продолжением работы [#!gurevich-GrPoSaRu!#] и посвящена топологической классификации градиентно-подобных потоков, заданных на гладком замкнутом ориентируемом многообразии M n размерности n>3, с использованием энергетической функции. Рассмотрен класс G(M n)градиентно-подобных потоков без гетероклинических пересечений, все седловые состояния равновесия которых имеют индекс Морса 1 или (n-1). Показано, что необходимое и достаточное условие топологической эквивалентности потоков из класса G(M n) состоит в ...
Added: October 14, 2014
Grines V., Носкова М.К., Pochinka O., Динамические системы 2015 Т. 5 № 1-2 С. 31-37
В настоящей работе устанавливается существование энергетической функции для А-диффеоморфизмов, заданных на замкнутых ориентируемых двумерных многообразиях, не имеющих циклов и имеющих нетривиальные базисные множества только размерности один. Известно, что каждое базисное множество такого эндоморфизма является либо аттрактором, репеллером и локально устроено как декартово произведение канторова множества на интервал. Несмотря на сложную топологию неблуждающего множества, построенная энергетическая ...
Added: January 26, 2016
Barinova M., Grines V., Pochinka O., Труды Математического института им. В.А. Стеклова РАН 2023 Т. 321 С. 45-61
C. Conley’s fundamental theorem of the theory of dynamical systems states that every dynamical system, even a nonsmooth one (i.e., a continuous flow or a discrete dynamical system generated by a homeomorphism), admits a continuous Lyapunov function. A Lyapunov function is strictly decreasing along the trajectories of the dynamical system outside the chain recurrent set ...
Added: September 7, 2023
Grines V., Kruglov E. V., Pochinka O., Proceedings of the Steklov Institute of Mathematics 2020 Vol. 308 P. 141-154
The Smale surgery on the three-dimensional torus allows one to obtain a so-called DA diffeomorphism from the Anosov automorphism. The nonwandering set of a DA diffeomorphism consists of a single two-dimensional expanding attractor and a finite number of source periodic orbits. As shown by V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, the ...
Added: October 14, 2019
Barinova M., Lobachevskii Journal of Mathematics 2021 Vol. 42 No. 14 P. 3317-3323
If the chain recurrent set of a diffeomorphism f given on a closed n-manifold M^n is
hyperbolic (equivalently, f is an Ω-stable) then it coincides with the closure of the periodic points set
Perf and its chain recurrent components coincide with the basic sets. Due to C. Conley for such a
diffeomorphism there is a Lyapunov function which ...
Added: March 17, 2022
Vassiliev V.A, Bulletin of the London Mathematical Society 2012 Vol. 44 No. 4 P. 637-641
The Arnold theorem (generalizing a consideration by Jacobi) states that on a generic Riemannian surface, which is sufficiently close to a sphere, the kth caustic of a generic point has at least four semi-cubical vertices. We prove this fact by the methods of the Morse theory; in particular, we replace the previous analytical condition of ...
Added: February 4, 2013
Grines V., Noskova M. K., Pochinka O., Transactions of the Moscow Mathematical Society 2015 Vol. 76 No. 2 P. 237-249
In this paper we establish the existence of an energy function for structurally stable diffeomorphisms of closed three-dimensional manifolds whose nonwandering set contains a two-dimensional expanding attractor. ...
Added: May 12, 2016
Pochinka O., Носкова М.К., Grines V., Журнал Средневолжского математического общества 2014 Т. 16 № 2 С. 20-26
В настоящей работе устанавливается существование энергетической функции для структурно устойчивых диффеоморфизмов замкнутых трехмерных многообразий, неблуждающее множество которых содержит двумерный растягивающийся аттрактор ...
Added: October 15, 2015
Gurevich E., Grines V., Pochinka O., Mathematical notes 2014 Vol. 96 No. 6 P. 921-927
For gradient-likeflows without heteroclinic intersections of the stable and unstable manifolds of saddle periodic points all of whose saddle equilibrium states have Morse index 1 or n−1, the notion of consistent equivalence of energy functions is introduced. It is shown that the consistent equivalence of energy functions is necessary and sufficient for topological equivalence. ...
Added: December 8, 2014
Barinova M., Шустова Е. К., Журнал Средневолжского математического общества 2023 Т. 25 № 2 С. 11-21
This paper is devoted to the construction of an energy function, i.e. a smooth Lyapunov function, whose set of critical points coincides with the chain-recurrent set of a dynamical system — for a cascade that is a direct product of two systems. One of the multipliers is a structurally stable diffeomorphism given on a two-dimensional ...
Added: August 2, 2023
Kolobianina A., Kruglov V., Журнал Средневолжского математического общества 2019 Т. 21 № 4 С. 460-468
The paper is devoted to the study of the class of Ω-stable flows without limit cycles on
surfaces, i.e. flows on surfaces with non-wandering set consisting of a finite number of hyperbolic
fixed points. This class is a generalization of the class of gradient-like flows, differing by forbiddance
of saddle points connected by separatrices. The results of the ...
Added: October 22, 2019
Barinova M., Гогулина Е. Ю., Pochinka O., Огарёв-Online 2020 № 13 С. 1-10
Smale diagram of Ω-stable diffeomorfism is a graph, which vertices correspond to basic sets and oriented edges sequentially connect vertices of maximal chain. Smale formulated description of acceptable diagrams, which are realized by a diffeomorfism. Present paper proves that an acyclic graph is implemented by an Ω-stable diffeomorfism on surface with nontrivial basic sets. ...
Added: November 23, 2020
Barinova M., Шустова Е. К., Журнал Средневолжского математического общества 2022 Т. 24 № 1 С. 21-30
A natural way for creating new dynamical systems is to consider direct products of already known systems. The paper studies some dynamical properties of direct products of homeomorphisms and diffeomorphisms. In particular, authors prove that a chain-recurrent set of the direct product of homeomorphisms is a direct product of the chain-recurrent sets. Another result established ...
Added: April 1, 2022
Barinova M., Pochinka O., Yakovlev E., Discrete and Continuous Dynamical Systems 2024 Vol. 44 No. 1 P. 1-17
This paper belongs to a series of papers devoted to the study of the structure of the non-wandering set of an A-diffeomorphism. We study such set NW(f) for an Ω-stable diffeomorphism f, given on a closed connected 3-manifold M^3. Namely, we prove that if all basic sets in NW(f) are trivial except attractors, then every non-trivial attractor is ...
Added: November 13, 2023
Pochinka O., Гринес В. З., Успехи математических наук 2013 Т. 68 № 1 (409) С. 129-188
Исследования связаны с каскадами Морса-Смейла на ориентируемых 3-многообразиях и включают в себя их полную топологическую классификацию, установление взаимосвязи их динамики с топологией объемлющего многообразия, критерий включения в топологический поток, а также необходимые и достаточные условия существования для таких каскадов энергетической функции. ...
Added: March 25, 2014
Kruglov V., Pochinka O., Журнал СВМО 2014 Т. 16 № 3 С. 57-61
В настоящей работе рассматриваются динамические системы с дискретным временем, порожденные итерациями градиентно-подобного диффеоморфизма поверхности, неблуждающее множество которого состоит из неподвижных точек положительного типа ориентации. Доказывается, что класс топологической сопряженности такой системы полностью определяется классом эквивалентности ее энергетической функции Морса ...
Added: January 27, 2015
Kolobianina A., Kruglov V., Журнал Средневолжского математического общества 2020 Т. 22 № 4 С. 434-441
In this paper, we consider the class of Ω-stable flows on surfaces, i.e. flows on surfaces with the non-wandering set consisting of a finite number of hyperbolic fixed points and a finite number of hyperbolic limit cycles. The class of Ω-stable flows is a generalization of the class of Morse-Smale flows, admitting the presence of ...
Added: November 27, 2020
191574970, Functional Analysis and Its Applications 2006 Vol. 40 No. 2 P. 81-90
It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...
Added: September 23, 2016