?
О топологической классификации диффеоморфизмов на 3-многообразиях с поверхностными двумерными аттракторами и репеллерами
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 a modeling diffeomorphism of the manifold, which is a locally trivial bundle over the circle with torus as a leaf. Under some restrictions on the asymptotic behavior of two-dimensional invariant manifolds of points of basis sets obtained the topological classification of structurally stable diffeomorphisms of the class.
Grines V., Kurenkov E., Журнал Средневолжского математического общества 2018 Т. 20 № 2 С. 159-174
The present paper is devoted to the topological classification of one-dimensional basiс sets of diffeomorphisms satisfying ещ the Smale's axiom A and given on orientable surfaces of negative Euler characteristic equipped with a metric of constant negative curvature. Using Lobachevsky's methods of geometry, each perfect one-dimensional attractor of A-diffeomorphism is uniquely associated with a geodesic ...
Added: June 5, 2018
Grines V., Kurenkov E., / Cornell University. Series math "arxiv.org". 2017.
Added: November 13, 2017
Nozdrinova E., Pochinka O., Tsaplina E., / Cornell University. Series arXiv "math". 2022.
The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other trajectories of the system. If there is a way to choose this pair so that the space orbits in ...
Added: December 30, 2022
Nozdrinova E., Pochinka O., / Cornell University. Серия arXiv "math". 2022.
The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other trajectories of the system. If there is a way to choose this pair so that the space orbits in ...
Added: December 30, 2022
Kurenkov E., Журнал Средневолжского математического общества 2017 Т. 19 № 1 С. 60-66
In the article we construct an axiom $A$ endomorphism $f$ of 2-torus with nonwondering set that contains one-dimensional contracting repeller satisfying following properties:
1) $f(\Lambda)= \Lambda$, $f^{-1}(\Lambda)= \Lambda$;
2) $\Lambda$ is locally homeomorphic to the product of the Cantor set and the interval;
3) $T^2\setminus\Lambda$ consist of a countable family of disjoint open disks.
The key idea of construction ...
Added: October 16, 2017
Pochinka O., Grines V., Zhuzhoma E. V., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2014 Vol. 24 No. 8 P.
In the survey, we consider bifurcations of attracting (or repelling) invariant sets of some classical dynamical systems with a discrete time. ...
Added: September 11, 2014
Fedotov A., Mathematical notes 2013 Vol. 94 No. 5 P. 681-691
Sufficient conditions for a generalized solenoid to be realized as a hyperbolic attractor of shere diffeomorphisms are obtained. The main theorem and its corollaries allow one to construct examples of attractors with various properties. ...
Added: March 25, 2014
On construction of axiom A 3-diffeomorphism with one-dimensional surface attractor-repeller dynamics
Grines V., Pochinka O., Barinova M., Динамические системы 2018 Vol. 8 No. 4 P. 299-305
In this paper we construct an omega-stable diffeomorphism $f$ on closed 3-manifold $M$ so that non-wandering set of $f$ consists of exactly one-dimensional attractor and repeller. All known examples were constructed by Ch. Bonatti, N. Guilman, Sh. Yi. We suggest a new model of the construction of such diffeomorphism. ...
Added: November 21, 2018
Grines V., Kurenkov E., Нелинейная динамика 2017 Т. 13 № 4 С. 557-571
Хорошо известно, что топологическая классификация динамических систем с гиперболической динамикой существенным образом определяется динамикой на неблуждающем множестве. Ф.~Пшетыцким было дано обобщение аксиомы $A$, ранее введенной С.~Смейлом для диффеоморфизмов, на случай гладких эндоморфизмов, а также доказана теорема о спектральном разложении, утверждающая, что неблуждающее множество $A$-эндоморфизма представляется в виде объединения базисных множеств.
В настоящей работе приводится критерий того, ...
Added: November 16, 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
Н.И. Жукова, Mathematical Notes (Rusian Federation) 2013 Т. 93 № 6 С. 994-996
In this paper a unified method for studying foliations with transversal psrsbolic geometry of rank one is presented.
Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations
are developed. ...
Added: September 28, 2014
Volk D., Kleptsyn V., Nonlinearity 2014 Vol. 27 No. 7 P. 1595-1601
In this paper we consider a class of skew products over transitive subshifts of finite type with interval fibres. For a natural class of 1-parameter families we prove that for all but countably many parameter values the nonwandering set (in particular, the union of all attractors and repellers) has zero measure. As a consequence, the ...
Added: December 22, 2015
Gurevich E., Макаров А. А., Труды Средневолжского математического общества 2020 Т. 22 № 3 С. 261-267
We consider a class $H(\mathbb{R}^n)$ of orientation preserving homeomorphisms of Euclidean space $\mathbb{R}^n$ such that for any homeomorphism $h\in H(\mathbb{R}^n)$ and for any point $x\in \mathbb{R}^n$ a condition $\lim \limits_{n\to +\infty}h^n(x)\to O$ holds, were $O$ is the origin. It is provided that for any $n\geq 1$ an arbitrary homeomorphism $h\in H(\mathbb{R}^n)$ is topologically conjugated with ...
Added: September 14, 2020
Kazakov A., Bakhanova Y., Козлов А. Д. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2019 Т. 27 № 5 С. 7-52
The main goal of the present paper is an explanation of topical issues of the theory of spiral chaos of three-dimensional flows, i.e. the theory of strange attractors associated with the existence of homoclinic loops to the equilibrium of saddle-focus type, based on the combination of its two fundamental principles, Shilnikov’s theory and universal scenarios ...
Added: October 18, 2019
Nozdrinova E., Pochinka O., / Cornell University. Series arXiv "math". 2022.
The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other trajectories of the system. If there is a way to choose this pair so that the space orbits in ...
Added: November 22, 2022
Zhukova N., Математический сборник 2012 Т. 203 № 3 С. 79-106
Доказано, что любое полное конформное слоение (M,F) коразмерности q> 2 является либо римановым, либо (Conf(S^q),S^q)-слоением. Если (M,F) не является римановым слоением, то оно имеет глобальный аттрактор, представляющий собой либо нетривиальное минимальное множество, либо один замкнутый слой или объединение двух замкнутых слоев. При этом компактность многообразия M не предполагается. В частности, каждое собственное полное конформное не риманово ...
Added: September 28, 2014
Kazakov A., Gonchenko S. V., Turaev D. V. et al., Physica D: Nonlinear Phenomena 2017 Vol. 350 P. 45-57
A one-parameter family of time-reversible systems on three-dimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits ...
Added: October 13, 2017
Grines V., Zhuzhoma E. V., Kurenkov E., Математический сборник 2021 Т. 212 № 5 С. 102-132
It is proved that in each homotopy class of continuous mappings of the two-dimensional torus that induce a hyperbolic action in the fundamental group and do not contain expanding mappings, there exists an A-endomorphism f whose non-wandering set consists of an attracting hyperbolic sink and a nontrivial one-dimensional contracting repeller, which is a one-dimensional orientable ...
Added: April 29, 2021
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
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
Chepyzhov V. V., Bekmaganbetov K., Chechkin G., Applicable Analysis 2019 Vol. 98 No. 1-2 P. 256-271
We consider reaction–diffusion systems with random rapidly oscillating coefficient. We do not assume any Lipschitz condition for the nonlinear function in the system, so, the uniqueness theorem for the corresponding initial-value problem may not hold for the considered reaction–diffusion system. Under the assumption that the random function is ergodic and statistically homogeneous in space variables we prove that the trajectory attractors ...
Added: November 11, 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
Romanov A., Известия РАН. Серия математическая 2006 Т. 70 № 5 С. 163-178
<img /> Для эволюционных уравнений параболического типа c гильбертовым фазовым пространством E рассмотрена проблема эффективной (с липшицевой оценкой) конечной параметризации множеств K в E функционалами из E*, или, в иных терминах, проблема линейного липшицева вложения K в конечномерное евклидово пространство. Если K - глобальный аттрактор уравнения, то такого рода параметризация оказывается равносильной конечномерности динамики на K. Получен ряд признаков параметризации (в различных ...
Added: December 6, 2012
Митрякова Т. М., Pochinka O., Журнал Средневолжского математического общества 2014 Т. 16 № 2 С. 76-79
In present paper we consider a class of 3-manifolds' diffeomorphisms lying on the border of a set of gradient-like systems and different from the last not more than one tangencies' orbits of two-dimensional separatrices. It is proved that for studying diffeomorphisms necessary and sufficient condition for topological conjugacy of two diffeomorphisms from this class is ...
Added: August 9, 2014