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## There are No Structural Stable Axiom A 3-Diffeomorphisms with Dynamics “One-dimensional Surfaced Attractor-repeller”

Results in Mathematics. 2023. Vol. 78. No. 2. Article 45.

In this paper, we study the structural stability of three-dimensional diffeomorphisms with source-sink dynamics. Here the role of source and sink is played by one-dimensional hyperbolic repeller and attractor. It is well known that in the case when the repeller and the attractor are solenoids (not embedded in the surface), the diffeomorphism is not structurally stable. The author proves that in the case when the attractor and the repeller are canonically embedded in a surface, the diffeomorphism is also not structurally stable.

Publication based on the results of:

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

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

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

Bekmaganbetov K., Chechkin G., Chepyzhov V. V., Chaos, Solitons and Fractals 2020 Vol. 140 P. 110208

We consider reaction–diffusion equation in perforated domain, with rapidly oscillating coefficient in boundary conditions. We do not assume any Lipschitz condition for the nonlinear function in the equa- tion, so, the uniqueness theorem for the corresponding initial boundary value problem may not hold for the considered reaction-diffusion equation. We prove that the trajectory attractors of ...

Added: November 11, 2020

Grines V., Kurenkov E., / Cornell University. Series math "arxiv.org". 2017.

Added: November 13, 2017

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

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

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

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

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

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

Grines V., Kruglov E. V., Pochinka O., Lobachevskii Journal of Mathematics 2021 Vol. 42 No. 14 P. 3372-3381

The paper is devoted to the topological classification of structurally stable diffeomorphisms
on three-dimensional manifolds whose non-wandering set contains a 2-dimensional
expanding attractor. V.Z. Grines and E.V. Zhuzhoma obtained the topological classification of
similar cascades in the dimension greater than 3. They proposed that the embedding of frames of
saddle separatrices is tame for dimension equals 3. The tameness ...

Added: March 14, 2022

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., Tsaplina E., Moscow Mathematical Journal 2024 Vol. 24 No. 1 P. 21-39

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: March 31, 2024

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

Vyacheslav Z. Grines, Olga V. Pochinka, Ekaterina E. Chilina, Regular and Chaotic Dynamics 2024 Vol. 29 No. 1 P. 156-173

The present paper is devoted to a study of orientation-preserving homeomorphisms on three-dimensional manifolds with a non-wandering set consisting of a finite number of surface attractors and repellers. The main results of the paper relate to a class of homeomorphisms for which the restriction of the map to a connected component of the non-wandering set ...

Added: March 8, 2024

Nozdrinova E., Pochinka O., / Cornell University. Серия arXiv "math". 2022.

Added: December 30, 2022

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

Chigarev V., Kazakov A., Pikovsky A., Chaos 2021 Vol. 31 No. 8 Article 083127

We apply the concepts of relative dimensions and mutual singularities to characterize the fractal properties of overlapping attractor and repeller in chaotic dynamical systems. We consider one analytically solvable example (a generalized baker’s map); two other examples, the Anosov–Möbius and the Chirikov–Möbius maps, which possess fractal attractor and repeller on a two-dimensional torus, are explored ...

Added: October 20, 2021

Gonchenko A. S., Gonchenko S., Lobachevskii Journal of Mathematics 2021 Vol. 42 No. 14 P. 3352-3364

We give a short review on discrete homoclinic attractors. Such strange attractors contain only one saddle fixed point and, hence, entirely its unstable invariant manifold. We discuss the most important peculiarities of these attractors such as their geometric and homoclinic structures, phenomenological scenarios of their appearance, pseudohyperbolic properties etc. ...

Added: February 10, 2023

Leonov G. A., Alexeeva T.A., Vestnik St. Petersburg University: Mathematics 2014 Vol. 47 No. 4 P. 154-158

Generalization of one of the classical Rцssler systems are considered. It is shown that, to estimate the dimensions of the attractors of these systems, Lyapunov functions can be effectively used. By using these functions, estimates of the Lyapunov dimensions of the attractors of generalized Rцssler systems are obtained. For the local Lyapunov dimensions of the ...

Added: February 26, 2015

N. I. Zhukova, , in : Progress in Analysis. Proceedings of the 8th congress of the International Society for Analysis, its Applications, and Computation (ISAAC), Moscow, Russia, August 22--27, 2011. Vol. 2.: M. : RUDN, 2012. P. 238-247.

We investigated conformal foliations $(M,F)$ of
codimension $q\geq 3$ and proved a criterion for them to be
Riemannian. In particular, the application of this criterion allowed
us to proof the existence of an attractor that is a minimal set for
each non-Riemannian conformal foliation. Moreover, if foliated
manifold is compact then non-Riemannian conformal foliation $(M,F)$
is $(Conf(S^q),S^q)$-foliation with finitely many minimal ...

Added: October 14, 2014

Nina. I. Zhukova, Galaev A., / Cornell University. Series math "arxiv.org". 2017.

The paper is focused on the existence problem of attractors for foliations. Since the existence of an attractor is a transversal property of the foliation, it is natural to consider foliations admitting transversal geometric structures. As transversal structures are chosen Cartan geometries due to their universality. The existence problem of an attractor on a complete ...

Added: March 23, 2017

Zhukova N. I., Mathematical notes 2013 Vol. 93 No. 5-6 P. 928-931

In this paper a unified method for studying foliations with transversal parabolic 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: October 19, 2014