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## Characteristic space of orbits of Morse-Smale diffeomorphisms on surfaces

Cornell University
,
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 its complement (the characteristic space of orbits) is connected, this creates prerequisites for finding complete topological invariants of the dynamical system. It is known that such a pair always exists for arbitrary Morse-Smale diffeomorphisms given on any manifolds of dimension n⩾3. Whereas for n=2 the existence of a connected characteristic space has been proved only for orientation-preserving gradient-like (without heteroclinic points) diffeomorphisms defined on an orientable surface. In the present work, it is constructively shown that the violation of at least one of the above conditions (absence of heteroclinic points, orientability of a surface, orientability of a diffeomorphism) leads to the existence of Morse-Smale diffeomorphisms on surfaces that do not have a connected characteristic space of orbits.

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

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., 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

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

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

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

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

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

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

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

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

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

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

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

Zhukova N., Applied Mathematics and Nonlinear Sciences 2020 Vol. 5 No. 2 P. 279–292

The purpose of this article is to review the author's results on the existence and
structure of minimal sets and attractors of conformal foliations of codimension $q,$ ${q\geq 3.}$
Results on strong transversal equivalence of conformal foliations are also presented.
Connections with works of other authors are indicated. Examples of conformal foliations with
exceptional, exotic and regular minimal sets ...

Added: December 30, 2019

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

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

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

Morozov A., Pochinka O., Журнал Средневолжского математического общества 2020 Т. 22 № 1 С. 71–80

In the present paper, we consider the class of orientation-preserving Morse-Smale diffeomorphisms f defined on an orientable surface M2. The work of A. A. Beznezhennykh and V. Z. Grines showed that such diffeomorphisms have a finite number of heteroclinic orbits. In addition, the classification problem of the diffeomorphisms under consideration is reduced to the problem ...

Added: June 22, 2020

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., 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., Pochinka O., Kruglov E. V., Russian Journal of Nonlinear Dynamics 2020 Vol. 16 No. 4 P. 595–606

This paper is devoted to the topological classification of structurally stable diffeomorphisms of the two-dimensional torus whose non-wandering set consists of an orientable one-dimensional attractor and finitely many of isolated source and saddle periodic points, under the assumption that the closure of the union of the stable manifolds of isolated periodic points consists of simple ...

Added: December 15, 2020

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

Added: November 13, 2017

Grines V., Pochinka O., Chilina E., / Cornell University. Series math "arxiv.org". 2023.

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: December 20, 2023