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## Estimates of the Lyapunov Dimension of Attractors for Generalized Rössler Systems

Vestnik St. Petersburg University: Mathematics. 2014. Vol. 47. No. 4. P. 154-158.

Leonov G. A., Alexeeva T.A.

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 attractors of these systems, exact expressions are given. In the limit case, the coincidence of the topological, Hausdorff, fractal, and Lyapunov dimensions of attractors is proved. It is shown that, for standard values of Rцssler parameters, the values given by expressions for local Lyapunov dimensions at zero coincide with those obtained in numerical experiments.

Leonov G. A., Alexeeva Tatyana A., Doklady Mathematics 2015 Vol. 91 No. 1 P. 5-8

Lyapunov functions can be effectively used for estimate the dimensions of the attractors of Rossler systems. In this paper, estimates of the Lyapunov dimensions of the attractors of generalized Rossler systems are obtained. For the local Lyapunov dimensions of the attractors of these systems exact expressions are given. It is shown that, for standard values ...

Added: February 23, 2016

Chigarev V., Kazakov A., Пиковский А., Chaos 2020 Vol. 30 No. 7 Article 073114

We consider several examples of dynamical systems demonstrating overlapping attractor and repeller. These systems are constructed via introducing controllable dissipation to prototypic models with chaotic dynamics (Anosov cat map, Chirikov standard map, and incompressible three-dimensional flow of the ABC-type on a three-torus) and ergodic non-chaotic behavior (skew-shift map). We employ the Kantorovich–Rubinstein–Wasserstein distance to characterize the ...

Added: October 31, 2020

Kuznetsov N. V., Mokaev T. N., Alexeeva T. A., Ekaterinburg : Институт математики и механики УрО РАН им. Н.Н. Красовского, 2019

Added: October 30, 2019

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

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

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

Nozdrinova E., Pochinka O., Tsaplina E., Characteristic space of orbits of Morse-Smale diffeomorphisms on surfaces / 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

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

Leonov G., Tatyana A. Alexeeva, Kuznetsov N., Entropy 2015 Vol. 17 No. 7 P. 5101-5116

In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the
Lyapunov dimension under diffeomorphism is demonstrated in the general case. This fact is used to obtain ...

Added: September 20, 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

Nozdrinova E., Pochinka O., Characteristic space of orbits of Morse-Smale diffeomorphisms on surfaces / 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

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

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

СПб. : Издательский дом Федоровой Г.В., 2015

Proceedings of the III International Conference in memory of V.I. Zubov "Stability and Control Processes (SCP 2015)". ...

Added: October 14, 2015

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

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

Nozdrinova E., Pochinka O., Characteristic space of orbits of Morse-Smale diffeomorphisms on surfaces / 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

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

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

Nina. I. Zhukova, Galaev A., Attractors of Cartan foliation / 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

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

Protasov V., Cicone A., Guglielmi N., Nonlinear Analysis: Hybrid Systems 2018 Vol. 29 P. 165-186

We consider linear dynamical systems with a structure of a multigraph. The vertices are associated to linear spaces and the edges correspond to linear maps between those spaces. We analyse the asymptotic growth of trajectories (associated to paths along the multigraph), the stability and the stabilizability problems. This generalizes the classical linear switching systems and ...

Added: September 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