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## Kantorovich–Rubinstein–Wasserstein distance between overlapping attractor and repeller

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 difference between the attractor and the repeller, in dependence on the dissipation level.

Publication based on the results of:

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

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

Kazakov A., On a scenario of onset of strongly dissipative mixed dynamics / Cornell University. Series math "arxiv.org". 2017. No. 1801.00150.

n this paper we present the scenario of the occurrence of strongly dissipative mixed dynamics in two-dimensional reversible diffeomorphisms, using as an example the system describing a motion of two point vortices under the influence of wave perturbation and shear flow. For mixed dynamics of this type the chaotic attractor intersects with the chaotic repeller, ...

Added: January 15, 2018

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., 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

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

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

Garashchuk I., Sinelshchikov D., Chaos 2021 Vol. 31 No. 2 Article 023130

We study the process of the destruction of synchronous oscillations in a model of two interacting microbubble contrast agents exposed to an external ultrasound field. Completely synchronous oscillations in this model are possible in the case of identical bubbles when the governing system of equations possess a symmetry leading to the existence of a synchronization ...

Added: May 27, 2021

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

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

Barinova M., Grines V., Pochinka O. et al., Chaos 2021 Vol. 31 No. 6 Article 063112

This paper is a continuation of research in the direction of energy function (a smooth Lyapunov function whose set of critical points coincides with the chain recurrent set of a system) construction for discrete dynamical systems. The authors established the existence of an energy function for any AA-diffeomorphism of a three-dimensional closed orientable manifold whose non-wandering ...

Added: June 10, 2021

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

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

Grines V., Kurenkov E., On hyperbolic attractors and repellers of endomorphisms / Cornell University. Series math "arxiv.org". 2017.

Added: November 13, 2017

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

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

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

Kroshnin A., Sobolevski A., Fréchet Barycenters and a Law of Large Numbers for Measures on the Real Line / Cornell University. Series arXiv "math". 2015. No. 1512.08421.

Endow the space P(R) of probability measures on R with a transportation cost J(mu, nu) generated by a translation-invariant convex cost function. For a probability distribution on P(R) we formulate a notion of average with respect to this transportation cost, called here the Fréchet barycenter, prove a version of the law of large numbers for ...

Added: December 31, 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

Zaev D., Decomposition of the Kantorovich problem and Wasserstein distances on simplexes / Cornell University. Series math "arxiv.org". 2015.

We consider L^p-Wasserstein distances on a subset of probability measures. If the subset of interest appears to be a simplex, these distances are determined by their values on extreme points of the simplex. We show that this fact is a corollary of the following decomposition result: an optimal transport plan can be represented as a mixture ...

Added: May 25, 2015

Malkin M., Gonchenko S., Li M. -., Dynamical Systems 2018 Vol. 33 No. 3 P. 441-463

Consider (m + 1)-dimensional, m ≥ 1, diffeomorphisms that have a saddle fixed point O with m-dimensional stable manifold Ws(O), one-dimensional unstable manifold Wu(O), and with the saddle value σ different from 1. We assume that Ws(O) and Wu(O) are tangent at the points of some homoclinic orbit and we let the order of tangency be arbitrary. In the case when σ < 1, we ...

Added: March 12, 2020

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

Zaev D., L^p-Wasserstein distances on state and quasi-state spaces of C*-algebras / Cornell University. Series math "arxiv.org". 2015.

We construct an analogue of the classical p-Wasserstein distance for the state space of a C*-algebra. Given an abstract Lipschitz gauge on a C*-algebra A in the sense of Rieffel, one can define the classical p-Wasserstein distance on the state space of each commutative C*-subalgebra of A. We consider a projective limit of these metric spaces, which appears to be the ...

Added: May 25, 2015

Karatetskaia E., Шыхмамедов А. И., Kazakov A., Chaos 2021 Vol. 31 Article 011102

A Shilnikov homoclinic attractor of a three-dimensional diffeomorphism contains a saddle-focus fixed point with a two-dimensional unstable invariant manifold and homoclinic orbits to this saddle-focus. The orientation-reversing property of the diffeomorphism implies a symmetry between two branches of the one-dimensional stable manifold. This symmetry leads to a significant difference between Shilnikov attractors in the orientation-reversing ...

Added: September 8, 2021