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## On a scenario of onset of strongly dissipative mixed dynamics

Cornell University
,
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, but their intersection forms a "thin" set. The main stage of this scenario is the appearance of homoclinic structures for a symmetric saddle orbit which arise after crisis of a homoclinic attractor and repeller.

Publication based on the results of:

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

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

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., / 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

Trifonov K., / Cornell University. Series arXiv "math". 2020. No. 3454820.

An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian and two non-symmetric heteroclinic orbits permuted by the involution. This is a co- dimension one structure and therefore ...

Added: December 26, 2020

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

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

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

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

Grines V., Zhuzhoma E. V., Russian Journal of Nonlinear Dynamics 2021 Vol. 17 No. 3 P. 335-345

The paper is devoted to an investigation of the genus of an orientable closed surface M2
which admits A-endomorphisms whose nonwandering set contains a one-dimensional strictly
invariant contracting repeller Λr with a uniquely defined unstable bundle and with an admissible
boundary of finite type. First, we prove that, if M2 is a torus or a sphere, then M2 ...

Added: October 19, 2021

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

Zhukova N., Differential Geometry and its Application 2021 Vol. 74 Article 101699

The existence problem for attractors of foliations with transverse linear connection
is investigated. In general foliations with transverse linear connection do not
admit attractors. A conditions that implies the existence of a global attractor which is
a minimal set, is specified. An application to transversely similar pseudo-Riemannian
foliations is obtained. The global structure of transversely similar Riemannian
foliations is described. ...

Added: October 20, 2020

О структуре резонансов 1:3 и 1:4 при обратимых возмущениях консервативных кубических отображений Эно

Samylina E., Shykhmamedov A., Kazakov A., Динамические системы 2017 Т. 7(35) № 3 С. 229-244

The paper is devoted to the study of local bifurcations of symmetry breaking which arise under reversible perturbations of conservative reversible systems. We chose a perturbed conservative cubic diffeomorphism of a plane as an example of the model on which such bifurcations were investigated. It is shown that the main symmetry breaking bifurcations here are ...

Added: April 5, 2018

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., / Cornell University. Series math "arxiv.org". 2017.

Added: November 13, 2017

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., 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., / 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

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

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