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## Неустойчивые по Ляпунову аттракторы Милнора

Доклады Академии наук. 2016. Т. 469. № 3. С. 287-290.

Труды Математического института им. В.А. Стеклова РАН 2012 Т. 277 С. 91-100

В теории динамических систем существуют разные, неэквивалентные между собой определения аттракторов. Наиболее распространенными являются два определения: максимальный аттрактор и аттрактор Милнора. Максимальный аттрактор по определению устойчив по Ляпунову, однако зачастую он является в некотором смысле избыточным. Определение аттрактора Милнора более реалистично с физической точки зрения. Однако аттрактор Милнора может быть неустойчив по Ляпунову. Один из центральных вопросов ...

Added: October 14, 2018

Annales de l'Institut Fourier 2015 Vol. 65 No. 5 P. 1881-1896

We study the billiard on a square billiard table with a one-sided vertical mirror.
We associate trajectories of these billiards with double rotations and study orbit behavior and questions of complexit ...

Added: March 2, 2016

L'Enseignement Mathématique 2014

We consider 3 -periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law of reflection. ...

Added: December 25, 2014

Morrison-Kawamata cone conjecture for hyperkahler manifolds / Cornell University. Series math "arxiv.org". 2014.

Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kahler cone with finitely many orbits. This is a version of the Morrison-Kawamata cone conjecture for ...

Added: September 5, 2014

Combinatorial models for spaces of cubic polynomials / Cornell University. Series math "arxiv.org". 2014.

To construct a model for a connectedness locus of polynomials of degree $d\ge
3$ (cf with Thurston's model of the Mandelbrot set), we define \emph{linked}
geolaminations $\mathcal{L}_1$ and $\mathcal{L}_2$. An \emph{accordion} is
defined as the union of a leaf $\ell$ of $\mathcal{L}_1$ and leaves of
$\mathcal{L}_2$ crossing $\ell$. We show that any accordion behaves like a gap
of one lamination ...

Added: February 11, 2015

Groups with infinitely many ends acting analytically on the circle / Cornell University. Series math "arxiv.org". 2015.

Added: June 22, 2016

Smart criticality / Cornell University. Series math "arxiv.org". 2014.

A crucial fact established by Thurston in his 1985 preprint is that distinct \emph{minors} of quadratic laminations do not cross inside the unit disk; this led to his construction of a combinatorial model of the Mandelbrot set. Thurston's argument is based upon the fact that \emph{majors} of a quadratic lamination never enter the region between ...

Added: February 11, 2015

NY: Springer, 2012

The volume is dedicated to Stephen Smale on the occasion of his 80th birthday. Besides his startling 1960 result of the proof of the Poincaré conjecture for all dimensions greater than or equal to five, Smale’s ground breaking contributions in various fields in Mathematics have marked the second part of the 20th century and beyond. ...

Added: December 19, 2012

Qualitative Theory of Dynamical Systems 2021 Vol. 20 No. 3 Article 77

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows ϕ on S^2 for which the attractor of the Cartesian square of ϕ is smaller than the Cartesian square of the attractor of ϕ. In the example for the minimal attractors, the flow ϕ also has a global physical measure such that its ...

Added: September 16, 2021

Sparkling saddle loops of vector fields on surfaces / Cornell University. Series math "arxiv.org". 2019. No. arXiv:1903.01933.

An orientation-preserving non-contractible separatrix loop of a hyperbolic saddle of a vector field on a two-dimensional surface may be accumulated by a separatrix of the same saddle. We study the unfolding of such loops in generic one-parameter families of vector fields as a semi-local bifurcation. As a byproduct, we construct a countable family of pairwise ...

Added: November 12, 2020

Milnor attractors of circle skew products / Cornell University. Series math "arxiv.org". 2015. No. 1508.02132.

We prove that for a generic skew product with circle fiber over an Anosov diffeomorphism the Milnor attractor (also called the likely limit set) coincides with the statistical attractor, is Lyapunov stable, and either has zero Lebesgue measure or coincides with the whole phase space. As a consequence we conclude that such skew product is ...

Added: November 24, 2015

Systems and Control Letters 2016 Vol. 90 P. 54-60

We prove the existence of positive linear switching systems (continuous time), whose trajectories grow to infinity, but slower than a given increasing function. This implies that, unlike the situation with linear ODE, the maximal growth of trajectories of linear systems may be arbitrarily slow. For systems generated by a finite set of matrices, this phenomenon ...

Added: February 22, 2016

Attractors of direct products / Cornell University. Series math "arxiv.org". 2020. No. arXiv:2011.04824.

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows φ on S^2 for which the attractor of the Cartesian square of φ is smaller than the Cartesian square of the attractor of φ. In the example for the minimal attractors, the flow φ also has an SRB-measure such that its square does ...

Added: November 12, 2020

Journal of Computational and Nonlinear Dynamics 2020 Vol. 15 No. 11 P. 111001

Using an example a system of two coupled generators of quasiperiodic oscillations, we study the occurrence of chaotic dynamics with one positive, two zero and several negative Lyapunov exponents. It is shown that such dynamic arises as a result of a sequence of bifurcations of two-frequency torus doubling and involve saddle tori occurring at their ...

Added: September 4, 2020

Nonlinearity 2012 Vol. 25 No. 12 P. 3389-3408

We study ergodic properties of a family of traffic maps acting in
the space of bi-infinite sequences of real numbers. The corresponding
dynamics mimics the motion of vehicles in a simple traffic flow, which
explains the name. Using connections to topological Markov chains we obtain
nontrivial invariant measures, prove their stochastic stability, and
calculate the topological entropy. Technically these results ...

Added: November 26, 2014

Nonlinearity 2014 Vol. 27 No. 6 P. 1205-1223

We study possible one-end finitely presented subgroups of , acting without finite orbits. Our main result, theorem 1, establishes that any such action possesses the so-called property (), that allows one to make distortion-controlled expansion and is thus sufficient to conclude that the action is Lebesgue-ergodic. We also propose a path towards full characterization of ...

Added: October 23, 2014

Теория. Практика. Инновации 2017 № 9 (21)

In this paper we consider the calculation of a dynamical system described by a second-order differential equation in which a fundamental system of solutions consisting of functions of exponential type is replaced by bounded functions of the Verhulst model. The time dependence of the forces acting on the dynamical system is analyzed, and the obtained ...

Added: September 6, 2017

Discrete and Continuous Dynamical Systems 2020 Vol. 40 No. 1 P. 441-465

We present an example of a C1 Anosov diffeomorphism of a two-torus with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure. ...

Added: October 21, 2019

Regular and Chaotic Dynamics 2016 Vol. 21 No. 3 P. 351-366

Polynomial dynamical systems describing interacting particles in the plane are
studied. A method replacing integration of a polynomial multi-particle dynamical system
by finding polynomial solutions of partial differential equations is introduced. The method
enables one to integrate a wide class of polynomial multi-particle dynamical systems. The
general solutions of certain dynamical systems related to linear second-order partial differential
equations are ...

Added: October 5, 2018

Функциональный анализ и его приложения 2017 Т. 51 № 2 С. 83-86

Мы приводим пример C1-диффеоморфизма Аносова двумерного тора, у которого есть SRB-мера, носитель которой — подкова нулевой меры, а бассейн притяжения имеет полную меру Лебега. ...

Added: October 14, 2018

The parameter space of cubic laminations with a fixed critical leaf / Cornell University. Series math "arxiv.org". 2015.

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by ...

Added: February 11, 2015

Springer, 2009

Vladimir Arnold is one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This first volume of his Collected Works focuses on representations of functions, celestial mechanics, and ...

Added: February 20, 2013

Moscow Mathematical Journal 2017 Vol. 17 No. 3 P. 511-553

We prove that for every smooth compact manifold M and any r>=1, whenever there is an open domain in Diff^r (M) exhibiting a persistent homoclinic tangency related to a basic set with a sectionally dissipative periodic saddle, topologically generic diffeomorphisms in this domain have Lyapunov unstable Milnor attractors. This implies, in particular, that the instability ...

Added: October 14, 2018

Ergodic Theory and Dynamical Systems 2014 Vol. 34 No. 2 P. 693-704

For a smooth manifold of any dimension greater than one, we present an open set of smooth endomorphisms such that any of them has a transitive attractor with a non-empty interior. These maps are m-fold non-branched coverings,m≥3. The construction applies to any manifold of the form S 1×M, where S 1 is the standard circle and Mis an arbitrary manifold. ...

Added: December 28, 2015