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Sparkling saddle loops of vector fields on surfaces
Cornell University
,
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 non-equivalent germs of bifurcation diagrams that appear in locally generic one-parameter families.
Skripchenko A., Troubetzkoy S., 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
Volk D., 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
Blank M., 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
Blokh A., Oversteegen L., Ptacek R. et al., / 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
Shilin I., Доклады Академии наук 2016 Т. 469 № 3 С. 287-290
В работе показано, что неустойчивость аттракторов Милнора по Ляпунову является локально топологически типичным динамическим явлением, которое наблюдается в присутствии устойчивых гомоклинических касаний для 2-сжимающих периодических седел. ...
Added: October 14, 2018
Yu. Ilyashenko, Kudryashov Y., I. Schurov, Inventiones Mathematicae 2018 Vol. 213 No. 2 P. 461-506
We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below “families” are “families of vector fields in the two-sphere”. This result disproves an Arnold’s conjecture of 1985. Then we construct an ...
Added: February 6, 2018
Blokh A., Oversteegen L., Ptacek R. et al., / 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
Stanislav Minkov, Shilin I., 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
Ekaterina Amerik, Misha Verbitsky, / 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
Stanislav Minkov, Ivan Shilin, / 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
Protasov V., 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
Гончарук Н. Б., Ilyashenko Y., Труды Математического института им. В.А. Стеклова РАН 2020 Т. 310 С. 86-106
Обсуждаются различные определения эквивалентности для бифуркаций векторных полей на сфере, и приводится большое количество примеров (как известных, так и новых), которые иллюстрируют достоинства и недостатки разных определений. Кроме классических определений сильной и слабой эквивалентности, рассматриваются новые понятия Sing-эквивалентности и умеренной эквивалентности. Эти определения представляются более подходящими и соответствующими интуитивному понятию эквивалентных бифуркаций. Они были введены и использованы для описания структурной неустойчивости ...
Added: May 27, 2021
Stankevich N., Shchegoleva N. A., Sataev I. R. et al., 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
Romaskevich O. L., 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
Filimonov D., Клепцын В. А., Nonlinearity 2014 Vol. 27 No. 6 P. 1205-1223
We study possible one-end finitely presented subgroups of <img />, acting without finite orbits. Our main result, theorem 1, establishes that any such action possesses the so-called property (<img />), 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 ...
Added: October 23, 2014
Kleptsyn V., Alvarez S., Malicet D. et al., / Cornell University. Series math "arxiv.org". 2015.
Added: June 22, 2016
Mikheev A. V., 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
Demina M.V., Kudryashov N. A., 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
Ilyashenko Y., Chaos 2021 Vol. 31 Article 013103
We study the geometry of the bifurcation diagrams of the families of vector fields in the plane. Countable number of pairwise non-equivalent germs of bifurcation diagrams in the two-parameter families is constructed. Previously, this effect was discovered for three parameters only. Our example is related to so-called saddle node (SN)–SN families: unfoldings of vector fields with one ...
Added: May 27, 2021
Blokh A., Oversteegen L., Ptacek R. et al., / 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
Burov A. A., Якушев И. А., Прикладная математика и механика 2014 Т. 78 № 5 С. 645-655
Рассматривается скольжение тяжелой бусинки, нанизанной на тонкий круговой обруч, вращающийся с постоянной угловой скоростью вокруг вертикальной оси, расположенной в его плоскости и, в общем случае, не проходящей через его вертикальный диаметр. Предполагается, что между бусинкой и обручем действует сила сухого трения. Находятся множества неизолированных положений относительного равновесия бусинки на обруче, исследуется их зависимость от параметров ...
Added: November 27, 2014
Blokh A., Oversteegen L., Ptacek R. et al., / Cornell University. Series math "arxiv.org". 2014.
We study the closure of the cubic Principal Hyperbolic Domain and its
intersection $\mathcal{P}_\lambda$ with the slice $\mathcal{F}_\lambda$ of the
space of all cubic polynomials with fixed point $0$ defined by the multiplier
$\lambda$ at $0$. We show that any bounded domain $\mathcal{W}$ of
$\mathcal{F}_\lambda\setminus\mathcal{P}_\lambda$ consists of $J$-stable
polynomials $f$ with connected Julia sets $J(f)$ and is either of \emph{Siegel
capture} ...
Added: February 11, 2015