Energy function of gradient-like flows and the problem of topological classification

Гринес Вячеслав Зигмундович; Гуревич Елена Яковлевна; Починка Ольга Витальевна

Article

Энергетическая функция градиентно-подобных потоков и проблема топологической классификации

Математические заметки. 2014. Т. 96. № 6. С. 856-863.

Гринес В. З., Гуревич Е. Я., Починка О. В.

For gradient-like flows without heteroclinic intersections of the stable and unstable manifolds of saddle periodic points all

of whose saddle equilibrium states have Morse index 1 or n − 1, the notion of consistent equivalence of energy functions

is introduced. It is shown that the consistent equivalence of energy functions is necessary and sufficient for topological

equivalence.

Research target: Mathematics

Language: Russian

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Keywords: топологическая классификация поток Морса-Смейла структурно-устойчивая динамическая система

О топологической классификации градиентно-подобных потоков без гетероклинических пересечений посредством энергетической функции

Гуревич Е. Я., Сахаров А. Н., Трегубова Е. В. Журнал Средневолжского математического общества. 2013. Т. 15. № 4. С. 91-100.

Added: Oct 14, 2014

О классификации диффеоморфизмов Морса–Смейла с одномерным множеством неустойчивых сепаратрис

Гринес В. З., Гуревич Е. Я., Медведев В. С. В кн.: Дифференциальные уравнения и динамические системы. Сборник статей. Т. 270. М.: МАИК, 2010.

Added: Sep 29, 2014

An open set of structurally unstable families of vector fields in the two-sphere

Ilyashenko Y., Kudryashov Y., Schurov I. arxiv.org. math. Cornell University, 2015. No. 1506.06797.

This is the first part of a two parts paper dedicated to global bifurcations in the plane. In this part we construct an open set of three parameter families whose topological classification has a numerical invariant that may take an arbitrary positive value. In the second part we construct an open set of six parameter families whose topological classification has a functional invariant. Any germ of a monotonically increasing function may be realized as this invariant. Here "families" are "families of vector fields in the two-sphere".

Added: Jun 24, 2015

Граф Пейкшото диффеоморфизмов Морса–Смейла на многообразиях размерности большей трех

Гринес В. З., Гуревич Е. Я., Медведев В. С. В кн.: Дифференциальные уравнения и динамические системы. Сборник статей. Т. 261. М.: МАИК, 2008.

Added: Sep 29, 2014

Деловой климат в оптовой торговле в I квартале 2012 года

Лола И. С., Лукашина Ж. Е., Пахомова О. П. М.: НИУ ВШЭ, 2012.

Added: May 21, 2012

Model for organizing cargo transportation with an initial station of departure and a final station of cargo distribution

Khachatryan N., Akopov A. S. Business Informatics. 2017. No. 1(39). P. 25-35.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Added: Apr 18, 2017

Nullstellensatz over quasi-fields

Trushin D. Russian Mathematical Surveys. 2010. Vol. 65. No. 1. P. 186-187.

Added: Feb 20, 2018

Laminations from the Main Cubioid

Timorin V., Blokh A., Oversteegen L. et al. arxiv.org. math. Cornell University, 2013. No. 1305.5788.

According to a recent paper \cite{bopt13}, polynomials from the closure $\ol{\phd}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\cu$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\cu^c$ of laminations which can be associated to polynomials from $\cu$.

Added: Oct 6, 2013

Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles

Bufetov A. I., Mkrtchyan S., Scherbina M. et al. arxiv.org. math. Cornell University, 2013. No. 1301.0342.

Added: Feb 21, 2013

Математическая статистика

Энатская Н. Ю., Хакимуллин Е. Р. М.: МИЭМ, 2011.

Added: Apr 26, 2012

Деловой климат в оптовой торговле во II квартале 2014 года и ожидания на III квартал

Лола И. С., Остапкович Г. В. Современная торговля. 2014. № 10.

Added: Jan 29, 2015

Введение в математическую статистику

Ивченко Г. И., Медведев Ю. И. М.: ЛКИ, 2010.

Added: Apr 12, 2012

Bounded limit cycles of polynomial foliations of ℂP²

Goncharuk N. B., Kudryashov Y. arxiv.org. math. Cornell University, 2015. No. 1504.03313.

In this article we prove in a new way that a generic polynomial vector field in ℂ² possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.

Added: Apr 15, 2015