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Regular version of the site

­­­Topological methods in dynamics

Priority areas of development: mathematics
2016
Department: Laboratory of Theory and Practice of Decision-Making Support Systems (Nizhny Novgorod)
The project has been carried out as part of the HSE Program of Fundamental Studies.

Goal of research: Development of methods of the qualitative theory of dynamical systems on manifolds for obtaining  new fundamental results and their application to solving  problems in mechanics, astrophysics, neural networks, meteorology, and others.

Methodology: methods of the qualitative theory of dynamical systems and topology.

Results of research: during the work on the project results in several scientific fields were obtained. Namely,

In direction 1) Regular dynamics and its applications to the study of magnetic fields in electrically conducting flows.

- For omega-stable flows without periodic orbits on surfaces it was introduced a new topological invariant, which is a four-color graph, it is proved that the isomorphic class of the graph is a complete topological invariant, and obtained an efficient algorithm to distinguish such graphs.

- A criterion for topological equivalence of continuous Morse-Smale flows whose non-wandering set consists of exactly three equilibrium states, and also studied the topological structure of manifolds admitting such flows.

- It was obtained the topological classification of Morse-Smale diffeomorphisms without heteroclinic trajectories on the sphere of the dimension four and more through a colored graph. It was shown that the colored graph is the most effective invariant for the classification of diffeomorphisms of considered class, as there exists an optimal linear algorithm for distinguishing between graphs of such diffeomorphisms.

- For Morse-Smale diffeomorphisms without heteroclinic intersections that are defined on the sphere of the dimension four and higher it was resolved the Palis problem for the embedding to a topological flow. It was established that sufficient conditions for the embedding to a topological flow coincide with the Palis conditions (which contrasts with the three-dimensional case).

- It was studied one-parameter family of trajectories of the points of the reuleaux triangle, located on its axis of symmetry, rolling him in the square. It was find all bifurcation values of the parameter, with the passage through which the qualitative change of the trajectory. Installed components that make up the trajectory, determines their number and their equations. At all values of the parameter greater than the first bifurcation value, the calculated ratio of the area of the figure bounded by a closed curve, to the area described around the shape of a square. Found the best approximation to a square.

 - It was described the domain structure in the solar corona using a multicolor graph whose isomorphic class is a complete invariant for the topology of domains and gives information about the number of separators.

In the direction 2. Construction of energy functions and the topological classification of systems with a chaotic behavior.

- For the endomorphism f:Mn→Mn satisfying axiom A and having a basic set of codimension 1, which is a compact submanifold was established the dynamics and the presence a smooth structure on the basic sets.

 - It was studied the topology of 3-manifolds and dynamics of A-diffeomorphisms having the non-wandering set located at a finite number of pairwise disjoint mutually disjoint  invariant two-dimensional tori, each of which is a union of the unstable or stable manifolds of one-dimensional basic sets and finite number of periodic points with the same Morse index. It was proved that the structurally stable diffeomorphisms f from the considered class is locally topologically conjugate to the direct product of the generalized DA-diffeomorphisms and a rough transformation of the circle. For such diffeomorphisms it was found a complete system of topological invariants and in each class of topological conjugacy built a standard representative.

- It was found necessary and sufficient conditions of topological conjugacy of 3-diffeomorphisms with a finite hyperbolic chain recurrent set with unique orbit of the heteroclinic tangency.

- It was established the dynamics in the neighborhood of one-dimensional basic sets of two-dimensional endomorphisms satisfying axiom A,.

- For a class of Morse-Smale diffeomorphisms with a surface dynamics it was established that the ambient orientable three-dimensional manifold of such diffeomorphisms is a locally-trivial bundle over the circle and it was obtained the resulting lower estimate of the number of heteroclinic curves of such diffeomorphisms.

- In the framework of the project it was published a book "Dynamical systems on 2- and 3-manifolds", which is the introduction to the topological classification of the structurally stable smooth cascades given on closed oriented manifolds of dimension two and three.

In the direction of 3) - Theory of bifurcations in systems with regular and chaotic dynamics

- It was studied scenarios for the birth of strange attractors in the non-holonomic model of Chaplygin gyroscope. It was established that the model shows a typical dissipative systems scenarios of transition to chaos, such as: (1) the destruction of quasiperiodic regime; (2) a cascade of period-doubling bifurcations for Feigenbaum; (3) doubling of tori.

- In the model of coupled oscillators was previously discovered an interesting phenomenon, which is manifested in the change of the coupling parameter. At small values of the parameter, the system demonstrates conservative behavior. However, with increasing parameter, the average divergence of the system becomes negative. The distribution of the invariant measure of the forward and reverse time indistinguishable for small values of the parameter, is becoming more and more different with the further increase of the coupling parameter. We were able to explain the phenomenon occurring in the system of the mixed dynamics, the recently opened third type of chaos, a characteristic reversible (reverse) smooth systems without invariant measure.

- In slow non-Autonomous periodically time-dependent two-dimensional system of the type Duffing shown that within a certain region, the system dynamics is chaotic. In addition, in this region coexist as chaotic behavior, and island sustainability (elliptical orbit) high periods.

- Proposed and studied a model of two neurons on the basis of two related elements Fitzhugh-Nagumo.

In the direction of 4) - Geometry and dynamics of foliations, agreed with additional structures

- Found necessary and sufficient conditions in order that the lamination of codimension q on an n-dimensional manifold with transversal linear connection are allowed psevdoriemannian transversely invariant metric of a given signature, relative to this parallel connection.

- Obtained the criterion of Riemannian of a foliation with transversal linear connection.

- For carcinophaga lamination (M,F) of arbitrary codimension q, allowing for the connectivity of Eresman, all the layers which are embedded submanifolds of M we prove the existence of open everywhere dense, generally speaking, a disjointed, saturated subset M0 of M and the manifold L0 such that the induced foliation (M0, {M0}), formed by layers locally the trivial bundle with standard layer L0 above, it is possible by smooth q-dimensional manifold. In the case of a codimension one foliation induced on each connected component of the manifold M0 is formed by layers of locally trivial lamination over the circle or trivial bundles over the line.

Publications:


Kazakov A., Коротков А. Г., Осипов Г. В. Dynamics of ensemble of excitatory coupled FitzHugh-Nagumo elements // Regular and Chaotic Dynamics. 2017
Гринес В. З., Жужома Е. В., Медведев В. С., Тарасова Н. А. О существовании периодических траекторий для непрерывных потоков Морса-Смейла // Труды Средневолжского математического общества. 2016. Т. 18. № 1. С. 12-16.
Борисов А. В., Kazakov A., Сатаев И. Р. Spiral Chaos in the Nonholonomic Model of a Chaplygin Top // Regular and Chaotic Dynamics. 2016. Vol. 21. No. 7-8. P. 939-954. doi
N. I. Zhukova. Typical Properties of Leaves of Cartan Foliations with Ehresmann Connection / Пер. с рус. // Journal of Mathematical Sciences. 2016. Vol. 219. No. 1. P. 112-124. doi
Grines V., Pochinka O., Van Strien S. On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli // Moscow Mathematical Journal. 2016. Vol. 16. No. 4. P. 727-749.
Kazakov A., Levanova T., Osipov G. V., Kurths J. Dynamics of ensemble of inhibitory coupled Rulkov maps // European Physical Journal. Special Topics. 2016. Vol. 225. No. 1. P. 147-157. doi
Kazakov A., Борисов А. В., Пивоварова Е. Н. Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top // Regular and Chaotic Dynamics. 2016. Vol. 21. No. 7-8. P. 885-901. doi
Куренков Е. Д., Починка О. В. Динамика точек треугольника Рёло // Огарев-online. Россия.. 2016. № 20
Гринес В. З., Жужома Е. В., Починка О. В. Системы Морса-Смейла и топологическая структура несущих многообразий // Современная математика. Фундаментальные направления. 2016. Т. 61. С. 5-40.
Kazakov A., Лерман Л. М., Кулагин Н. Е. Relaxation Oscillations and Chaos in a Duffing Type Equation: A Case Study // Discontinuity, Nonlinearity, and Complexity. 2016. Vol. 5. No. 4. P. 437-454.
Sheina K. I., Zhukova N. The groups of basic automorphisms of complete Cartan foliations // Lobachevskii Journal of Mathematics. 2016
Kazakov A., Ветчанин Е. В. Bifurcations and Chaos in the Dynamics of Two Point Vortices in an Acoustic Wave // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2016. Vol. 26. No. 4. P. 1650063-1-1650063-13. doi
Сатаев И. Р., Казаков А. О. Сценарии перехода к хаосу в неголономной модели волчка Чаплыгина // Нелинейная динамика. 2016. Т. 12. № 2. С. 235-250. doi
Гринес В. З., Починка О. В., Шиловская А. А. Диффеоморфизмы 3-многообразий с одномерными базисными множествами просторно расположенными на 2-торах. // Труды Средневолжского математического общества. 2016. Т. 18. № 1. С. 17-26.
Жукова Н. И., Шеина К. И. Критерий псевдоримановости слоения с трансверсальной линейной связностью. // Журнал Средневолжского математического общества. 2016. Т. 18. № 2. С. 30-40.
Malyshev D., Pochinka O. Description of domain structures in the Solar Corona by means multi-color graphs // Динамические системы. 2016. Vol. 6(34). No. 1. P. 3-14.
Гринес В. З., Куренков Е. Д. О структуре одномерных базисных множеств эндоморфизмов поверхностей // Труды Средневолжского математического общества. 2016. Т. 18. № 2. С. 16-24.
Grines V., Malyshev D., Pochinka O., Zinina S. Efficient algorithms for the recognition of topologically conjugate gradient-like diffeomorhisms // Regular and Chaotic Dynamics. 2016. Vol. 21. No. 2. P. 189-203. doi
Zhukova N., Dolgonosova A. Pseudo-Riemannian foliations and their graphs / Cornell University. Series math "arxiv.org". 2016.