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Of all publications in the section: 8
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Article
Malyshev D., Pochinka O. Динамические системы. 2016. Vol. 6(34). No. 1. P. 3-14.

Magnetic charging topology explains many energy processes (flares, prominences, etc.) in the solar corona by changing the domain structure associated with the appearance or disappearance of the separators. It is known that at most of the nulls of the magnetic field are prone. In this paper it is proved that a topology of the domains of a field with the prone nulls is completely described by a multi-color graph. In addition, we give an efficient algorithm for distinguishing of these graphs.

Added: Oct 5, 2016
Article
Grines V., Pochinka O., Barinova M. Динамические системы. 2018. Vol. 8. No. 4. P. 299-305.

In this paper we construct an omega-stable diffeomorphism $f$ on closed 3-manifold $M$ so that non-wandering set of $f$ consists of exactly one-dimensional attractor and repeller. All known examples were constructed by Ch. Bonatti, N. Guilman, Sh. Yi. We suggest a new model of the construction of such diffeomorphism.

Added: Nov 21, 2018
Article
Pochinka O., Nozdrinova E., Dolgonosova A. Динамические системы. 2017. Vol. 7(35). No. 2. P. 103-111.

В настоящей работе рассматриваются диффеоморфизмы Морса-Смейла, заданные на неодносвязном замкнутом многообразии M^n,n>3. Для таких систем вводится понятие тривиальной (нетривиальной) связанности их периодических орбит. Устанавливается, что изотопные тривиальные и нетривиальные диффеоморфизмы не могут быть соединены дугой с бифуркациями коразмерности один. Построены примеры таких каскадов Морса-Смейла на многообразии S^(n-1)xS^1.

Added: Nov 16, 2017
Article
V. Kruglov. Динамические системы. 2018. Vol. 8(36). No. 1. P. 15-21.

The class of C^1-smooth gradient-like flows (Morse flows) on closed surface is the subclass of the Morse-Smale flows class, which are rough. Their non-wandering set consists of a finite number of hyperbolic fixed points and a finite number of hyperbolic limit cycles, and they does not have trajectories connecting saddle points. It is well known that the topological equivalence class of a Morse- Smale flow on a surface can be described combinatorially, for example, by the directed Peixoto graph, or by the Oshemkov-Sharko molecule. However, the description of the class of the topological conjugacy of such a system already requires the introduction of continuous invariants (moduli), corresponding to the periods of limit cycles at least. Thus, one class of the equivalence contains continuum classes of the topological conjugacy. Gradient-like flows are Morse-Smale flows without limit cycles. In this paper we prove that gradient-like flows on a closed surface are topologically conjugate iff they are topologically equivalent.

Added: Oct 20, 2018
Article
Гуревич Е. Я., Зинина С. Х. Динамические системы. 2015. Т. 5. № 1-2. С. 25-30.

We obtain  topological classification of  3 dimensional manifolds admitting gradient - like flows whose non-wandering set belongs to  attracting and repelling closed  invariant  surfaces. We show that such manifolds are mapping tori (that are factor spaces of direct product of a surface  $\mathbb{S}_g $ and the interval  $[0,1]$ via  equivalence relation $(z,1)\sim (\tau,0)$, where  $\tau\colon \mathbb{S}_g\to \mathbb{S}_g$ is a homeomorphism). We obtain sufficient conditions for  $\tau$  to be isotopic to periodic map.

Added: Feb 18, 2016
Article
Куренков Е. Д., Рязанова К. А. Динамические системы. 2017. Т. 7. № 2. С. 113-118.
Added: Nov 15, 2017
Article
Гуревич Е. Я., Смирнова А. С. Динамические системы. 2018. Т. 2. № 15. С. 159-172.

We consider a class $G$ of Morse-Smale diffeomorphisms  on the sphere $S^n$ of dimension $n\geq 4$ such that invariant manifolds of different saddle periodic points of any diffeomorphisms from $G$ have no intersection. Dynamics of an arbitrary  diffeomorphism $f\in G$ can be represented as ``sink-source'' dynamics where the  ``sink'' $A_f$ (the ``source'' $R_f$) is  the connected unions of one- and zero-dimensional   unstable (stable) manifolds.  We  study a structure of  the space  $V_f=S^n\setminus (A_f\cup R_f)$ and  the topology of embedding in $V_f$ of separatrices of dimension $(n-1)$.  We prove that the orbit space $\widehat{V}_f=V_f/_f$ is homeomorphic to the direct product $\mathbb{S}^{n-1}\times \mathbb{S}^1$, and the projection $l_\sigma\subset \widehat{V}_f$ of $(n-1)$-dimensional separatrix of a  saddle periodic point $\sigma$ is   either homeomorphic  to the direct product $\mathbb{S}^{n-2}\times \mathbb{S}^1$ and bounds in $\widehat{V}_f$  a manifold homeomorphic to  $\mathbb{B}^{n-1}\times \mathbb{S}^1$, or homeomorphic to a non-oriented locally-trivial fiber bundle under the circle $\mathbb{S}^1$ with the fiber $\mathbb{S}^{n-2}$, and  such the manifold may by only one amount all projection of the separatrices.

Added: Nov 2, 2018
Article
Казаков А. О., Самылина Е. А., Шыхмамедов А. И. Динамические системы. 2017. Т. 7(35). № 3. С. 299-244.

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 the so-called reversible pitch-fork bifurcations due to which a symmetric elliptic point becomes a symmetric saddle point and a pair of asymptotically stable and completely unstable points (one point is symmetric to another) appears in its neighborhood. The mechanism of destruction of conservative dynamics is demonstrated on the example of 1:3 and 1:4 resonances, which appear near the elliptic point of conservative cubic Henon map. In addition, in this paper we present an algorithm for constractig reversible perurbations. which break down the conservative dynamics in two-dimension maps.

Added: Apr 5, 2018