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Of all publications in the section: 55
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Article
Казаков А. О., Баханова Ю. В., Коротков А. Г. Журнал Средневолжского математического общества. 2017. Т. 19. № 2. С. 13-24.

Investigations of spiral chaos in generalized Lotka-Volterra systems and Rosenzweig-MacArthur systems that describe the interaction of three species are made in this work. It is shown that in systems under study the spiral chaos appears in agreement with Shilnikov's scenario, that is when changing a parameter in system a stable limit cycle and a saddle-focus born from stable equilibrium. Then the unstable invariant manifold of saddle-focus winds on the stable limit cycle and forms a whirlpool. For some parameter's value the unstable invariant manifold touches one-dimensional stable invariant manifold and forms homoclinic trajectory to saddle-focus. If in this case the limit cycle loses stability (for example, as result of sequence of period doubling bifurcations) and saddle value of saddle-focus is negative then strange attractor appears on base of homoclinic trajectory.

Added: Oct 13, 2017
Article
Жукова Н. И. Журнал Средневолжского математического общества. 2018. Т. 20. № 4. С. 395-407.

It is shown that the structural theory of Molino for Riemannian foliations on compact manifolds and complete Riemannian manifolds is generalized to Riemannian foliations with Ehresmann connection. There are no restrictions on the codimension of the foliation and the dimension of the foliated manifold. For a Riemannian foliation $(M, F)$ with Ehresmann connection it is proved that the closure of any leaf forms a minimal set, the family of all such closures forms a singular Riemannian foliation $(M, \overline{F})$. It is shown that in $M$ there exists a connected open dense $\overline{F}$-saturated subset $M_0$ such that the induced foliation $(M_0, \overline{F}|_{M_0})$ is formed by fibers of a locally trivial bundle over some smooth Hausdorff manifold. It is proved the equivalence of a number of properties of Riemannian foliations $(M, F)$ with Ehresmann connection. In particular, it is shown that the structural Lie algebra of $(M, F)$ is equal to zero if and only if the leaf space of $(M, F)$ is naturally endowed with a smooth orbifold structure. Constructed examples show that for foliations with transversally linear connection and conformal foliations the similar statements are not true in general.

Added: Dec 27, 2019
Article
Ноздринова Е. В. Журнал Средневолжского математического общества. 2017. Т. 19. № 2. С. 91-97.

In this paper we consider the class GG orientation preserving gradient-like diffeomorphisms ff defined on a smooth oriented closed surface M2. Establishes that for any such pair of diffeomorphisms there is a dual attractor-repeller Af,Rf, which have a topological dimension not greater than 1 and the orbit space in their Supplement Vf (characteristic space) is homeomorphic to the two-dimensional torus. The immediate consequence of this result is, for example, the same period all of separatrices of a saddle of diffeomorphisms f∈G On the possibility of such representation of the dynamics of the system in the form `source-drain" founded a number of classification results for a structurally stable dynamical systems with dabloidami a set consisting of a finite number of orbits of systems of Morse-Smale. For example, for systems in dimension three, there is always a coherent characteristic space associated with the choice of a one-dimensional dual pairs of attractor-repeller. In dimension two this is not true even in the gradient-like case, however, the paper shows that there is a one-dimensional dual pair, the characteristic of the orbit space which is connected.

Added: Jun 19, 2017
Article
Ноздринова Е. В. Журнал Средневолжского математического общества. 2020. Т. 22. № 3. С. 306-318.
Added: Nov 18, 2020
Article
Сироткин Д. В. Журнал Средневолжского математического общества. 2017. Т. 19. № 2. С. 98-104.
Added: Aug 23, 2017
Article
Гринес В. З., Починка О. В., Левченко Ю. Журнал Средневолжского математического общества. 2015. Т. 17. № 1. С. 30-37.
Added: Oct 14, 2015
Article
Гринес В. З., Починка О. В., Шиловская А. А. Журнал Средневолжского математического общества. 2015. Т. 17. № 2. С. 27-34.
Added: Oct 14, 2015
Article
Долгоносова А. Ю., Жукова Н. И. Журнал Средневолжского математического общества. 2015. Т. 17. № 4. С. 14-23.

We prove the equivalence of three different approaches to the definition of completeness of a foliation with transverse linear connection. It is shown that for the transverse ane foliations (M, F) of codimension q, q > 1, each of the mentioned above conditions are equivalent to fulllment of the following two conditions: 1) there exists an Ehresmann connection to (M, F) ; 2) the induced foliation on the universal covering space is formed by bres of submersion onto q -dimensional ane space.

Added: Mar 12, 2016
Article
Починка О. В., Митрякова Т., Гринес В. З. Журнал Средневолжского математического общества. 2013. Т. 15. № 4. С. 9-14.
Added: Mar 25, 2014
Article
Колобянина А. Е., Круглов В. Е. Журнал Средневолжского математического общества. 2019. Т. 21. № 4. С. 460-468.

The paper is devoted to the study of the class of Ω-stable flows without limit cycles on surfaces, i.e. flows on surfaces with non-wandering set consisting of a finite number of hyperbolic fixed points. This class is a generalization of the class of gradient-like flows, differing by forbiddance of saddle points connected by separatrices. The results of the work are the proof of the existence of a Morse energy function for any flow from the considered class and the construction of such a function for an arbitrary flow of the class. Since the results are a generalization of the corresponding results of K. Meyer for Morse-Smale flows and, in particular, for gradient-like flows, the methods for constructing the energy function for the case of this article are a further development of the methods used by K. Meyer, taking in sense the specifics of Ω-stable flows having a more complex structure than gradient-like flows due to the presence of the so-called “chains” of saddle points connected by their separatrices.

Added: Oct 22, 2019
Article
Круглов В. Е., Починка О. В. Журнал Средневолжского математического общества. 2014. Т. 16. № 3. С. 57-61.
Added: Jan 27, 2015
Article
Колобянина А. Е., Круглов В. Е. Журнал Средневолжского математического общества. 2020. Т. 22. № 4. С. 434-441.

In this paper, we consider the class of Ω-stable flows on surfaces, i.e. flows on surfaces with the non-wandering set consisting of a finite number of hyperbolic fixed points and a finite number of hyperbolic limit cycles. The class of Ω-stable flows is a generalization of the class of Morse-Smale flows, admitting the presence of saddle connections that do not form cycles. The authors have constructed the Morse-Bott energy function for any such flow. The results obtained are an ideological continuation of the classical works of S. Smale, who proved the existence of the Morse energy function for gradient-like flows, and K. Meyer, who established the existence of the Morse-Bott energy function for Morse-Smale flows. The specificity of Ω-stable flows takes them beyond the framework of structural stability, but the decrease along the trajectories of such flows is still tracked by the regular Lyapunov function.

Added: Nov 27, 2020