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2013
Impact Factor

Vyacheslav Grines

ul. Gagarina 23, Nizhny Novgorod, 603950 Russia
Nizhnii Novgorod State University

Publications:

Grines V. Z., Gurevich E. Y., Pochinka O. V.
On the Number of Heteroclinic Curves of Diffeomorphisms with Surface Dynamics
2017, vol. 22, no. 2, pp.  122-135
Abstract
Separators are fundamental plasma physics objects that play an important role in many astrophysical phenomena. Looking for separators and their number is one of the first steps in studying the topology of the magnetic field in the solar corona. In the language of dynamical systems, separators are noncompact heteroclinic curves. In this paper we give an exact lower estimation of the number of noncompact heteroclinic curves for a 3-diffeomorphism with the so-called “surface dynamics”. Also, we prove that ambient manifolds for such diffeomorphisms are mapping tori.
Keywords: separator in a magnetic field, heteroclinic curves, mapping torus, gradient-like diffeomorphisms
Citation: Grines V. Z., Gurevich E. Y., Pochinka O. V.,  On the Number of Heteroclinic Curves of Diffeomorphisms with Surface Dynamics, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 122-135
DOI:10.1134/S1560354717020022
Grines V. Z., Malyshev D. S., Pochinka O. V., Zinina S. K.
Efficient Algorithms for the Recognition of Topologically Conjugate Gradient-like Diffeomorhisms
2016, vol. 21, no. 2, pp.  189-203
Abstract
It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration algorithm. However, an efficient algorithm (i. e., polynomial in the number of vertices) has not yet been developed for it, and the problem has not been proved to be intractable (i. e., NPcomplete). We give polynomial-time algorithms for recognition of the corresponding graphs for two gradient-like systems. Moreover, we present efficient algorithms for determining the orientability and the genus of the ambient surface. This result, in particular, sheds light on the classification of configurations that arise from simple, point-source potential-field models in efforts to determine the nature of the quiet-Sun magnetic field.
Keywords: Morse – Smale diffeomorphism, gradient-like diffeomorphism, topological classification, three-color graph, directed graph, graph isomorphism, surface orientability, surface genus, polynomial-time algorithm, magnetic field
Citation: Grines V. Z., Malyshev D. S., Pochinka O. V., Zinina S. K.,  Efficient Algorithms for the Recognition of Topologically Conjugate Gradient-like Diffeomorhisms, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 189-203
DOI:10.1134/S1560354716020040
Grines V. Z., Levchenko Y. A., Medvedev V. S., Pochinka O. V.
On the Dynamical Coherence of Structurally Stable 3-diffeomorphisms
2014, vol. 19, no. 4, pp.  506-512
Abstract
We prove that each structurally stable diffeomorphism $f$ on a closed 3-manifold $M^3$ with a two-dimensional surface nonwandering set is topologically conjugated to some model dynamically coherent diffeomorphism.
Keywords: structural stability, surface basic set, partial hyperbolicity, dynamical coherence
Citation: Grines V. Z., Levchenko Y. A., Medvedev V. S., Pochinka O. V.,  On the Dynamical Coherence of Structurally Stable 3-diffeomorphisms, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 506-512
DOI:10.1134/S1560354714040066
Grines V. Z., Pochinka O. V.
Energy functions for dynamical systems
2010, vol. 15, no. 2-3, pp.  185-193
Abstract
The paper contains exposition of results devoted to the existence of an energy functions for dynamical systems.
Keywords: Lyapunov function, energy function, Morse–Smale system
Citation: Grines V. Z., Pochinka O. V.,  Energy functions for dynamical systems, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 185-193
DOI:10.1134/S1560354710020073
Grines V. Z., Zhuzhoma V. S.
Expanding attractors
2006, vol. 11, no. 2, pp.  225-246
Abstract
The article is a survey on local and global structures (including classification results) of expanding attractors of diffeomorphisms $f : M \to M$ of a closed smooth manifold $M$. Beginning with the most familiar expanding attractors (Smale solenoid; DA-attractor; Plykin attractor; Robinson–Williams attractors), one reviews the Williams theory, Bothe's classification of one-dimensional solenoids in 3-manifolds, Grines–Plykin–Zhirov's classification of one-dimensional expanding attractors on surfaces, and Grines–Zhuzhoma's classification of codimension one expanding attractors of structurally stable diffeomorphisms. The main theorems are endowed with ideas of proof
Keywords: Axiom A diffeomorphisms, (codimension one) expanding attractors, structurally stable diffeomorphisms, hyperbolic automorphisms
Citation: Grines V. Z., Zhuzhoma V. S.,  Expanding attractors , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 225-246
DOI:10.1070/RD2006v011n02ABEH000347

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