Vyacheslav Grines

Vyacheslav Grines
ul. Bolshaya Pecherskaya 25/12, Nizhnii Novgorod, 603155, Russia
National Research University Higher School of Economics

Professor: HSE Campus in Nizhny Novgorod, Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod), Department of Funda- mental Mathematics;
Chief Research Fellow: HSE Campus in Nizhny Novgorod / Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod), International Laboratory of Dynamical Systems and Applications

Born: December 13, 1946 in Isyaslavl', Ukraina.

Positions held:
2015-Present: Professor: HSE Campus in Nizhny Novgorod, Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod), Department of Fundamental Mathematics;
Chief Research Fellow: HSE Campus in Nizhny Novgorod, Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod), International Laboratory of Dynamical Systems and Applications;
2013-2015: Professor of department of numerical and functional analysis, Lobachevskii State University, Nizhnii Novgorod;
1977-2013: Professor of Mathematics, Head of department of mathematics of Nizhny Novgorod State Agriculture Academy;
1969-1977: Researcher, Res.Inst. of Appl. Math.&Cybernetics, State University, N.Novgorod

Scientific degrees:
1976: candidate of physical and mathematical sciences.
1998: doctor of physical and mathematical sciences.

Area of expertise:
Dynamical Systems and Foliations on Manifolds.


Barinova M. K., Grines V. Z., Pochinka O. V., Zhuzhoma E. V.
Hyperbolic Attractors Which are Anosov Tori
2024, vol. 29, no. 2, pp.  369-375
We consider a topologically mixing hyperbolic attractor $\Lambda\subset M^n$ for a diffeomorphism $f:M^n\to M^n$ of a compact orientable $n$-manifold $M^n$, $n>3$. Such an attractor $\Lambda$ is called an Anosov torus provided the restriction $f|_{\Lambda}$ is conjugate to Anosov algebraic automorphism of $k$-dimensional torus $\mathbb T^k$. We prove that $\Lambda$ is an Anosov torus for two cases: 1) $\dim{\Lambda}=n-1$, $\dim{W^u_x}=1$, $x\in\Lambda$; 2) $\dim\,\Lambda=k,\,\dim\, W^u_x=k-1,\,x\in\Lambda$, and $\Lambda$ belongs to an $f$-invariant closed $k$-manifold, $2\leqslant k\leqslant n$, topologically embedded in $M^n$.
Keywords: hyperbolic attractor, Anosov diffeomorphism, $\Omega$-stable diffeomorphism, chaotic attractor
Citation: Barinova M. K., Grines V. Z., Pochinka O. V., Zhuzhoma E. V.,  Hyperbolic Attractors Which are Anosov Tori, Regular and Chaotic Dynamics, 2024, vol. 29, no. 2, pp. 369-375
Grines V. Z., Pochinka O. V., Chilina E. E.
The present paper is devoted to a study of orientation-preserving homeomorphisms on three-dimensional manifolds with a non-wandering set consisting of a finite number of surface attractors and repellers. The main results of the paper relate to a class of homeomorphisms for which the restriction of the map to a connected component of the non-wandering set is topologically conjugate to an orientation-preserving pseudo-Anosov homeomorphism. The ambient $\Omega$-conjugacy of a homeomorphism from the class with a locally direct product of a pseudo-Anosov homeomorphism and a rough transformation of the circle is proved. In addition, we prove that the centralizer of a pseudo-Anosov homeomorphisms consists of only pseudo- Anosov and periodic maps.
Keywords: pseudo-Anosov homeomorphism, two-dimensional attractor
Citation: Grines V. Z., Pochinka O. V., Chilina E. E.,  On Homeomorphisms of Three-Dimensional Manifolds with Pseudo-Anosov Attractors and Repellers, Regular and Chaotic Dynamics, 2024, vol. 29, no. 1, pp. 156-173
Grines V. Z., Medvedev V. S., Zhuzhoma E. V.
Let $\mathbb{G}_k^{cod 1}(M^n)$, $k\geqslant 1$, be the set of axiom A diffeomorphisms such that the nonwandering set of any $f\in\mathbb{G}_k^{cod 1}(M^n)$ consists of $k$ orientable connected codimension one expanding attractors and contracting repellers where $M^n$ is a closed orientable $n$-manifold, $n\geqslant 3$. We classify the diffeomorphisms from $\mathbb{G}_k^{cod 1}(M^n)$ up to the global conjugacy on nonwandering sets. In addition, we show that any $f\in\mathbb{G}_k^{cod 1}(M^n)$ is $\Omega$-stable and is not structurally stable. One describes the topological structure of a supporting manifold $M^n$.
Keywords: axiom A diffeomorphism, expanding attractor, contracting repeller
Citation: Grines V. Z., Medvedev V. S., Zhuzhoma E. V.,  Classification of Axiom A Diffeomorphisms with Orientable Codimension One Expanding Attractors and Contracting Repellers, Regular and Chaotic Dynamics, 2024, vol. 29, no. 1, pp. 143-155
Grines V. Z., Mints D. I.
In P.D.McSwiggen’s article, it was proposed Derived from Anosov type construction which leads to a partially hyperbolic diffeomorphism of the 3-torus. The nonwandering set of this diffeomorphism contains a two-dimensional attractor which consists of one-dimensional unstable manifolds of its points. The constructed diffeomorphism admits an invariant onedimensional orientable foliation such that it contains unstable manifolds of points of the attractor as its leaves. Moreover, this foliation has a global cross section (2-torus) and defines on it a Poincar´e map which is a regular Denjoy type homeomorphism. Such homeomorphisms are the most natural generalization of Denjoy homeomorphisms of the circle and play an important role in the description of the dynamics of aforementioned partially hyperbolic diffeomorphisms. In particular, the topological conjugacy of corresponding Poincaré maps provides necessary conditions for the topological conjugacy of the restrictions of such partially hyperbolic diffeomorphisms to their two-dimensional attractors. The nonwandering set of each regular Denjoy type homeomorphism is a Sierpiński set and each such homeomorphism is, by definition, semiconjugate to the minimal translation of the 2-torus. We introduce a complete invariant of topological conjugacy for regular Denjoy type homeomorphisms that is characterized by the minimal translation, which is semiconjugation of the given regular Denjoy type homeomorphism, with a distinguished, no more than countable set of orbits.
Keywords: topological classification, Denjoy type homeomorphism, Sierpiński set, partial hyperbolicity
Citation: Grines V. Z., Mints D. I.,  On Partially Hyperbolic Diffeomorphisms and Regular Denjoy Type Homeomorphisms, Regular and Chaotic Dynamics, 2023, vol. 28, no. 3, pp. 295-308
Grines V. Z., Medvedev V. S., Zhuzhoma E. V.
Let $M^n$, $n\geqslant 3$, be a closed orientable $n$-manifold and $\mathbb{G}(M^n)$ the set of A-diffeomorp\-hisms $f: M^n\to M^n$ whose nonwandering set satisfies the following conditions: $(1)$ each nontrivial basic set of the nonwandering set is either an orientable codimension one expanding attractor or an orientable codimension one contracting repeller; $(2)$ the invariant manifolds of isolated saddle periodic points intersect transversally and codimension one separatrices of such points can intersect only one-dimensional separatrices of other isolated periodic orbits. We prove that the ambient manifold $M^n$ is homeomorphic to either the sphere $\mathbb S^n$ or the connected sum of $k_f \geqslant 0$ copies of the torus $\mathbb T^n$, $\eta_f\geqslant 0$ copies of $\mathbb S^{n-1}\times \mathbb S^1$ and $l_f\geqslant 0$ simply connected manifolds $N^n_1, \dots, N^n_{l_f}$ which are not homeomorphic to the sphere. Here $k_f\geqslant 0$ is the number of connected components of all nontrivial basic sets, $\eta_{f}=\frac{\kappa_f}{2} -k_f+\frac{\nu_f - \mu_f +2}{2},$ $ \kappa_f\geqslant 0$ is the number of bunches of all nontrivial basic sets, $\mu_f\geqslant 0$ is the number of sinks and sources, $\nu_f\geqslant 0$ is the number of isolated saddle periodic points with Morse index $1$ or $n-1$, $0\leqslant l_f\leqslant \lambda_f$, $\lambda_f\geqslant 0$ is the number of all periodic points whose Morse index does not belong to the set $\{0,1,n-1,n\}$ of diffeomorphism $f$. Similar statements hold for gradient-like flows on $M^n$. In this case there are no nontrivial basic sets in the nonwandering set of a flow. As an application, we get sufficient conditions for the existence of heteroclinic intersections and periodic trajectories for Morse – Smale flows.
Keywords: Decomposition of manifolds, axiom A systems, Morse – Smale systems, heteroclinic intersections
Citation: Grines V. Z., Medvedev V. S., Zhuzhoma E. V.,  On the Topological Structure of Manifolds Supporting Axiom A Systems, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 613-628
Grines V. Z., Gurevich E. Y., Pochinka O. V.
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
Grines V. Z., Malyshev D. S., Pochinka O. V., Zinina S. K.
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
Grines V. Z., Levchenko Y. A., Medvedev V. S., Pochinka O. V.
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
Grines V. Z., Pochinka O. V.
Energy functions for dynamical systems
2010, vol. 15, nos. 2-3, pp.  185-193
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, nos. 2-3, pp. 185-193
Grines V. Z., Zhuzhoma E. V.
Expanding attractors
2006, vol. 11, no. 2, pp.  225-246
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 E. V.,  Expanding attractors , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 225-246

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