Professor: HSE Campus in Nizhny Novgorod, Faculty of Informatics,
Mathematics, and Computer Science (HSE Nizhny Novgorod), Department of Funda-
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.
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
1976: candidate of physical and mathematical sciences.
1998: doctor of physical and mathematical sciences.
Area of expertise:
Dynamical Systems and Foliations on Manifolds.
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
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.
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
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.
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
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.
Grines V. Z., Pochinka O. V.
Energy functions for dynamical systems
2010, vol. 15, no. 2-3, pp. 185-193
The paper contains exposition of results devoted to the existence of an energy functions for dynamical systems.
Grines V. Z., Zhuzhoma E. V.
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