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., Kruglov E. V., Pochinka O. V.
The Topological Classification of Diffeomorphisms of the Two-Dimensional Torus with an Orientable Attractor
2020, Vol. 16, no. 4, pp. 595-606
This paper is devoted to the topological classification of structurally stable diffeomorphisms of the two-dimensional torus whose nonwandering set consists of an orientable one-dimensional attractor and finitely many isolated source and saddle periodic points, under the assumption that the closure of the union of the stable manifolds of isolated periodic points consists of simple pairwise nonintersecting arcs. The classification of one-dimensional basis sets on surfaces has been exhaustively obtained in papers by V. Grines. He also obtained a classification of some classes of structurally stable diffeomorphisms of surfaces using combined algebra-geometric invariants. In this paper, we distinguish a class of diffeomorphisms that admit purely algebraic differentiating invariants.
Grines V. Z., Kurenkov E. D.
On hyperbolic attractors and repellers of endomorphisms
2017, Vol. 13, No. 4, pp. 557–571
It is well known that the topological classification of dynamical systems with hyperbolic dynamics is significantly defined by dynamics on a nonwandering set. F. Przytycki generalized axiom $A$ for smooth endomorphisms that was previously introduced by S. Smale for diffeomorphisms, and proved the spectral decomposition theorem which claims that the nonwandering set of an $A$-endomorphism is a union of a finite number of basic sets.
In the present paper the criterion for a basic set of an $A$-endomorphism to be an attractor is given. Moreover, dynamics on basic sets of codimension one is studied. It is shown that if an attractor is a topological submanifold of codimension one of type $(n − 1, 1)$, then it is smoothly embedded in the ambient manifold, and the restriction of the endomorphism to this basic set is an expanding endomorphism. If a basic set of type $(n, 0)$ is a topological submanifold of codimension one, then it is a repeller, and the restriction of the endomorphism to this basic set is also an expanding endomorphism.
Grines V. Z., Gurevich E. Y., Zhuzhoma E. V., Zinina S. K.
Heteroclinic curves of Morse – Smale cascades and separators in magnetic field of plasma
2014, Vol. 10, No. 4, pp. 427-438
We obtain properties of three-dimensional phase space and dynamics of Morse–Smale diffeomorphism that led to existence of at least one heteroclinical curve in non-wandering set of the diffeomorphism. We apply this result to solve a problem of existence of separators in magnetic field of plasma.
Grines V. Z., Levchenko Y. A., Pochinka O. V.
On topological classification of diffeomorphisms on 3-manifolds with two-dimensional surface attractors and repellers
2014, Vol. 10, No. 1, pp. 17-33
We consider a class of diffeomorphisms on 3-manifolds which satisfy S. Smale’s axiom A such that their nonwandering set consists of two-dimensional surface basic sets. Interrelation between dynamics of such diffeomorphism and topology of the ambient manifold is studied. Also we establish that each considered diffeomorphism is Ω-conjugated with a model diffeomorphism of mapping torus. Under certain assumptions on asymptotic properties of two-dimensional invariant manifolds of points from the basic sets, we obtain necessary and sufficient conditions of topological conjugacy of structurally stable diffeomorphisms from the considered class.