Vyacheslav Grines
ul. Bolshaya Pecherskaya 25/12, Nizhnii Novgorod, 603155, Russia
National Research University Higher School of Economics
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. 122135
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 3diffeomorphism with the socalled “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 Gradientlike Diffeomorhisms
2016, vol. 21, no. 2, pp. 189203
Abstract
It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradientlike 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 polynomialtime algorithms for recognition of the corresponding graphs for two gradientlike 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, pointsource potentialfield models in efforts to determine the nature of the quietSun magnetic field.

Grines V. Z., Levchenko Y. A., Medvedev V. S., Pochinka O. V.
On the Dynamical Coherence of Structurally Stable 3diffeomorphisms
2014, vol. 19, no. 4, pp. 506512
Abstract
We prove that each structurally stable diffeomorphism $f$ on a closed 3manifold $M^3$ with a twodimensional 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. 23, pp. 185193
Abstract
The paper contains exposition of results devoted to the existence of an energy functions for dynamical systems.

Grines V. Z., Zhuzhoma E. V.
Expanding attractors
2006, vol. 11, no. 2, pp. 225246
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; DAattractor; Plykin attractor; Robinson–Williams attractors), one reviews the Williams theory, Bothe's classification of onedimensional solenoids in 3manifolds, Grines–Plykin–Zhirov's classification of onedimensional 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
