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2013
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# 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