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Olga Pochinka

Gagarina av., 23, Nizhny Novgorod, 603950 Russia
Nizhny Novgorod State University


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.
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.
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.
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.
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.
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, no. 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, no. 2-3, pp. 185-193

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