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2013
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# Olga Pochinka

B. Pecherskaya 25, Nizhny Novgorod, 603105 Russia
Higher School of Economics

## Publications:

 Medvedev T. V., Nozdrinova E. V., Pochinka O. V. Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms 2022, vol. 27, no. 1, pp.  77-97 Abstract In 1976 S.Newhouse, J.Palis and F.Takens introduced a stable arc joining two structurally stable systems on a manifold. Later in 1983 they proved that all points of a regular stable arc are structurally stable diffeomorphisms except for a finite number of bifurcation diffeomorphisms which have no cycles, no heteroclinic tangencies and which have a unique nonhyperbolic periodic orbit, this orbit being the orbit of a noncritical saddle-node or a flip which unfolds generically on the arc. There are examples of Morse – Smale diffeomorphisms on manifolds of any dimension which cannot be joined by a stable arc. There naturally arises the problem of finding an invariant defining the equivalence classes of Morse – Smale diffeomorphisms with respect to connectedness by a stable arc. In the present review we present the classification results for Morse – Smale diffeomorphisms with respect to stable isotopic connectedness and obstructions to existence of stable arcs including the authors’ recent results. Keywords: stable arc, Morse – Smale diffeomorphism Citation: Medvedev T. V., Nozdrinova E. V., Pochinka O. V.,  Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 77-97 DOI:10.1134/S1560354722010087
 Pochinka O. V., Zinina S. K. Construction of the Morse – Bott Energy Function for Regular Topological Flows 2021, vol. 26, no. 4, pp.  350-369 Abstract In this paper, we consider regular topological flows on closed $n$-manifolds. Such flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale flows, which are closely related to the topology of the supporting manifold. This connection is provided by the existence of the Morse – Bott energy function for the Morse – Smale flows. It is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds, on which dynamical systems can be considered only in a continuous category. The existence of continuous analogs of regular flows on any topological manifolds is an open question, as is the existence of energy functions for such flows. In this paper, we study the dynamics of regular topological flows, investigate the topology of the embedding and the asymptotic behavior of invariant manifolds of fixed points and periodic orbits. The main result is the construction of the Morse – Bott energy function for such flows, which ensures their close connection with the topology of the ambient manifold. Keywords: energy function, Morse – Bott energy function, regular topological flow, chain recurrent set, ambient manifold Citation: Pochinka O. V., Zinina S. K.,  Construction of the Morse – Bott Energy Function for Regular Topological Flows, Regular and Chaotic Dynamics, 2021, vol. 26, no. 4, pp. 350-369 DOI:10.1134/S1560354721040031
 Kruglov V., Malyshev D. S., Pochinka O. V., Shubin D. On Topological Classification of Gradient-like Flows on an $n$-sphere in the Sense of Topological Conjugacy 2020, vol. 25, no. 6, pp.  716-728 Abstract In this paper, we study gradient-like flows without heteroclinic intersections on an $n$-sphere up to topological conjugacy. We prove that such a flow is completely defined by a bicolor tree corresponding to a skeleton formed by codimension one separatrices. Moreover, we show that such a tree is a complete invariant for these flows with respect to the topological equivalence also. This result implies that for these flows with the same (up to a change of coordinates) partitions into trajectories, the partitions for elements, composing isotopies connecting time-one shifts of these flows with the identity map, also coincide. This phenomenon strongly contrasts with the situation for flows with periodic orbits and connections, where one class of equivalence contains continuum classes of conjugacy. In addition, we realize every connected bicolor tree by a gradient-like flow without heteroclinic intersections on the $n$-sphere. In addition, we present a linear-time algorithm on the number of vertices for distinguishing these trees. Keywords: gradient-like flow, topological classification, topological conjugacy, $n$-sphere, lineartime algorithm Citation: Kruglov V., Malyshev D. S., Pochinka O. V., Shubin D.,  On Topological Classification of Gradient-like Flows on an $n$-sphere in the Sense of Topological Conjugacy, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 716-728 DOI:10.1134/S1560354720060143
 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