Impact Factor

    Olga Pochinka

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


    Pochinka O. V., Nozdrinova E. V.
    Stable Arcs Connecting Polar Cascades on a Torus
    2021, Vol. 17, no. 1, pp.  23-37
    The problem of the existence of an arc with at most countable (finite) number of bifurcations connecting structurally stable systems (Morse – Smale systems) on manifolds was included in the list of fifty Palis – Pugh problems at number 33.
    In 1976 S. Newhouse, J.Palis, F.Takens introduced the concept of a stable arc connecting two structurally stable systems on a manifold. Such an arc does not change its quality properties with small changes. In the same year, S.Newhouse and M.Peixoto proved the existence of a simple arc (containing only elementary bifurcations) between any two Morse – Smale flows. From the result of the work of J. Fliteas it follows that the simple arc constructed by Newhouse and Peixoto can always be replaced by a stable one. For Morse – Smale diffeomorphisms defined on manifolds of any dimension, there are examples of systems that cannot be connected by a stable arc. In this connection, the question naturally arises of finding an invariant that uniquely determines the equivalence class of a Morse – Smale diffeomorphism with respect to the relation of connection by a stable arc (a component of a stable isotopic connection).
    In the article, the components of the stable isotopic connection of polar gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.
    Keywords: stable arc, saddle-node, gradient-like diffeomorphism, two-dimensional torus
    Citation: Pochinka O. V., Nozdrinova E. V.,  Stable Arcs Connecting Polar Cascades on a Torus, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp.  23-37
    Grines V. Z., Kruglov E. V., Pochinka O. V.
    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.
    Keywords: A-diffeomorphisms of a torus, topological classification, orientable attractor
    Citation: Grines V. Z., Kruglov E. V., Pochinka O. V.,  The Topological Classification of Diffeomorphisms of the Two-Dimensional Torus with an Orientable Attractor, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp.  595-606
    Medvedev T. V., Nozdrinova E. V., Pochinka O. V., Shadrina E. V.
    We consider the class $G$ of gradient-like orientation-preserving diffeomorphisms of the 2-sphere with saddles of negative orientation type. We show that the for every diffeomorphism $f\in G$ every saddle point is fixed. We show that there are exactly three equivalence classes (up to topological conjugacy) $G=G_1\cup G_2\cup G_3$ where a diffeomorphism $f_1\in G_1$ has exactly one saddle and three nodes (one fixed source and two periodic sinks); a diffeomorphism $f_2\in G_2$ has exactly two saddles and four nodes (two periodic sources and two periodic sinks) and a diffeomorphism $f_3\in G_3$ is topologically conjugate to a diffeomorphism $f_1^{-1}$. The main result is the proof that every diffeomorphism $f\in G$ can be connected to the ``source-sink'' diffeomorphism by a stable arc and this arc contains at most finitely many points of period-doubling bifurcations.
    Keywords: sink-source map, stable arc
    Citation: Medvedev T. V., Nozdrinova E. V., Pochinka O. V., Shadrina E. V.,  On a Class of Isotopic Connectivity of Gradient-like Maps of the 2-sphere with Saddles of Negative Orientation Type, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp.  199-211
    Pochinka O. V., Loginova A. S., Nozdrinova E. V.
    This article presents a number of models that arise in physics, biology, chemistry, etc., described by a one-dimensional reaction-diffusion equation. The local dynamics of such models for various values of the parameters is described by a rough transformation of the circle. Accordingly, the control of such dynamics reduces to the consideration of a continuous family of maps of the circle. In this connection, the question of the possibility of joining two maps of the circle by an arc without bifurcation points naturally arises. In this paper it is shown that any orientation-preserving source-sink diffeomorphism on a circle is joined by such an arc. Note that such a result is not true for multidimensional spheres.
    Keywords: reaction-diffusion equation, source-sink arc
    Citation: Pochinka O. V., Loginova A. S., Nozdrinova E. V.,  One-Dimensional Reaction-Diffusion Equations and Simple Source-Sink Arcs on a Circle, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp.  325-330
    Pochinka O. V., Kruglov E. V., Dolgonosova A.
    In this paper, one of the possible scenarios for the creation of heteroclinic separators in the solar corona is described and realized. This reconnection scenario connects the magnetic field with two zero points of different signs, the fan surfaces of which do not intersect, with a magnetic field with two zero points which are connected by two heteroclinic separators. The method of proof is to create a model of the magnetic field produced by the plasma in the solar corona and to study it using the methods of dynamical systems theory. Namely, in the space of vector fields on the sphere $S^3$ with two sources, two sinks and two saddles, we construct a simple arc with two saddle-node bifurcation points that connects the system without heteroclinic curves to a system with two heteroclinic curves. The discretization of this arc is also a simple arc in the space of diffeomorphisms. The results are new.
    Keywords: reconnections, separators, bifurcations
    Citation: Pochinka O. V., Kruglov E. V., Dolgonosova A.,  Scenario of reconnection in the solar corona with a simple discretization, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 4, pp.  573–578
    Grines V. Z., Levchenko Y. A., Pochinka O. V.
    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.
    Keywords: diffeomorphism, basic set, topological conjugacy, attractor, repeller
    Citation: Grines V. Z., Levchenko Y. A., Pochinka O. V.,  On topological classification of diffeomorphisms on 3-manifolds with two-dimensional surface attractors and repellers, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 1, pp.  17-33
    Pochinka O. V.
    In this paper class $MS(M^3)$ of Morse–Smale diffeomorphisms (cascades) given on connected closed orientable 3-manifolds are considered. For a diffeomorphism $f \in MS(M^3)$ it is introduced a notion scheme $S_f$, which contains an information on the periodic data of the cascade and a topology of embedding of the sepsrstrices of the saddle points. It is established that necessary and sufficient condition for topological conjugacy of diffeomorphisms $f$, $f’ \in MS(M^3)$ is the equivalence of the schemes $S_f$, $S_f’$.
    Keywords: Morse–Smale diffeomorphism (cascade), topological conjugacy, space orbit
    Citation: Pochinka O. V.,  Necessary and sufficient conditions for topological classification of Morse–Smale cascades on 3-manifolds, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 2, pp.  227-238
    Mitryakova T. M., Pochinka O. V.
    In this paper diffeomorphisms on orientable surfaces are considered, whose non-wandering set consists of a finite number of hyperbolic fixed points and the wandering set contains a finite number of heteroclinic orbits of transversal and non-transversal intersections. We investigate substantial class of diffeomorphisms for which it is found complete topological invariant — a scheme consisting of a set of geometrical objects equipped by numerical parametres (moduli of topological conjugacy).
    Keywords: orbits of heteroclinic tangency, one-sided tangency, topological conjugacy, moduli of topological conjugacy
    Citation: Mitryakova T. M., Pochinka O. V.,  To a question on classification of diffeomorphisms of surfaces with a finite number of moduli of topological conjugacy, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 1, pp.  91-105

    Back to the list