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2013
Impact Factor

    Elena Nozdrinova

    ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150, Russia
    National Research University Higher School of Economics (HSE)

    Publications:

    Medvedev T. V., Nozdrinova E. V., Pochinka O. V., Shadrina E. V.
    Abstract
    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
    DOI:10.20537/nd190209
    Nozdrinova E. V.
    Abstract
    The problem of the existence of a simple arc connecting two structurally stable systems on a closed manifold is included in the list of the fifty most important problems of dynamical systems. This problem was solved by S. Newhouse and M. Peixoto for Morse – Smale flows on an arbitrary closed manifold in 1980. As follows from the works of Sh. Matsumoto, P. Blanchard, V. Grines, E.Nozdrinova, and O.Pochinka, for the Morse – Smale cascades, obstructions to the existence of such an arc exist on closed manifolds of any dimension. In these works, necessary and sufficient conditions for belonging to the same simple isotopic class for gradient-like diffeomorphisms on a surface or a three-dimensional sphere were found. This article is the next step in this direction. Namely, the author has established that all orientation-reversing diffeomorphisms of a circle are in one component of a simple connection, whereas the simple isotopy class of an orientation-preserving transformation of a circle is completely determined by the Poincar´e rotation number.
    Keywords: rotation number, simple arc
    Citation: Nozdrinova E. V.,  Rotation Number as a Complete Topological Invariant of a Simple Isotopic Class of Rough Transformations of a Circle, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  543-551
    DOI:10.20537/nd180408
    Pochinka O. V., Loginova A. S., Nozdrinova E. V.
    Abstract
    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
    DOI:10.20537/nd180303

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