Elena Nozdrinova
ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150, Russia
National Research University Higher School of Economics (HSE)
Publications:
Pochinka O. V., Nozdrinova E. V.
Stable Arcs Connecting Polar Cascades on a Torus
2021, Vol. 17, no. 1, pp. 2337
Abstract
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 gradientlike diffeomorphisms on a twodimensional torus are found under the assumption that all nonwandering points are fixed and have a positive orientation type. 
Medvedev T. V., Nozdrinova E. V., Pochinka O. V., Shadrina E. V.
On a Class of Isotopic Connectivity of Gradientlike Maps of the 2sphere with Saddles of Negative Orientation Type
2019, Vol. 15, no. 2, pp. 199211
Abstract
We consider the class $G$ of gradientlike orientationpreserving diffeomorphisms of the 2sphere 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 ``sourcesink'' diffeomorphism by a stable arc and this arc contains at most finitely many points of perioddoubling bifurcations.

Nozdrinova E. V.
Rotation Number as a Complete Topological Invariant of a Simple Isotopic Class of Rough Transformations of a Circle
2018, Vol. 14, no. 4, pp. 543551
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 gradientlike diffeomorphisms
on a surface or a threedimensional sphere were found. This article is the next step
in this direction. Namely, the author has established that all orientationreversing diffeomorphisms
of a circle are in one component of a simple connection, whereas the simple isotopy class
of an orientationpreserving transformation of a circle is completely determined by the Poincar´e
rotation number.

Pochinka O. V., Loginova A. S., Nozdrinova E. V.
OneDimensional ReactionDiffusion Equations and Simple SourceSink Arcs on a Circle
2018, Vol. 14, no. 3, pp. 325330
Abstract
This article presents a number of models that arise in physics, biology, chemistry, etc.,
described by a onedimensional reactiondiffusion 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
orientationpreserving sourcesink diffeomorphism on a circle is joined by such an arc. Note that
such a result is not true for multidimensional spheres.
