ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150, Russia
National Research University Higher School of Economics (HSE)
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. 543-551
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.
Pochinka O. V., Loginova A. S., Nozdrinova E. V.
One-Dimensional Reaction-Diffusion Equations and Simple Source-Sink Arcs on a Circle
2018, Vol. 14, no. 3, pp. 325-330
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.