Nonlinear physics and mechanics
Stable Arcs Connecting Polar Cascades on a Torus
O. V. Pochinka, E. V. Nozdrinova
Higher School of Economics Nizhny Novgorod,
ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150 Russia
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.
stable arc, saddle-node, gradient-like diffeomorphism, two-dimensional torus
PDF file (438 kB)
O. V. Pochinka, E. V. Nozdrinova, “Stable Arcs Connecting Polar Cascades on a Torus”, Rus. J. Nonlin. Dyn., 17:1 (2021), 23–37
Citation in format AMSBIB
\by O. V. Pochinka, E. V. Nozdrinova
\paper Stable Arcs Connecting Polar Cascades on a Torus
\jour Rus. J. Nonlin. Dyn.
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