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Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2021, Volume 8, Issue 2, Pages 295–304
DOI: https://doi.org/10.21638/spbu01.2021.209
(Mi vspua116)
 

MATHEMATICS

Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit

E. V. Vasil'eva

St. Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
Abstract: A diffeomorphism of the plane into itself with a fixed hyperbolic point is considered; the presence of a nontransverse homoclinic point is assumed. Stable and unstable manifolds touch each other at a homoclinic point; there are various ways of touching a stable and unstable manifold. In the works of Sh. Newhouse, L. P. Shilnikov and other authors, studied diffeomorphisms of the plane with a nontranverse homoclinic point, under the assumption that this point is a tangency point of finite order. It follows from the works of these authors that an infinite set of stable periodic points can lie in a neighborhood of a homoclinic point; the presence of such a set depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point at which the tangency of a stable and unstable manifold is a tangency of finite order. Allocate a countable number of types of periodic points lying in the vicinity of a homoclinic point; points belonging to the same type are called n-pass (multi-pass), where n is a natural number. In the present paper, it is shown that if the tangency is not a tangency of finite order, the neighborhood of a nontransverse homolinic point can contain an infinite set of stable single-pass, double-pass, or three-pass periodic points with characteristic exponents separated from zero.
Keywords: diffeomorphism, nontransverse homoclinic point, stability, characteristic exponents.
Funding agency Grant number
Russian Foundation for Basic Research 19-01-00388
The work is supported by Russian Foundation for Basic Research (grant no. 19-01-00388).
Received: 23.10.2020
Revised: 13.11.2020
Accepted: 17.12.2020
English version:
Vestnik St. Petersburg University, Mathematics, 2021, Volume 8, Issue 3, Pages 180–186
DOI: https://doi.org/10.1134/S106345412102014X
Document Type: Article
UDC: 517.925.53
MSC: 37C75, 37C29, 34C37
Language: Russian
Citation: E. V. Vasil'eva, “Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit”, Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8:2 (2021), 295–304; Vestn. St. Petersbg. Univ., Math., 8:3 (2021), 180–186
Citation in format AMSBIB
\Bibitem{Vas21}
\by E.~V.~Vasil'eva
\paper Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit
\jour Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy
\yr 2021
\vol 8
\issue 2
\pages 295--304
\mathnet{http://mi.mathnet.ru/vspua116}
\crossref{https://doi.org/10.21638/spbu01.2021.209}
\transl
\jour Vestn. St. Petersbg. Univ., Math.
\yr 2021
\vol 8
\issue 3
\pages 180--186
\crossref{https://doi.org/10.1134/S106345412102014X}
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