Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit
DOI:
https://doi.org/10.21638/spbu01.2021.209Abstract
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
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References
Литература
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References
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7. Vasileva E.V. Stability of periodic points of diffeomorphism of a plane in the case of a homoclinic orbit. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 6 (64), iss. 1, 44–52 (2019). https://doi.org/10.21638/11701/spbu01.2019.103 (In Russian) [Engl. transl.: Vestnik St. Petersb. Univ., Math. 52, iss. 1, 30–35 (2019). https://doi.org/10.3103/S1063454119010138].
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Published
2021-07-21
How to Cite
Vasil’eva, E. V. (2021). 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), 295–304. https://doi.org/10.21638/spbu01.2021.209
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
