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This article is cited in 1 scientific paper (total in 1 paper)
MATHEMATICS
Multi-pass 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 a plane into itself with a fixed hyperbolic point and a nontransversal point homoclinic to it is studied. There are various ways of touching a stable and unstable manifold at a homoclinic point. Periodic points whose trajectories do not leave the vicinity of the trajectory of a homoclinic point are divided into a countable set of types. Periodic points of the same type are called n-pass periodic points if their trajectories have n turns that lie outside a sufficiently small neighborhood of the hyperbolic point. Earlier in the articles of Sh. Newhouse, L. P. Shil'nikov, B. F. Ivanov and other authors, diffeomorphisms of the plane with a nontransversal homoclinic point were studied, it was assumed that this point is a tangency point of finite order. In these papers, it was shown that in a neighborhood of a homoclinic point there can be infinite sets of stable two-pass and three-pass periodic points. The presence of such sets depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point with a finite order of tangency of a stable and unstable manifold. It is shown in the paper that for any fixed natural number n, a neighborhood of a nontransversal homolinic point can contain an infinite set of stable n-pass periodic points with characteristic exponents separated from zero.
Keywords:
diffeomorphism of plane, nontransversal homoclinic point, stability, characteristic exponents.
Received: 16.02.2020 Revised: 14.03.2020 Accepted: 19.03.2020
Citation:
E. V. Vasil'eva, “Multi-pass stable periodic points of diffeomorphism of a plane with a homoclinic orbit”, Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8:3 (2021), 406–416; Vestn. St. Petersbg. Univ., Math., 8:4 (2021), 227–235
Linking options:
https://www.mathnet.ru/eng/vspua91 https://www.mathnet.ru/eng/vspua/v8/i3/p406
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