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Abstract: A diffeomorphism of the plane into itself with a fixed hyperbolic point is considered; it is assumed that there is a non-transversal homoclinic point. Stable and unstable manifolds are tangent to each other at the homoclinic point; there are various ways of tangency of stable and unstable manifolds. Diffeomorphisms of the plane with a non-transversal homoclinic point were analyzed in the studies of Sh. Newhouse, L.P. Shil’nikov, and other authors under the assumption that this point is a tangency point of finite order. It follows from the studies 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, we assume that a homoclinic point is not a point at which the tangency of a stable and an unstable manifold is a tangency of finite order. There is a countable number of types of periodic points lying in the neighborhood of the homoclinic point; points belonging to the same type are called n-pass, where n is a natural number. In this paper, it is shown that, if the tangency is not finite-order, the neighborhood of a non-transversal homoclinic point can contain an infinite set of stable single-pass, two-pass, or three-pass periodic points with characteristic exponents bounded away from zero.
Original language | English |
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Pages (from-to) | 180-186 |
Number of pages | 7 |
Journal | Vestnik St. Petersburg University: Mathematics |
Volume | 54 |
Issue number | 2 |
State | Published - Apr 2021 |
ID: 86573693