Different Types of Stable Periodic Points of Diffeomorphism of a Plane with a Homoclinic Orbit

Autor: E. V. Vasil’eva
Rok vydání: 2021
Předmět:
Zdroj: Vestnik St. Petersburg University, Mathematics. 54:180-186
ISSN: 1934-7855
1063-4541
Popis: 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.
Databáze: OpenAIRE
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