TY - JOUR

T1 - Different Types of Stable Periodic Points of Diffeormorphism of a Plane with a Homoclinic Point

AU - Vasil’eva, E. V.

N1 - Vasil’eva, E.V. Different Types of Stable Periodic Points of Diffeomorphism of a Plane with a Homoclinic Orbit. Vestnik St.Petersb. Univ.Math. 54, 180–186 (2021). https://proxy.library.spbu.ru:2060/10.1134/S106345412102014X

PY - 2021/4

Y1 - 2021/4

N2 - 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.

AB - 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.

KW - characteristic exponents

KW - diffeomorphism

KW - non-transversal homoclinic point

KW - stability

UR - http://www.scopus.com/inward/record.url?scp=85108173676&partnerID=8YFLogxK

UR - https://link.springer.com/article/10.1134%2FS106345412102014X

M3 - Article

AN - SCOPUS:85108173676

VL - 54

SP - 180

EP - 186

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 2

ER -