Multi-pass stable periodic points of diffeomorphism of a plane with a homoclinic orbit

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.

Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit

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.

Stable and unstable manifold structures in the Hénon family

For parameter values $a$ where the quadratic map $f_a(x) = 1-ax^2$ has an attracting periodic point, and small values of $b$, the Hénon map $H_{a,b}(x, y) = (1 + y - ax^2, bx)$ has a periodic attractor and an attracting set that is homeomorphic with the inverse limit of $f_a|_{[1-a, 1]}$. This attracting set consists of the collection of unstable manifolds of a hyperbolic invariant set together with the attracting periodic orbit. In the case in which $f_a$ is also not renormalizable with smaller period and $b < 0$, we give a symbolic description of the collection of stable manifolds of this hyperbolic set and show that this collection is homeomorphic with the collection of unstable manifolds precisely when the attracting periodic orbit is accessible from the complement of the attracting set, a condition that can be characterized in terms of the kneading sequence of the quadratic map. As an application, we answer a question raised by Hubbard and Oberste-Vorth by proving that the basin boundaries corresponding to three distinct period five sinks in the Hénon family are non-homeomorphic.

Applications of the Melnikov method to twist maps in higher dimensions using the variational approach

We work with symplectic diffeomorphisms of the $n$-annulus ${\Bbb{A}}^n=T^*({\Bbb{R}}^n/{\Bbb{Z}}^n)$. Using the variational approach of Aubry and Mather, we are able to give a local description of the stable (and unstable) manifold for a hyperbolic fixed point. We use this in order to get a Melnikov-like formula for exact symplectic twist maps. This formula involves an infinite series that could be computed in some specific cases. We apply our formula to prove the existence of heteroclinic orbits for a family of twist maps in ${\Bbb{R}}^4$.