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

2021 ◽  
Vol 8 (3) ◽  
pp. 406-416
Author(s):  
Ekaterina V. Vasil’eva ◽  
Keyword(s):  
Finite Order ◽  
Unstable Manifold ◽  
Small Neighborhood ◽  
Periodic Points ◽  
Homoclinic Point ◽  
Hyperbolic Point ◽  
Stable Periodic ◽  
Nontransversal Homoclinic Point ◽  
Stable And Unstable Manifold ◽  
Infinite Set

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.

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

2021 ◽  
Vol 8 (2) ◽  
pp. 295-304
Author(s):  
Ekaterina V. Vasil’eva ◽  
Keyword(s):  
Finite Order ◽  
Unstable Manifold ◽  
Periodic Points ◽  
Homoclinic Point ◽  
Double Pass ◽  
Stable And Unstable Manifolds ◽  
Hyperbolic Point ◽  
Stable Periodic ◽  
Stable And Unstable Manifold ◽  
Infinite Set

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.

Horseshoes in the measure-preserving Hénon map

1995 ◽  
Vol 15 (6) ◽  
pp. 1045-1059 ◽  
Author(s):  
Ray Brown
Keyword(s):  
Fixed Point ◽  
Unstable Manifold ◽  
Recursive Formula ◽  
Homoclinic Point ◽  
Hénon Map ◽  
Henon Map ◽  
Broad Array ◽  
Hyperbolic Fixed Point ◽  
Elementary Methods

AbstractWe show, using elementary methods, that for 0 < a the measure-preserving, orientation-preserving Hénon map, H, has a horseshoe. This improves on the result of Devaney and Nitecki who have shown that a horseshoe exists in this map for a ≥ 8. For a > 0, we also prove the conjecture of Devaney that the first symmetric homoclinic point is transversal.To obtain our results, we show that for a branch, Cu, of the unstable manifold of a hyperbolic fixed point of H, Cu crosses the line y = − x and that this crossing is a homoclinic point, χc. This has been shown by Devaney, but we obtain the crossing using simpler methods. Next we show that if the crossing of Wu(p) and Ws(p) at χc is degenerate then the slope of Cu at this crossing is one. Following this we show that if χc is a degenerate homoclinic its x-coordinate must be greater than l/(2a). We then derive a contradiction from this by showing that the slope of Cu at H-1(χc) must be both positive and negative, thus we conclude that χc is transversal.Our approach uses a lemma that gives a recursive formula for the sign of curvature of the unstable manifold. This lemma, referred to as ‘the curvature lemma’, is the key to reducing the proof to elementary methods. A curvature lemma can be derived for a very broad array of maps making the applicability of these methods very general. Further, since curvature is the strongest differentiability feature needed in our proof, the methods work for maps of the plane which are only C2.

Homeomorphic restrictions of smooth endomorphisms of an interval

1992 ◽  
Vol 12 (3) ◽  
pp. 429-439 ◽  
Author(s):  
Karen M. Brucks ◽  
Maria Victoria Otero-Espinar ◽  
Charles Tresser
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Periodic Points ◽  
Limit Set ◽  
Asymptotic Dynamics ◽  
Infinite Set

AbstractWe describe the asymptotic dynamics of homeomorphisms obtained as restrictions of generic C2 endomorphisms of an interval with finitely many critical points, all of which are non-flat, and with all periodic points hyperbolic. The ω -limit set of such a restricted endomorphism cannot be infinite, except when the restriction of the endomorphism to the closure of the orbit of some critical point is a minimal homeomorphism of an infinite set.

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