scholarly journals Stable periodic points for smooth diffeomorphisms of multidimensional space

2013 ◽  
pp. 27-35
Author(s):  
E.V. Vasil'eva ◽  
Keyword(s):  
Periodic Points ◽  
Multidimensional Space ◽  
Stable Periodic

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.

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