Homoclinic tangencies in \(\mathbb{R}^n\)
type: Item
instance of: scholarly article
title: Homoclinic tangencies in \(\mathbb{R}^n\)
zbMATH Open document ID: 1072.37013
author: Victoria Rayskin
published in: Discrete and Continuous Dynamical Systems
publication date: 2005-04-11T00:00:00Z
review text: The celebrated theorem of Birkhoff-Smale provides a method for rigorously concluding the existence of a horseshoe structure. The basic assumption of this theorem is the presence of a transverse homoclinic point. The assumption of transversality is not easy to verify for concrete dynamical systems. This paper is one of many attempts to remove this assumption in the following situation: Let \(M\) be an \(n\)-dimensional smooth manifold, and let \(f: M\to M\) be a diffeomorphism with a hyperbolic fixed-point \(p\in M\) and with stable and unstable invariant manifolds \(W^s\) and \(W^u\) at \(p\). The author proves that if \(f\) is locally \(C^1\)-linearizable in a neighborhood of \(p\), \(W^u\) and \(W^s\) have complementary dimensions \(n-1\) and \(1\), \(W^u\) and \(W^s\) have a finite tangential contact at an isolated homoclinic point \(B\) and \(W^u\) contains a graph portion \(\Lambda\) associated with \(B\), then the invariant manifolds have a point of transversal intersection arbitrary close to the point \(B\). Therefore, \(f\) has a horseshoe. The paper is well written and organized.
reviewed by: A. Klíč
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 37D10
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 37C05
zbMATH DE Number: 2155032
zbMATH Keywords: Homoclinic tangency
zbMATH Keywords: invariant manifolds
zbMATH Keywords: \(\lambda\)-Lemma
zbMATH Keywords: order of contact
zbMATH Keywords: horseshoe structure
MaRDI profile type: Publication
full work available at URL: dcds.2005.12.465
OpenAlex ID: W2040878071
DOI: 10.3934/DCDS.2005.12.465
description: scientific article; zbMATH DE number 2155032