Arithmetic properties of topological invariants of systems with nonstructurally stable homoclinic trajectories
type: Item
instance of: scholarly article
title: Arithmetic properties of topological invariants of systems with nonstructurally stable homoclinic trajectories
zbMATH Open document ID: 0635.58025
DOI: 10.1007/BF01056417
author: L. P. Shil'nikov
author: S. V. Gonchenko
published in: Ukrainian Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: This interesting paper deals with the C r-diffeomorphism \({\mathcal T}\) (r\(\geq 3)\), defined on a two-dimensional manifold, possessing a saddle fixed point A with the eigenvalues \(\lambda\), \(\gamma\), such that \(0<| \lambda | <1<| \gamma |\), \(\sigma =| \lambda \gamma | \leq 1\). The existence of the structurally unstable homoclinic trajectory \(\Gamma\) of the saddle fixed point is assumed. A special coordinate system in a neighbourhood of A exists, such that the diffeomorphism \({\mathcal T}\) can be represented in the following form: \(\bar x=\lambda x+f(x,y)x\) 2y, \(\bar y=\gamma y+g(x,y)xy\) 2, \(A=(0,0)\). Let us choose the pair of the points \({\mathcal M}\) \(+,{\mathcal M}\)-\(\in \Gamma\), \({\mathcal M}\) \(+=(x\) \(+,0)\in W\) \(s_{loc}\), \({\mathcal M}\) \(-=(0,y\) \(-)\in W\) \(u_{loc}\), such that \({\mathcal T}\) mM \(-=M\) \(+\) for some \(m\in N\). Let us put \(\vartheta =-\ln | \lambda | /\ln | \gamma |\), \(\tau =(\ln | \gamma |)^{-1}\times \ln | {\mathfrak c}x\) \(+/y\)- \(|\), where \(c\) is a quantity depending on \({\mathcal T}\). The authors show that the quantities \(\vartheta\) and \(\tau\) are topological invariants of the two-dimensional diffeomorphisms, i.e. if \({\mathcal T}_ 1\) is topologically conjugated with \({\mathcal T}_ 2\), then \(\vartheta_ 1=\vartheta_ 2\).
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 4034714
zbMATH Keywords: stable manifold
zbMATH Keywords: saddle quantity
zbMATH Keywords: topological conjugacy
zbMATH Keywords: C r- diffeomorphism
zbMATH Keywords: saddle fixed point
zbMATH Keywords: eigenvalues
reviewed by: A. Klíč
MaRDI profile type: Publication
cites work: Q4179563
cites work: Bifurcations and stability of families of diffeomorphisms
description: scientific article; zbMATH DE number 4034714