The reduction principle for discrete dynamical and semidynamical systems in metric spaces
type: Item
instance of: scholarly article
title: The reduction principle for discrete dynamical and semidynamical systems in metric spaces
zbMATH Open document ID: 0824.34049
DOI: 10.1007/BF00952086
author: Andrejs Reinfelds
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1995-11-09T00:00:00Z
review text: The author extends the results of the classical Hartman-Grobman theorem, which states that a system of autonomous differential equations of the form \(\dot x= Ax+ f(x, y)\), \(\dot y= By+ g(x, y)\), is dynamically equivalent to the linear system \(\dot x= Ax\), \(\dot y= By\), if the spectra of \(A\) and \(B\) are separated by the imaginary axis in the complex plane, and the mappings \(f\) and \(g\) are uniformly Lipschitzian with sufficiently small Lipschitz constant and vanish at the origin. The author considers the case of a mapping \(T\) in a complete metric space, defined by \(T(x, y)= (f(x, y), g(x, y))\) and having a fixed point. He generalizes the results of \textit{A. Aulbach} and \textit{B. M. Garay} [Z. Angew. Math. Phys. 44, 469-494 (1993; Zbl 0803.46045), and 45, 505-542 (1994; Zbl 0813.47075)], concerning the decoupling and linearization for a mapping in a Banach space. By relaxing the hypothesis, he proves the more general statement that there are mappings \(u\), \(v\) and \(q\) such that \(T\) is topologically conjugate to the mapping \(R\) defined by \(R(x, y)= (f(x, u(x)), g(q(x, y), y))\). A theorem of conjugacy of a given, possibly non-invertible mapping which admits an invariant set, to a simpler mapping than the given one in terms of decoupling is proven in the paper too. The results are valid in an arbitrary complete metric space.
reviewed by: Emilio Elizalde
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 754596
zbMATH Keywords: reduction principle
zbMATH Keywords: discrete dynamics
zbMATH Keywords: Hartman-Grobman theorem
zbMATH Keywords: autonomous differential equations
zbMATH Keywords: complete metric space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 754596