A Hartman--Grobman theorem for a class of retarded functional differential equations
type: Item
instance of: scholarly article
title: A Hartman--Grobman theorem for a class of retarded functional differential equations
zbMATH Open document ID: 0779.34061
author: Natalia Sternberg
published in: Journal of Mathematical Analysis and Applications
publication date: 1993-09-05T00:00:00Z
review text: The author proves the following local stability theorem for a class of retarded functional differential equations: Let \(f\) be a retarded functional differential equation on \(\mathbb{R}^ n\) with a compact attractor \(A(f)\). Let \(\varphi_ 0\) be a hyperbolic equilibrium solution of \(f\). Let \(\Phi\) be the solution map of \(f\) and let \(L\) be the solution map of its linear variational equation. Suppose that \(\Phi\) is one-to-one on \(A(f)\). Then there exists an open neighbourhood \(W\) of \(\varphi_ 0\) in \(A(f)\), an \(\varepsilon>0\) and a map \(h\) from \(A(f)\) into \(C^ 0([- r,0],\mathbb{R}^ n)\), \(r>0\), such that \(h| W\) is a homeomorphism from \(W\) into \(h(W)\), \(h(\varphi_ 0)=0\) and \(L(t)\circ h=h\circ\Phi(t)\) on \(W\), for all \(t\in (-\varepsilon,\varepsilon)\).
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 34D45
zbMATH DE Number: 399059
zbMATH Keywords: local stability theorem
zbMATH Keywords: retarded functional differential equations
zbMATH Keywords: compact attractor
zbMATH Keywords: linear variational equation
MaRDI profile type: Publication
full work available at URL: jmaa.1993.1206
OpenAlex ID: W2068976085
DOI: 10.1006/JMAA.1993.1206
Recommended article: A Hartman-Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria
Recommended article: A HARTMAN-GROBMAN THEOREM FOR SOME PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
Recommended article: The Hartman-Wintner theorem for functional-differential equations
Recommended article: An existence theorem for a retarded functional differential equation
Recommended article: The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spaces
Recommended article: Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals
Recommended article: RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES AND HENSTOCK-KURZWEIL INTEGRALS
Recommended article: On general properties of retarded functional differential equations on manifolds
Recommended article: Regularity Results and Solution Semigroups for Retarded Functional Differential Equations
Recommended article: The Hartman-Grobman theorem for semilinear hyperbolic evolution equations
description: scientific article; zbMATH DE number 399059