Homoclinic orbits on invariant manifolds of a functional differential equation
type: Item
instance of: scholarly article
title: Homoclinic orbits on invariant manifolds of a functional differential equation
zbMATH Open document ID: 0967.34065
author: Jiongyu Wu
author: Weinian Zhang
published in: Journal of Differential Equations
publication date: 2000-11-05T00:00:00Z
review text: The authors consider general linear functional-differential equations. In section 2 generalized exponential dichotomies are discussed. For a fixed generalized exponential dichotomy, the existence of corresponding invariant manifolds is obtained in section 3. Section 4 is devoted to a special class of invariant manifolds, called weak hyperbolic manifolds. In section 5 the authors apply their general results to the following delay equation \[ x'(t)= (\text{cosh }1)(\text{sinh }1)^{-1} x(t)- (\text{sinh }1)^{-1}(1+ x_2(t)) x(t- 1),\tag{1} \] considered by Lin, which has the homoclinic solution \[ x(t)= (\text{sinh }1)(\text{cosh }t)^{-1}.\tag{2} \] They prove that the problem of the homoclinic orbit for (1) is a finite-dimensional one, i.e. the orbit of (2) lies in a finite-dimensional invariant manifold.
Mathematics Subject Classification ID: 34K19
Mathematics Subject Classification ID: 34C30
Mathematics Subject Classification ID: 34K12
Mathematics Subject Classification ID: 34D09
zbMATH DE Number: 1525503
zbMATH Keywords: functional-differential equation
zbMATH Keywords: invariant manifold
zbMATH Keywords: homoclinic orbit
zbMATH Keywords: exponential dichotomies
reviewed by: V. Petrov
MaRDI profile type: Publication
full work available at URL: jdeq.2000.3774
OpenAlex ID: W1964529135
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DOI: 10.1006/JDEQ.2000.3774
description: scientific article; zbMATH DE number 1525503