Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback
type: Item
instance of: scholarly article
title: Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback
zbMATH Open document ID: 1004.34002
author: Tibor Krisztin
author: Hans-Otto Walther
author: Jianhong Wu
published in: Fields Institute Monographs
publication date: 2000-08-28T00:00:00Z
review text: The authors study the scalar delay differential equation \[ \dot x(t)= -\mu x(t)+ f(x(t- 1)),\tag{\(*\)} \] where \(\mu\geq 0\) and \(f(x)\) is an increasing bounded \(C^1\) function with \(f(0)= 0\). A set of additional assumptions is imposed such that (1) Equation \((*)\) has two more equilibria (positive and negative) both stable and attracting; (2) The corresponding linearization at zero \[ \dot y(t)= -\mu y(t)+ f'(0) y(t-1) \] has at least a three-dimensional unstable subspace. An invariant subset of the original nonlinear equation is described in great detail (which takes the whole space of 165 pages of the paper). The authors provide the following brief description of a main part of their results: ``Under natural and mild additional conditions the leading 3-dimensional local unstable manifold at the stationary point \(0\) extends in forward time to a smooth solid spindle with singularities at its tip, that are further stationary points both stable and attractive. An invariant smooth disk of solution curves winding from \(0\) towards a bordering unstable periodic orbit splits the spindle into invariant halves each of which is attracted to one of its tips.'' The book also contains a set of eight appendices which represent an independent interest by themselves.
reviewed by: Anatoli F. Ivanov
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 34Kxx
Mathematics Subject Classification ID: 37Cxx
zbMATH DE Number: 1497415
zbMATH Keywords: scalar delay differential equation
zbMATH Keywords: equilibria
zbMATH Keywords: local unstable manifold
zbMATH Keywords: smooth solid spindle
zbMATH Keywords: singularities
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1497415