Invariant manifolds of singularly perturbed ordinary differential equations
type: Item
instance of: scholarly article
title: Invariant manifolds of singularly perturbed ordinary differential equations
zbMATH Open document ID: 0611.34049
DOI: 10.1007/BF00945464
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1985-01-01T00:00:00Z
review text: Consider an autonomous system \[ (1)\quad dx/dt=f(x,y,\epsilon),\quad \epsilon dy/dt=g(x,y,\epsilon), \] where \(f: D\to {\mathbb{R}}^ m\), \(g: D\to {\mathbb{R}}^ n\), \(D=D_ 1\times D_ 2\times (-\epsilon_ 0,\epsilon_ 0)\) is a bounded domain in \({\mathbb{R}}^{m+n+1}\), and \(D_ 1\) is star shaped with \(C^{\nu +1}\) boundary, \(\nu\geq 1\). The author imposes certain stability assumption on (1) and proves the existence of an asymptotically stable (unstable) invariant manifold. This invariant manifold is \(\epsilon\)-close to the so-called reduced manifold of (1). The result is applied to a three-dimensional autonomous system describing a model for nerve impulse suggested by E. C. Zeeman: \[ (2)\quad \dot x_ 1=-1-x_ 2,\quad \dot x_ 2=-2x_ 2-2y,\quad \epsilon \dot y=- (x_ 1+x_ 2y+y^ 3). \]
reviewed by: Li Bingxi
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34C30
Mathematics Subject Classification ID: 92C50
zbMATH DE Number: 3987745
zbMATH Keywords: first order differential equation
zbMATH Keywords: autonomous system
zbMATH Keywords: reduced manifold
author: Kaspar Nipp
MaRDI profile type: Publication
cites work: Dichotomies in stability theory
cites work: Geometric singular perturbation theory for ordinary differential equations
cites work: Q4769286
cites work: Breakdown of Stability in Singularly Perturbed Autonomous Systems II. Estimates for the Solutions and Application
description: scientific article; zbMATH DE number 3987745