Heteroclinic bifurcation and singularly perturbed boundary value problems
type: Item
instance of: scholarly article
title: Heteroclinic bifurcation and singularly perturbed boundary value problems
zbMATH Open document ID: 0708.34017
DOI: 10.1016/0022-0396(90)90082-Z
author: Xiao-Biao Lin
published in: Journal of Differential Equations
publication date: 1990-01-01T00:00:00Z
review text: A singularly perturbed boundary value problem \(\dot x=f(x,y,\epsilon),\epsilon \dot y=g(x,y,\epsilon)\) with boundary conditions \[ B_ 1(x(\omega_ 0),y(\omega_ 0),\epsilon)=0,\quad B_ 2(x(\omega_ 2+\omega),y(\omega_ 0+\omega),\epsilon)=0 \] is studied. Given a candidate for the 0th order approximation which exhibits boundary and interior layers, a complete procedure is presented to compute higher order expansions. Error estimates are given and the results are extended to periodic boundary conditions. Several ideas from dynamical systems are employed, e.g. exponential dichotomies, Fredholm's alternative, and heteroclinic bifurcations. The geometric interpretation of the solution structure is very helpful to understand the prepositions and conclusions.
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 4163281
zbMATH Keywords: singularly perturbed boundary value problem
zbMATH Keywords: Error estimates
zbMATH Keywords: periodic boundary conditions
zbMATH Keywords: exponential dichotomies
zbMATH Keywords: Fredholm's alternative
zbMATH Keywords: heteroclinic bifurcations
MaRDI profile type: Publication
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full work available at URL: 0022-0396(90)90082-z
OpenAlex ID: W2154894922
description: scientific article; zbMATH DE number 4163281