The Conley index for fast-slow systems. I: One-dimensional slow variable
type: Item
instance of: scholarly article
title: The Conley index for fast-slow systems. I: One-dimensional slow variable
zbMATH Open document ID: 0945.34029
DOI: 10.1023/A:1021961819853
author: Tomáš Gedeon
author: Hiroshi Kokubu
author: K. Mischaikow
author: Hiroe Oka
author: James F. Reineck
published in: Journal of Dynamics and Differential Equations
publication date: 2000-09-19T00:00:00Z
review text: The family of differential equations \[ \dot x=f(x,\lambda),\quad \dot \lambda=\varepsilon g(x,\lambda), \] with \(f:\mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^n\) and \(g:\mathbb{R}^n\times \mathbb{R}\to \mathbb{R}\), is considered. The main theorems of the paper are based on the Conley index theory and assert the existence of heteroclinic connections and periodic solutions for the system at \(\varepsilon >0\) small enough provided there exist isolating neighborhoods of some special forms for the system at \(\varepsilon=0\) and restricted values of \(\lambda\). The concepts introduced and the obtained results are explained using the Nagumo equation.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34D15
zbMATH DE Number: 1334590
zbMATH Keywords: fast-slow systems
zbMATH Keywords: isolating neighborhood
zbMATH Keywords: index pair
zbMATH Keywords: Conley index
zbMATH Keywords: Nagumo equation
reviewed by: Roman Srzednicki
MaRDI profile type: Publication
full work available at URL: a:1021961819853
OpenAlex ID: W3008691183
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description: scientific article; zbMATH DE number 1334590