Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation
type: Item
instance of: scholarly article
title: Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation
zbMATH Open document ID: 0872.34033
published in: Journal of Differential Equations
publication date: 1997-10-12T00:00:00Z
review text: The aim is to investigate the structure of the homoclinic and heteroclinic solutions of the scalar fourth order equation \[ \gamma u^{(4)}-\beta u''+F'(u)=0, \] with \(\gamma,\beta>0\). The two primary examples \[ F_1(u)= {\textstyle {\frac{1}{4}}} (u^2-1)^2 \quad\text{and}\quad F_2(u)= {\textstyle {\frac{2}{\pi^2}}} (1+\cos\pi u) \] are considered as a background of the theory. The authors construct a countable family of multitransition homoclinic and heteroclinic solutions. The glueing method applied in the paper can also be used to obtain periodic orbits which are in an arbitrarily small neighborhood of the heteroclinic loop and are structurally similar to those which would be found using dynamical methods of Devaney. Also, the existence of the family of homoclinic and heteroclinic solutions described is sufficient to show that the dynamics are chaotic.
reviewed by: Ding Tongren
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 951889
zbMATH Keywords: chaotic motions
zbMATH Keywords: homoclinic
zbMATH Keywords: heteroclinic solutions
zbMATH Keywords: scalar fourth order equation
zbMATH Keywords: glueing method
zbMATH Keywords: dynamical methods of Devaney
author: William D. Kalies
author: Robert Vandervorst
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0161
OpenAlex ID: W1970930935
DOI: 10.1006/JDEQ.1996.0161
description: scientific article; zbMATH DE number 951889