Chaos in the Duffing equation
type: Item
instance of: scholarly article
title: Chaos in the Duffing equation
zbMATH Open document ID: 0772.34040
author: F. Battelli
author: Kenneth Palmer
published in: Journal of Differential Equations
publication date: 1993-05-16T00:00:00Z
review text: The authors study the equation (1) \(z''+a^ 2g_ 0(z)=f(t)\), with \(f\) \(T\)-periodic, under the assumption that the equation \(z''+g_ 0(z)=0\) has either (i) an orbit homoclinic to a saddle point, or (ii) a heteroclinic orbit joining saddle points. In the homoclinic (resp. heteroclinic) case, they prove that if \(f'(t)\) (resp. \(f(t))\) has a simple zero, then for sufficiently large \(a\), equation (1) has a transverse homoclinic (resp. heteroclinic) orbit. This result follows from an abstract theorem the authors prove about existence of bounded solutions of the singularly perturbed equation \(\varepsilon x'+g(x)=\varepsilon^ 2h(t,x,\varepsilon)\), to which (1) can be converted by a change of variables. The proof of this theorem uses exponential dichotomies and scales of Banach spaces.
reviewed by: Stephen Schecter
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 70K50
zbMATH DE Number: 166901
zbMATH Keywords: chaos
zbMATH Keywords: Duffing equation
zbMATH Keywords: homoclinic
zbMATH Keywords: heteroclinic orbit
zbMATH Keywords: singularly perturbed equation
zbMATH Keywords: exponential dichotomies
MaRDI profile type: Publication
full work available at URL: jdeq.1993.1013
OpenAlex ID: W1967531003
DOI: 10.1006/JDEQ.1993.1013
description: scientific article; zbMATH DE number 166901