A commentary ``On the periodic solutions of a forced second-order equation'' by S. P. Hastings and J. B. McLeod
type: Item
instance of: scholarly article
title: A commentary ``On the periodic solutions of a forced second-order equation'' by S. P. Hastings and J. B. McLeod
zbMATH Open document ID: 0813.34040
DOI: 10.1007/BF01209068
author: Stephen Wiggins
author: Tasso J. Kaper
published in: Journal of Nonlinear Science
publication date: 1995-05-28T00:00:00Z
review text: In this short note, the authors comment on the relationship of the analytical methods (shooting and perturbation) used by Hastings and McLeod in the paper ``On the periodic solutions of a forced second-order equation'' [ibid. 225-245 (1991; Zbl 0801.34040)] to the geometrical approach of nonlinear dynamical systems theory. Their bridge between analysis and geometry is a global perturbation method known as Melnikov's method. Some questions arising from the methods of Hastings and McLeod and their relationship to the more geometrical methods of dynamical systems theory are given by the authors.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 563419
zbMATH Keywords: adiabatic dynamical system
zbMATH Keywords: smale horseshoe
zbMATH Keywords: homoclinic tangle
zbMATH Keywords: geometrical approach
zbMATH Keywords: nonlinear dynamical systems theory
zbMATH Keywords: global perturbation method
zbMATH Keywords: Melnikov's method
reviewed by: Maria Gousidou-Koutita
MaRDI profile type: Publication
cites work: Phase change between separatrix crossings
cites work: Q3769642
cites work: Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory
cites work: Q3031671
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Knotted periodic orbits in suspensions of Smale's horseshoe: torus knots and bifurcation sequences
cites work: Passage through a separatrix in a resonance problem with a slowly-varying parameter
cites work: Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system
cites work: Sustained Resonance for a Nonlinear System with Slowly Varying Coefficients
cites work: Periodic Orbits in Slowly Varying Oscillators
cites work: Homoclinic Orbits in Slowly Varying Oscillators
cites work: On the Detection and Dynamical Consequences of Orbits Homoclinic to Hyperbolic Periodic Orbits and Normally Hyperbolic Invariant Tori in a Class of Ordinary Differential Equations
cites work: Global bifurcations and chaos. Analytical methods
cites work: Q4692840
full work available at URL: https://doi.org/10.1007/bf01209068
OpenAlex ID: W2055803655
description: scientific article; zbMATH DE number 563419