Geometrical aspects of stability theory for Hill's equations
type: Item
instance of: scholarly article
title: Geometrical aspects of stability theory for Hill's equations
zbMATH Open document ID: 0840.34047
DOI: 10.1007/BF00382887
author: Mark Levi
author: Henk Broer
published in: Archive for Rational Mechanics and Analysis
publication date: 1996-01-10T00:00:00Z
review text: For the equation \(x''+ (a+ bp(t)) x= 0\), \(p(t)\equiv p(t+ 2\pi)\) (\(a\), \(b\) are real parameters) the authors introduce the Hill's map by \(H: (a, b)\mapsto P_{a, b}\), where \(P_{a, b}\) is the Poincaré (or period) matrix of the equation. They give a global geometric picture of this map, which explains the nature of the classical stability domains in the parameter plane.
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34B30
Mathematics Subject Classification ID: 70J40
zbMATH DE Number: 832137
zbMATH Keywords: Mathieu equation
zbMATH Keywords: Poincaré map
zbMATH Keywords: resonance
zbMATH Keywords: Hill's map
zbMATH Keywords: stability domains
reviewed by: László Hatvani
MaRDI profile type: Publication
cites work: Perturbation geometry for Mathieu's equation
cites work: LECTURES ON BIFURCATIONS IN VERSAL FAMILIES
cites work: Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields
cites work: Q4744562
cites work: Remarks on the perturbation theory for problems of Mathieu type
cites work: Q4012889
cites work: Stability of the Inverted Pendulum—A Topological Explanation
cites work: Instability intervals of Hill's equation
cites work: Q5793672
cites work: Hill’s Equation with a Large Potential
cites work: Asymptotic Behavior of Stability Regions for Hill’s Equation
description: scientific article; zbMATH DE number 832137