Asymptotic behaviour of the stability parameter for a family of singular-limit Hill's equation
type: Item
instance of: scholarly article
title: Asymptotic behaviour of the stability parameter for a family of singular-limit Hill's equation
zbMATH Open document ID: 1165.34031
author: Regina Martínez
author: Anna Samà
published in: Journal of Differential Equations
publication date: 2008-12-12T00:00:00Z
review text: The authors consider the following Hill's equation \[ \ddot{x} - (\lambda_1 + \lambda_2 g^{\alpha-2})x = 0 \] where \(\alpha \in (0, 2)\) and \(g=g(t; \delta)\) is a periodic function in \(t\), that depends on a small parameter \(\delta\), \(g(t; \delta) > 0\) for all \(t\) and \(g(0; t) \rightarrow 0\) as \(\delta \rightarrow 0\). This problem occurs in the study of the linear stability of homographic solutions in the planar three-body problem. The authors derive nondegeneracy conditions for the asymptotic behaviour of the stability parameter, which is the trace of the fundamental solution matrix. Using a blow-up of the singularity, they demonstrate that the degeneracies occur, if heteroclinic orbits exist in some related systems. The stability and instability regions are also computed numerically and show a very complicated pattern.
reviewed by: Alois Steindl
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 70F07
Mathematics Subject Classification ID: 70J25
zbMATH DE Number: 5382303
zbMATH Keywords: three-body problem
zbMATH Keywords: homographic solutions
zbMATH Keywords: heteroclinic orbits
zbMATH Keywords: blow-up
describes a project that uses: Taylor
MaRDI profile type: Publication
full work available at URL: j.jde.2008.04.003
OpenAlex ID: W2044777405
cites work: Resonance tongues in Hill's equations: A geometric approach
cites work: Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation
cites work: A Software Package for the Numerical Integration of ODEs by Means of High-Order Taylor Methods
cites work: Q5540675
cites work: Stability diagram for 4D linear periodic systems with applications to homographic solutions
cites work: Analysis of the stability of a family of singular-limit linear periodic systems in \(\mathbb {R}^4\). Applications
cites work: Introduction to Hamiltonian dynamical systems and the \(N\)-body problem
cites work: Coexistence of periodic solutions of Ince's equation
DOI: 10.1016/J.JDE.2008.04.003
description: scientific article; zbMATH DE number 5382303