Nonlocal criteria for the existence and stability of periodic oscillations in autonomous Hamiltonian systems
type: Item
instance of: scholarly article
title: Nonlocal criteria for the existence and stability of periodic oscillations in autonomous Hamiltonian systems
zbMATH Open document ID: 0618.70012
DOI: 10.1016/0021-8928(86)90056-0
published in: Journal of Applied Mathematics and Mechanics
publication date: 1986-01-01T00:00:00Z
review text: The conditions under which single-parameter families of periodic solutions (the existence in a sufficiently small neighbourhood of the origin of coordinates follows from the Lyapunov theorem can be continued in a parameter to the boundary of the given domain, in particular to a certain isoenergetic surface, are found. These conditions, which can be verified by the use of the Hessian of a Hamilton function, also ensure the orbital stability of solutions to a first approximation. Bilateral estimates of the oscillation periods are obtained, and it is established that any solution with a period which satisfies such an estimate belongs to the corresponding family. As an example, the nonlinear oscillations of a string with lumped asses are examined.
Mathematics Subject Classification ID: 70K20
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 4003577
zbMATH Keywords: equilibrium configuration
zbMATH Keywords: variational methods
zbMATH Keywords: single-parameter families of periodic solutions
zbMATH Keywords: Lyapunov theorem
zbMATH Keywords: isoenergetic surface
zbMATH Keywords: Hessian of a Hamilton function
zbMATH Keywords: orbital stability
zbMATH Keywords: first approximation
zbMATH Keywords: Bilateral estimates
zbMATH Keywords: oscillation periods
zbMATH Keywords: nonlinear oscillations
zbMATH Keywords: string with lumped asses
author: A. A. Zevin
MaRDI profile type: Publication
full work available at URL: 0021-8928(86)90056-0
OpenAlex ID: W2067357719
cites work: Q3231426
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cites work: Periodic Solutions of Hamiltonian Systems: A Survey
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cites work: Gravity gradient stabilization of large space stations
cites work: Some conditions for the existence and stability of periodic oscillations in non-linear non-autonomous Hamiltonian systems
description: scientific article; zbMATH DE number 4003577