The parametrically forced pendulum: A case study in \(1\frac12\) degree of freedom
type: Item
instance of: scholarly article
title: The parametrically forced pendulum: A case study in \(1\frac12\) degree of freedom
zbMATH Open document ID: 1061.37039
author: Igor Hoveijn
author: Carles Simó
author: Henk Broer
author: Martijn van Noort
author: Gert Vegter
published in: Journal of Dynamics and Differential Equations
publication date: 2005-02-18T00:00:00Z
review text: The global nonchaotic dynamics of the parametrically forced pendulum in a Hamiltonian \(1\frac{1}{2}\) degree of freedom setting with two parameters \(\ddot{x}=(\alpha+\beta \cos t)\sin x\) is investigated. Here, \(x\in \mathcal{S}^1\) is the deviation from the upper equilibrium with range over the whole circle, \(x=0\) corresponds to the upper equilibrium, \(x=\pi\) denotes the lower equilibrium, \(t\in \mathcal{S}^1\) is the independent variable. The parameters \(\alpha=\sqrt{\frac{g}{l}}\;\) (\(g\) is the gravitational acceleration, \(l\) is the length of the pendulum) and \(\beta\) correspond to the square of the eigenfrequency at the lower equilibrium and the amplitude of the forcing and they generally are nonnegative. At the absence of forcing the system is integrable. The authors are interested in all periodic and quasi-periodic dynamics and their bifurcations with special attention to orbits of low period as usually generating largest stability islands. Two main results are obtained. Firstly, at any parameters \((\alpha,\beta)\) there exists a bounded region outside of which the dynamics consists of a set of large measure invariant tori with thin layers of resonant dynamics between them. Secondly, the parameter plane can be divided into four different types of fundamental domains, based on the stability types of the upper and lower equilibria. The stability boundaries are curves of pitchfork or period doubling bifurcations, where the pitchfork bifurcation has codimension one due to spatial symmetry of the system. Analytical and numerical methods are used to study the dynamics in a parameter region away from integrality. The authors give a broad review of the predecessor's works.
reviewed by: Irina V. Konopleva
Mathematics Subject Classification ID: 37J20
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37M20
Mathematics Subject Classification ID: 70H08
zbMATH DE Number: 2134826
zbMATH Keywords: parametrically forced pendulum
zbMATH Keywords: Hamiltonian dynamics
zbMATH Keywords: bifurcations
zbMATH Keywords: numerical methods
zbMATH Keywords: KAM theory
MaRDI profile type: Publication
full work available at URL: s10884-004-7829-5
OpenAlex ID: W2156443431
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DOI: 10.1007/S10884-004-7829-5
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