Chaotic dynamics of a whirling pendulum
type: Item
instance of: scholarly article
title: Chaotic dynamics of a whirling pendulum
zbMATH Open document ID: 0658.70029
DOI: 10.1016/0167-2789(88)90076-0
author: S. W. Shaw
author: Stephen Wiggins
published in: Physica D
publication date: 1988-01-01T00:00:00Z
review text: A physical system is considered consisting of a rigid frame which is free to rotate about a vertical axis and to which is attached a planar simple pendulum. This system has ``one and a half'' degrees of freedom due to the fact that the frame and pendulum may freely rotate about the vertical axis, i.e., conservation of angular momentum holds for the ``ideal'', or unperturbed, system. Using a Hamiltonian formulation we reduce the unperturbed equations of motion to a conservative planar system in which the constant angular momentum plays the role of a parameter. This system is shown to possess one or two sets of homoclinic motions depending on the level of the angular momentum. When this system is perturbed by external excitations and dissipative forces these homoclinic motions can break into homoclinic tangles providing the conditions for chaotic motions of the horseshoe type to exist. The criteria for this to occur can be formulated using a variation of Melnikov's method developed for slowly varying oscillators. For the present problem, the angular momentum becomes a slowly varying parameter upon addition of the disturbances. These ideas are used to rigorously prove the existence of chaotic motions for this system and to compute, to first order, global bifurcation parameter conditions. Since two types of homoclinic motions can occur, two different chaotic modes of motion can result and physical interpretations of these motions are given. In addition, a limiting case is considered in which the system becomes a single degree of freedom oscillator with parametric excitation.
Mathematics Subject Classification ID: 70K50
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 4076719
zbMATH Keywords: planar simple pendulum
zbMATH Keywords: Hamiltonian formulation
zbMATH Keywords: conservative planar system
zbMATH Keywords: homoclinic motions
zbMATH Keywords: external excitations
zbMATH Keywords: dissipative forces
zbMATH Keywords: Melnikov's method
zbMATH Keywords: existence of chaotic motions
zbMATH Keywords: global bifurcation parameter conditions
Wikidata QID: Q59425585
MaRDI profile type: Publication
full work available at URL: 0167-2789(88)90076-0
OpenAlex ID: W2036494422
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description: scientific article; zbMATH DE number 4076719