Global dynamics of a rapidly forced cart and pendulum
type: Item
instance of: scholarly article
title: Global dynamics of a rapidly forced cart and pendulum
zbMATH Open document ID: 0897.70017
DOI: 10.1023/A:1008248704427
published in: Nonlinear Dynamics
publication date: 1997-09-09T00:00:00Z
review text: A free swinging plane pendulum's fixing hinge moves along an inclined line in accordance with a prescribed law \((q_0= \beta\cos \omega t)\). The pendulum is subjected to the gravitational force, and the classical case of an inverted pendulum excited by high frequency vertical force is included in this setting. The authors study periodic orbits associated with stable fixed points of the averaged system which are not fixed points of the forced system. Existence of such motions and existence of periodic orbits near saddle equilibrium of the averaged system are shown by the method of successive approximations; an analytical approximation is computed in the same way. The latter periodic orbits are related to saddle periodic orbits having stable and unstable manifolds forming homoclinic tangles. The authors apply the Melnikov's theory, the Poincaré maps and the enlarged theory of exponentially small splitting of separatrices. Three special limiting cases are analysed: a) \(q_0\to 0\), \(\omega\) is fixed; b) \(\omega\to \infty\), \(q_0\) is fixed; c) \(\omega\to \infty\), \(q_0\to 0\). Location and size of the chaotic regions are calculated, and effect of a small damping is investigated. Several diagrams visualize the phenomena. Results can be applied in the theory of nonlinear control systems.
Mathematics Subject Classification ID: 70K40
Mathematics Subject Classification ID: 37B99
zbMATH DE Number: 1043740
zbMATH Keywords: existence
zbMATH Keywords: gravitational force
zbMATH Keywords: stable fixed points
zbMATH Keywords: averaged system
zbMATH Keywords: periodic orbits
zbMATH Keywords: saddle equilibrium
zbMATH Keywords: method of successive approximations
zbMATH Keywords: homoclinic tangles
zbMATH Keywords: Melnikov's theory
zbMATH Keywords: Poincaré maps
zbMATH Keywords: theory of exponentially small splitting of separatrices
zbMATH Keywords: limiting cases
zbMATH Keywords: chaotic regions
author: Steven P. Weibel
author: Tasso J. Kaper
author: John Baillieul
reviewed by: Á. P. Bosznay
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1043740