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The Motion of the Spherical Pendulum Subjected to a D_n Symmetric Perturbation

Chossat, Pascal and Bou-Rabee, Nawaf M. (2005) The Motion of the Spherical Pendulum Subjected to a D_n Symmetric Perturbation. SIAM Journal on Applied Dynamical Systems, 4 (4). pp. 1140-1158. ISSN 1536-0040. http://resolver.caltech.edu/CaltechAUTHORS:20110609-132532272

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Abstract

The motion of a spherical pendulum is characterized by the fact that all trajectories are relative periodic orbits with respect to its circle group of symmetry (invariance by rotations around the vertical axis). When the rotational symmetry is broken by some mechanical effect, more complicated, possibly chaotic behavior is expected. When, in particular, the symmetry reduces to the dihedral group D_n of symmetries of a regular n-gon, n > 2, the motion itself undergoes dramatic changes even when the amplitude of oscillations is small, which we intend to explain in this paper. Numerical simulations confirm the validity of the theory and show evidence of new interesting effects when the amplitude of the oscillations is larger (symmetric chaos).


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1137/040616681DOIUNSPECIFIED
http://epubs.siam.org/siads/resource/1/sjaday/v4/i4/p1140_s1PublisherUNSPECIFIED
Additional Information:© 2005 Society for Industrial and Applied Mathematics. Received by the editors October 10, 2004; accepted for publication (in revised form) by P. Holmes June 22, 2005; published electronically November 22, 2005. The authors are grateful to Jerry Marsden for his interest in this work and for having suggested Nawaf Bou-Rabee to undertake the numerical part. P. Chossat is grateful to the Ambassador of France in India, Mr. Dominique Girard, for allowing him to take a picture of the centaur, and for his support. Both authors are grateful to Phil Holmes, Gabor Domokos, and Tim Healey for their insightful comments.
Subject Keywords:spherical pendulum, elasticity, dihedral symmetry, reduction, nonlinear oscillations, chaos
Classification Code:AMS subject classifications: 34Cxx, 37Jxx, 70Kxx, 74Hxx
Record Number:CaltechAUTHORS:20110609-132532272
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20110609-132532272
Official Citation:The Motion of the Spherical Pendulum Subjected to a D_n Symmetric Perturbation Pascal Chossat and Nawaf M. Bou-Rabee SIAM J. Appl. Dyn. Syst. 4, pp. 1140-1158
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:23961
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:09 Jun 2011 21:23
Last Modified:26 Dec 2012 13:18

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