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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. doi:10.1137/040616681.

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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).

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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
Issue or Number:4
Classification Code:AMS subject classifications: 34Cxx, 37Jxx, 70Kxx, 74Hxx
Record Number:CaltechAUTHORS:20110609-132532272
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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
Deposited By: Ruth Sustaita
Deposited On:09 Jun 2011 21:23
Last Modified:09 Nov 2021 16:19

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