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Tippe Top Inversion as a Dissipation-Induced Instability

Bou-Rabee, Nawaf M. and Marsden, Jerrold E. and Romero, Louis A. (2004) Tippe Top Inversion as a Dissipation-Induced Instability. SIAM Journal on Applied Dynamical Systems, 3 (3). pp. 352-377. ISSN 1536-0040. http://resolver.caltech.edu/CaltechAUTHORS:BOUsiamjads04

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Abstract

By treating tippe top inversion as a dissipation-induced instability, we explain tippe top inversion through a system we call the modified Maxwell--Bloch equations. We revisit previous work done on this problem and follow Or's mathematical model [SIAM J. Appl. Math., 54 (1994), pp. 597--609]. A linear analysis of the equations of motion reveals that the only equilibrium points correspond to the inverted and noninverted states of the tippe top and that the modified Maxwell--Bloch equations describe the linear/spectral stability of these equilibria. We supply explicit criteria for the spectral stability of these states. A nonlinear global analysis based on energetics yields explicit criteria for the existence of a heteroclinic connection between the noninverted and inverted states of the tippe top. This criteria for the existence of a heteroclinic connection turns out to agree with the criteria for spectral stability of the inverted and noninverted states. Throughout the work we support the analysis with numerical evidence and include simulations to illustrate the nonlinear dynamics of the tippe top.


Item Type:Article
Additional Information:© 2004 Society for Industrial and Applied Mathematics Received by the editors October 22, 2003; accepted for publication (in revised form) by M. Golubitsky January 22, 2004; published electronically July 6, 2004. This work was performed by an employee of the U.S. Government or under U.S. Government contract. The U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by these rights. CA 91125 (nawaf@acm.caltech.edu). The research of this author was supported by the U.S. DOE Computational Science Graduate Fellowship through grant DE-FG02- 97ER25308. ‡Control and Dynamical Systems, Caltech, Pasadena, CA 91125 The research of this author [N. M. B.-R.] was partially supported by the National Science Foundation. The research of this author [L. A. R.] was supported by Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. We wish to acknowledge Andy Ruina for helpful comments.
Subject Keywords:tippe top inversion, dissipation-induced instability, constrained rotational motion, axisymmetric rigid body
Record Number:CaltechAUTHORS:BOUsiamjads04
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:BOUsiamjads04
Alternative URL:http://dx.doi.org/10.1137/030601351
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1384
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:13 Jan 2006
Last Modified:26 Dec 2012 08:44

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