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Unformatted text preview: Dissipation-Induced Heteroclinic Orbits in Tippe Tops Nawaf Bou-Rabee * Jerrold E. Marsden Louis A. Romero 12 February 2008 Abstract This paper demontrates that the conditions for the existence of a dissipation-induced heteroclinic orbit between the inverted and nonin- verted states of a tippe top are determined by a complex version of the equations for a simple harmonic oscillator: the modified Maxwell Bloch equations. A standard linear analysis reveals that the modified MaxwellBloch equations describe the spectral instability of the non- inverted state and Liapunov stability of the inverted state. Standard nonlinear analysis based on the energy-momentum method gives neces- sary and sufficient conditions for the existence of a dissipation-induced connecting orbit between these relative equilibria. 1 Introduction Tippe tops come in a variety of forms. The most common geometric form is a cylindrical stem attached to a truncated ball, as shown in Figure 1.1 . On a flat surface, the tippe top will rest stably with its stem up. However, spun fast enough on its blunt end, the tippe top momentarily defies gravity, inverts, * Applied and Computational Mathematics, Caltech, Pasadena, 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 ( marsden@cds. caltech.edu ). The research of this author was partially supported by the National Science Foundation. Sandia National Laboratories, P.O. Box 5800, MS 1110, Albuquerque, NM 87185-1110 ( firstname.lastname@example.org ). The research of this author 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. 352 1 Introduction 353 and spins on its stem until dissipation causes it to slow down and then fall over. This spectacular sequence of events occurs because, and in spite of, dissipation. (a) noninverted (b) inverted (c) heteroclinic connection Figure 1.1: Tippe Top Relative Equilibria & Heteroclinic Orbit. The noninverted and inverted states of the tippe top, and a still of a numerical simulation of the heteroclinic connection between these states. For movies of numerical simulations with discussion the reader is referred to [ 5 ]. Tippe top inversion is a tangible illustration of dissipation-induced insta- bilities, relative equilibria, and the energy-momentum method. Tippe top inversion can be understood by analyzing a system known as the modified MaxwellBloch equations [ 5 ]. These equations are a complex version of the simple harmonic oscillator and a generalization of a previously derived normal form describing dissipation-induced instabilities in the neighborhood of the 1:1 resonance [ 16 ]....
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.
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