DissipationInduced Heteroclinic
Orbits in Tippe Tops
Nawaf BouRabee
*
Jerrold E. Marsden
†
Louis A. Romero
‡
12 February 2008
Abstract
This paper demontrates that the conditions for the existence of a
dissipationinduced 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
Maxwell–Bloch equations describe the spectral instability of the non
inverted state and Liapunov stability of the inverted state.
Standard
nonlinear analysis based on the energymomentum method gives neces
sary and sufficient conditions for the existence of a dissipationinduced
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 (
[email protected]
caltech.edu
). The research of this author was supported by the U.S. DOE Computational
Science Graduate Fellowship through grant DEFG0297ER25308.
†
Control
and
Dynamical
Systems,
Caltech,
Pasadena,
CA
91125
(
[email protected]
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 871851110
(
[email protected]
).
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
DEAC0494AL85000.
352
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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 dissipationinduced insta
bilities, relative equilibria, and the energymomentum method.
Tippe top
inversion can be understood by analyzing a system known as the
modified
Maxwell–Bloch equations
[
5
].
These equations are a complex version of
the simple harmonic oscillator and a generalization of a previously derived
normal form describing dissipationinduced instabilities in the neighborhood
of the 1:1 resonance [
16
].
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 Fall '10
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 Angular Momentum, Kinetic Energy, Tippe Tops, spherical tippe, tippe

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