SIGEST

Abstract

If you are not familiar with dynamical systems in general or tippe top inversion in particular, do not read the paper in the SIGEST section of this issue, “Dissipation-Induced Heteroclinic Orbits in Tippe Tops” by N. Bou-Rabee, J. Marsden, and L. Romero. Just kidding, of course, but to be more precise: don't read it until after you consider taking a few seconds to view the short movie that you can find by clicking on the link “Tippe Top Inversion” at Dr. Bou-Rabee's webpage, www.acm.caltech.edu/~nawaf or directly by typing in the longer address http://epubs.siam.org/mm/SJADAY/030601351/6013501.mpg. Then you will be fully motivated to read and enjoy this fascinating paper. Tippe top inversion is a classical problem in applied mathematics. (For an extensive scientific and historical background on tippe tops, including a picture of Nobel Prize winning physicists Wolfgang Pauli and Niels Bohr playing with a tippe top, see http://www.fysikbasen.dk/English.php?page=Vis&id=79.) A tippe top is a sphere, or a large section of one, with a small cylindrical stem attached (see cover photo for an example). Counterintuitively, the spinning tippe top can defy gravity and for a period of time invert from spinning on its large spherical portion (the “noninverted” state) to spinning on its stem (the “inverted” state). This inversion is possible because the center of mass of the tippe top does not coincide with the center of the spherical portion, and because of the tradeoff between gravitational potential energy and rotational kinetic energy. The inversion is possible only in the presence of friction, which means that for a mathematical model to simulate tippe top inversion, friction must be included. The behavior of the tippe top is perhaps the best known example of “dissipation-induced instability,” and this type of instability is at the heart of the behavior analyzed in this paper. The term “heteroclinic orbit” refers to a path in phase space that connects the two equilibrium points of a dynamical system; a dissipation-induced heteroclinic orbit is one that exists due to dissipation. The key contribution of this paper is to show that the stabilities of the noninverted state and the inverted state of the tippe top may be studied using modified Maxwell–Bloch equations, which also are important in a number of other fields of applied mathematics, including nonlinear optics. The authors show that the conditions for the existence of the heteroclinic orbit that connects these two states are described completely by the modified Maxwell–Bloch equations. This result quickly has become influential in the area of dynamical systems. This paper originally was published as “Tippe Top Inversion as a Dissipation-Induced Instability” in the SIAM Journal on Applied Dynamical Systems, 3 (2004), pp. 352–377. For the SIGEST publication, the authors have revised the paper significantly, augmenting the introductory section to make the paper more accessible to a general audience, streamlining some of the analysis, and adding an interesting final section on related work. This paper offers a fascinating glance into the world of dynamical systems via a classical problem, and a toy that has fascinated children and Nobel Prize winners alike.