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If you are not familiar with dynamical systems in general or tippe top inversionin particular, do not read the paper in the SIGEST section of this issue, “Dissipation-InducedHeteroclinicOrbits in Tippe Tops” byN. Bou-Rabee, J. Marsden, and L. Romero.Just kidding, of course, but to be more precise: don’t read it until after you considertaking 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 ordirectly by typing in the longer address http://epubs.siam.org/mm/SJADAY/030601351/60135 01.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 ex-tensive scientific and historical background on tippe tops, including a picture of NobelPrize winning physicists Wolfgang Pauli and Niels Bohr playing with a tippe top, seehttp://www.fysikbasen.dk/English.php?page=Vis&id=79.) A tippe top is a sphere, or alarge section of one, with a small cylindrical stem attached (see cover photo for anexample). Counterintuitively, the spinning tippe top can defy gravity and for a periodof time invert from spinning on its large spherical portion (the “noninverted” state) tospinning on its stem (the “inverted” state). This inversion is possible because the centerof mass of the tippe top does not coincide with the center of the spherical portion, andbecause of the tradeoff between gravitational potential energy and rotational kineticenergy. The inversion is possible only in the presence of friction, which means that fora mathematical model to simulate tippe top inversion, friction must be included. Thebehavior of the tippe top is perhaps the best known example of “dissipation-inducedinstability,” and this type of instability is at the heart of the behavior analyzed in thispaper. The term “heteroclinic orbit” refers to a path in phase space that connects thetwo equilibrium points of a dynamical system; a dissipation-induced heteroclinic orbitis one that exists due to dissipation.

The key contribution of this paper is to show that the stabilities of the nonin-verted state and the inverted state of the tippe top may be studied using modifiedMaxwell–Bloch equations, which also are important in a number of other fields of ap-plied mathematics, including nonlinear optics. The authors show that the conditionsfor the existence of the heteroclinic orbit that connects these two states are describedcompletely by the modified Maxwell–Bloch equations. This result quickly has becomeinfluential in the area of dynamical systems.

This paper originally was published as “Tippe Top Inversion as aDissipation-InducedInstability” in the SIAM Journal on Applied Dynamical Systems, 3 (2004), pp. 352–377. Forthe SIGEST publication, the authors have revised the paper significantly, augmentingthe introductory section to make the paper more accessible to a general audience,streamlining some of the analysis, and adding an interesting final section on relatedwork. This paper offers a fascinating glance into the world of dynamical systems via aclassical problem, and a toy that has fascinated children and Nobel Prize winners alike.

The Editors

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