Dissipation-Induced Heteroclinic Orbits in Tippe Tops

Creators: Bou-Rabee, Nawaf M.; Marsden, Jerrold E.; Romero, Louis A.

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Abstract

This paper demonstrates that the conditions for the existence of a dissipation-induced heteroclinic orbit between the inverted and noninverted 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 noninverted state and Lyapunov stability of the inverted state. Standard nonlinear analysis based on the energy momentum method gives necessary and sufficient conditions for the existence of a dissipation-induced connecting orbit between these relative equilibria.

Additional Information

©2008 Society for Industrial and Applied Mathematics. Published electronically May 5, 2008. This paper originally appeared in SIAM Journal on Applied Dynamical Systems, Volume 3, Number 3, 2004, pages 352–377. 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. The research of this author [J.E.M.] 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 U.S. Department of Energy under contract DE-AC04-94AL85000.

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