Regular and Chaotic Dynamics

, Volume 22, Issue 2, pp 180–196 | Cite as

The Hess–Appelrot case and quantization of the rotation number

  • Ivan A. Bizyaev
  • Alexey V. Borisov
  • Ivan S. Mamaev
Article

Abstract

This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.

Keywords

invariant submanifold rotation number Cantor ladder limit cycles 

MSC2010 numbers

70E17 37E45 

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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • Ivan A. Bizyaev
    • 1
  • Alexey V. Borisov
    • 1
  • Ivan S. Mamaev
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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