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Regular and Chaotic Dynamics

, Volume 21, Issue 5, pp 581–592 | Cite as

The integrable case of Adler–van Moerbeke. Discriminant set and bifurcation diagram

  • Pavel E. Ryabov
  • Andrej A. Oshemkov
  • Sergei V. Sokolov
Article

Abstract

The Adler–van Moerbeke integrable case of the Euler equations on the Lie algebra so(4) is investigated. For the LA pair found by Reyman and Semenov-Tian-Shansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler–van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to self-intersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler–van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on so(4).

Keywords

integrable Hamiltonian systems spectral curve bifurcation diagram 

MSC2010 numbers

70E05 70E17 37J35 34A05 

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Pavel E. Ryabov
    • 1
    • 2
    • 3
  • Andrej A. Oshemkov
    • 4
  • Sergei V. Sokolov
    • 2
  1. 1.Financial UniversityMoscowRussia
  2. 2.Institute of Machines Science, Russian Academy of SciencesMoscowRussia
  3. 3.Moscow Institute of Physics and Technology (State University)Dolgoprudny, Moscow RegionRussia
  4. 4.Lomonosov Moscow State UniversityLeninskie Gory, MoscowRussia

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