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 Regul. Chaotic Dyn., 2016, Volume 21, Issue 5, Pages 581–592 (Mi rcd211)

The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram

Pavel E. Ryabovabc, Andrej A. Oshemkovd, Sergei V. Sokolovb

a Moscow Institute of Physics and Technology (State University) Institutskiy per. 9, Dolgoprudny, Moscow Region, 141700 Russia
b Institute of Machines Science, Russian Academy of Sciences, Maly Kharitonyevsky Per. 4, Moscow, 101990 Russia
c Financial University, Leningradsky prosp. 49, Moscow, 125993 Russia
d Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991 Russia

Abstract: The Adler – van Moerbeke integrable case of the Euler equations on the Lie algebra $so(4)$ is investigated. For the $L-A$ 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

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-0011916-01-0017016-01-0080916-01-0037815-41-02049 Ministry of Education and Science of the Russian Federation 7962.2016.1 This work is partially supported by the grants of RFBR No. 14–01–00119, 16–01–00170, 16–01–00809, and 16–01–00378, common grant of RFBR and Volgograd Region Authorities No. 15–41–02049, and the grant of the President of the Russian Federation for State Support of Leading Scientific Schools No. 7962.2016.1.

DOI: https://doi.org/10.1134/S1560354716050087

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Bibliographic databases:

MSC: 70E05, 70E17, 37J35, 34A05
Accepted:14.09.2016
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Citation: Pavel E. Ryabov, Andrej A. Oshemkov, Sergei V. Sokolov, “The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram”, Regul. Chaotic Dyn., 21:5 (2016), 581–592

Citation in format AMSBIB
\Bibitem{RyaOshSok16} \by Pavel E. Ryabov, Andrej A. Oshemkov, Sergei V. Sokolov \paper The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram \jour Regul. Chaotic Dyn. \yr 2016 \vol 21 \issue 5 \pages 581--592 \mathnet{http://mi.mathnet.ru/rcd211} \crossref{https://doi.org/10.1134/S1560354716050087} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3556085} \zmath{https://zbmath.org/?q=an:06662686} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000385167300008} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84990876567} 

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• http://mi.mathnet.ru/eng/rcd/v21/i5/p581

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. V. Sokolov, “Integriruemyi sluchai Adlera–van Mërbeke. Vizualizatsiya bifurkatsii torov Liuvillya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:4 (2017), 532–539
2. A. A. Oshemkov, P. E. Ryabov, S. V. Sokolov, “Explicit determination of certain periodic motions of a generalized two-field gyrostat”, Russ. J. Math. Phys., 24:4 (2017), 517–525
3. S. V. Sokolov, P. E. Ryabov, “Bifurcation diagram of the two vortices in a Bose–Einstein condensate with intensities of the same signs”, Dokl. Math., 97:3 (2018), 286–290