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 Regul. Chaotic Dyn., 2016, Volume 21, Issue 1, Pages 1–17 (Mi rcd64)

Topological Analysis Corresponding to the Borisov–Mamaev–Sokolov Integrable System on the Lie Algebra $so(4)$

Department of Fundamental Sciences, Azarbaijan Shahid Madani University, 35 Km Tabriz-Maragheh Road, Tabriz, Iran

Abstract: In 2001, A. V. Borisov, I. S. Mamaev, and V. V. Sokolov discovered a new integrable case on the Lie algebra $so(4)$. This is a Hamiltonian system with two degrees of freedom, where both the Hamiltonian and the additional integral are homogenous polynomials of degrees 2 and 4, respectively. In this paper, the topology of isoenergy surfaces for the integrable case under consideration on the Lie algebra $so(4)$ and the critical points of the Hamiltonian under consideration for different values of parameters are described and the bifurcation values of the Hamiltonian are constructed. Also, a description of bifurcation complexes and typical forms of the bifurcation diagram of the system are presented.

Keywords: topology, integrable Hamiltonian systems, isoenergy surfaces, critical set, bifurcation diagram, bifurcation complex, periodic trajectory

DOI: https://doi.org/10.1134/S1560354716010019

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MSC: 37Jxx, 70H06, 70E50, 70G40, 70H14
Accepted:20.12.2015
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Citation: Rasoul Akbarzadeh, “Topological Analysis Corresponding to the Borisov–Mamaev–Sokolov Integrable System on the Lie Algebra $so(4)$”, Regul. Chaotic Dyn., 21:1 (2016), 1–17

Citation in format AMSBIB
\Bibitem{Akb16} \by Rasoul Akbarzadeh \paper Topological Analysis Corresponding to the Borisov–Mamaev–Sokolov Integrable System on the Lie Algebra $so(4)$ \jour Regul. Chaotic Dyn. \yr 2016 \vol 21 \issue 1 \pages 1--17 \mathnet{http://mi.mathnet.ru/rcd64} \crossref{https://doi.org/10.1134/S1560354716010019} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3457073} \zmath{https://zbmath.org/?q=an:06580139} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000373028300001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84957586219} 

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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. 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
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. P. E. Ryabov, “Explicit integration of the system of invariant relations for the case of M. Adler and P. van Moerbeke”, Dokl. Math., 95:1 (2017), 17–20
4. R. Akbarzadeh, “The topology of isoenergetic surfaces for the Borisov–Mamaev–Sokolov integrable case on the Lie algebra $so(3,1)$”, Theoret. and Math. Phys., 197:3 (2018), 1727–1736
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