ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Bifurcation Diagrams and Fomenko’s Surgery on Liouville Tori of a Rigid Body in the Goryachev-Chaplygin Case on Sokolov Terms

Discontinuity, Nonlinearity, and Complexity 7(4) (2018) 437--449 | DOI:10.5890/DNC.2018.12.008

Jaouad Kharbach$^{1}$, Mohammed Benkhali$^{1}$, Walid Chatar$^{1}$, Ahmed Sali$^{1}$, Abdellah Rezzouk$^{1}$, Mohammed Ouazzani-Jamil$^{2}$

$^{1}$ Department of Physics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, B.P.1796 Fez-Atlas, 30003, Morocco

$^{2}$ Private University of Fez, Lot. Quaraouiyine Route Ain Chkef, Fez, 30000, Morocco

Abstract

In this paper, by taking advantage of the richness of Fomenko’s theory of surgery on (bifurcations of) Liouville tori, we give a complete description of the topology and bifurcations of the invariant level sets of a heavy rigid body around a fixed point corresponding to the Goryachev-Chaplygin case on sokolov terms. In particular, for all values of the parameters of the system, the bifurcation diagrams of the momentum mapping are constructed, bifurcations of the common level sets of the first integrals are described, explicit periodic solutions were determined, the topology of the invariant manifolds and all generic bifurcations are illustrated numerically.

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