Geodesic
Bifurcation sets in the Kovalevskaya-Yehia problem
Sbornik. Mathematics, Tome 203 (2012) no. 4, pp. 459-499 Cet article a éte moissonné depuis la source Math-Net.Ru

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The two-parameter family of bifurcation diagrams $\Sigma$ of the moment map is investigated in the integrable Kovalevskaya-Yehia case for the motion of a rigid body. A method is developed which is useful for calculating the bifurcation set $\Theta$ in the parameter space which corresponds to bifurcations of diagrams in $\Sigma$ and for classifying these bifurcations. The properties of the sets $\Sigma$ and $\Theta$ are thoroughly investigated, and details of the modifications the bifurcation diagrams undergo as the value of the parameter crosses $\Theta$ are described. Illustrations which explain the structure of the different types of diagram and their interrelations are given. Bibliography: 22 titles.
Keywords: Kovalevskaya-Yehia problem, integrable systems, bifurcation diagrams. 
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P. P. Andreyanov; K. E. Dushin. Bifurcation sets in the Kovalevskaya-Yehia problem. Sbornik. Mathematics, Tome 203 (2012) no. 4, pp. 459-499. http://geodesic.mathdoc.fr/item/SM_2012_203_4_a0/

[1] A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, v. 1, 2, Chapman Hall, Boca Raton, FL, 2004 | MR | MR | Zbl | Zbl

[2] G. G. Appelrot, “Ne vpolne simmetrichnye tyazhelye giroskopy”, Dvizhenie tverdogo tela vokrug nepodvizhnoi tochki, Izd-vo AN SSSR, M.–L., 1940, 61–155

[3] M. P. Kharlamov, Topologicheskii analiz integriruemykh zadach dinamiki tverdogo tela, Izd-vo LGU, L., 1988 | MR

[4] A. T. Fomenko, “Morse theory of integrable Hamiltonian systems”, Soviet Math. Dokl., 33:2 (1986), 502–506 | MR | Zbl

[5] S. V. Matveev, A. T. Fomenko, “Morse-type theory for integrable Hamiltonian systems with tame integrals”, Math. Notes, 43:5 (1988), 382–386 | DOI | MR | Zbl | Zbl

[6] A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658 | DOI | MR | Zbl | Zbl

[7] A. T. Fomenko, Kh. Tsishang, “On typical topological properties of integrable Hamiltonian systems”, Math. USSR-Izv., 32:2 (1989), 385–412 | DOI | MR | Zbl | Zbl

[8] A. T. Fomenko, Kh. Tsishang, “A topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom”, Math. USSR-Izv., 36:3 (1991), 567–596 | DOI | MR | Zbl | Zbl

[9] A. T. Fomenko, “Topological invariants of Liouville integrable Hamiltonian systems”, Funct. Anal. Appl., 22:4 (1988), 286–296 | DOI | MR | Zbl

[10] A. T. Fomenko, “The symplectic topology of completely integrable Hamiltonian systems”, Russian Math. Surveys, 44:1 (1989), 181–219 | DOI | MR | Zbl | Zbl

[11] A. V. Brailov, A. T. Fomenko, “The topology of integral submanifolds of completely integrable Hamiltonian systems”, Math. USSR-Sb., 62:2 (1989), 373–383 | DOI | MR | Zbl | Zbl

[12] S. V. Matveev, A. T. Fomenko, V. V. Sharko, “Round Morse functions and isoenergy surfaces of integrable Hamiltonian systems”, Math. USSR-Sb., 63:2 (1989), 319–336 | DOI | MR | Zbl

[13] A. V. Bolsinov, V. S. Matveev, A. T. Fomenko, “Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry”, Sb. Math., 189:10 (1998), 1441–1466 | DOI | MR | Zbl

[14] H. M. Yehia, “New integrable cases in the dynamics of rigid bodies”, Mech. Res. Comm., 13:3 (1986), 169–172 | DOI | MR | Zbl

[15] Kh. M. Yakhya, “Novye integriruemye sluchai zadachi o dvizhenii girostata”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 4 (1987), 88–90

[16] M. P. Kharlamov, “Topological analysis of classical integrable systems in the dynamics of the rigid body”, Soviet Math. Dokl., 28:3 (1983), 802–805 | MR | Zbl

[17] M. P. Kharlamov, “Bifurcation of common levels of first integrals of the Kovalevskaya problem”, J. Appl. Math. Mech., 47:6 (1983), 737–743 | DOI | MR | Zbl

[18] A. V. Bolsinov, P. Kh. Rikhter, A. T. Fomenko, “Metod krugovykh molekul i topologiya volchka Kovalevskoi”, Matem. sb., 191:2 (2000), 3–42 | MR | Zbl

[19] P. E. Ryabov, M. P. Kharlamov, “Bifurkatsii pervykh integralov v sluchae Kovalevskoi–Yakhi”, Regulyarnaya i khaoticheskaya dinamika, 2:2 (1997), 25–40 | MR | Zbl

[20] P. E. Ryabov, Bifurkatsionnoe mnozhestvo zadachi o dvizhenii tverdogo tela vokrug nepodvizhnoi tochki v sluchae Kovalevskoi–Yakhi, Dis. ... kand. fiz.-matem. nauk, M., MGU, 1997

[21] P. V. Morozov, Liuvilleva klassifikatsiya nekotorykh integriruemykh sistem mekhaniki tverdogo tela, Dis. ... kand. fiz.-matem. nauk, MGU, M., 2006

[22] A. A. Oshemkov, “Vychislenie invariantov Fomenko dlya osnovnykh integriruemykh sluchaev dinamiki tverdogo tela”, Tr. sem. po vekt. i tenz. analizu, 25:2 (1993), 23–109 | Zbl