Geodesic
The Hess–Appelrot case and quantization of the rotation number
Russian journal of nonlinear dynamics, Tome 13 (2017) no. 3, pp. 433-452.

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This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.
Keywords: invariant submanifold, rotation number, Cantor ladder
Mots-clés : limit cycles. 
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I. A. Bizyaev; A. V. Borisov; I. S. Mamaev. The Hess–Appelrot case and quantization of the rotation number. Russian journal of nonlinear dynamics, Tome 13 (2017) no. 3, pp. 433-452. http://geodesic.mathdoc.fr/item/ND_2017_13_3_a9/

[1] Bolsinov A. V., Borisov A. V., Mamaev I. S., “Rolling of a ball without spinning on a plane: The absence of an invariant measure in a system with a complete set of integrals”, Regul. Chaotic Dyn., 17:6 (2012), 571–579 | DOI | MR | Zbl

[2] Bizyaev I. A., “Nonintegrability and obstructions to the Hamiltonianization of a nonholonomic Chaplygin top”, Dokl. Math., 90:2 (2014), 631–634 | DOI | DOI | MR | Zbl

[3] Borisov A. V., Mamaev I. S., “Symmetries and reduction in nonholonomic mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604 | DOI | MR | Zbl

[4] Engelbrecht J. R., Mirollo R., “Structure of long-term average frequencies for Kuramoto oscillator systems”, Phys. Rev. Lett., 109:3 (2012), 034103, 5 pp. | DOI

[5] Lloyd N. G., “The number of periodic solutions of the equation $\dot{z}= z^N + p_1(t)z^{N-1} + ... + p_N(t)$”, Proc. London Math. Soc. (3), 27:4 (1973), 667–700 | DOI | MR | Zbl

[6] Lubowiecki P., Żoła̧dek H., “The Hess – Appelrot system: 1. Invariant torus and its normal hyperbolicity”, J. Geom. Mech., 4:4 (2012), 443–467 | MR | Zbl

[7] Arnol'd V. I., “Small denominators: 1. Mapping the circle onto itself”, Izv. Akad. Nauk SSSR Ser. Mat., 25:1 (1961), 21–86 (Russian) | MR

[8] Bizyaev I. A., Borisov A. V., Mamaev I. S., “The Hess – Appelrot system and its nonholonomic analogs”, Proc. Steklov Inst. Math., 294 (2016), 252–275 | DOI | MR | MR | Zbl

[9] Borisov A. V., Mamayev I. S., “The Hess case in rigid-body dynamics”, J. Appl. Math. Mech., 67:2 (2003), 227–235 | DOI | MR | Zbl

[10] Borisov A. V., Mamaev I. S., Dynamics of a rigid body: Hamiltonian methods, integrability, chaos, 2nd ed., R Dynamics, Institute of Computer Science, Izhevsk, 2005 (Russian) | MR

[11] Bukhshtaber V. M., Karpov O. V., Tertychnyi S. I., “The rotation number quantization effect”, Theoret. and Math. Phys., 162:2 (2010), 211–221 | DOI | DOI | MR

[12] Glutsyuk A. A., Kleptsyn V. A., Filimonov D. A., Schurov I. V., “On the adjacency quantization in an equation modeling the Josephson effect”, Funct. Anal. Appl., 48:4 (2014), 272–285 | DOI | DOI | MR | Zbl

[13] Ziglin S. L., “Splitting of separatrices, branching of solutions and nonexistence of an integral in the dynamics of a solid body”, Trans. Moscow Math. Soc., 1982, no. 1, 283–298 | MR | Zbl

[14] Il'yashenko Yu. S., Ryzhov D. A., Filimonov D. A., “Phase lock for equations describing a resistive model of a Josephson junction and their perturbations”, Funct. Anal. Appl., 45:3 (2011), 192–203 | DOI | DOI | MR | Zbl

[15] Kozlov V. V., “Splitting of the separatrices in the perturbed Euler – Poinsot problem”, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 31:6 (1976), 99–104 (Russian) | Zbl

[16] Markeev A. P., Theoretical mechanics, R Dynamics, Institute of Computer Science, Izhevsk, 2007 (Russian)

[17] Nekrassov P. A., “Étude analytique d'un cas du mouvement d'un corps pesant autour d'un point fixe”, Mat. Sb., 18:2 (1896), 161–274 (Russian)

[18] Pliss V. A., Nonlocal problems of the theory of oscillations, Acad. Press, New York, 1966, 306 pp. | MR | MR | Zbl

[19] Zhukovsky N. E., “Hess' loxodromic pendulum”, Collected Works: Vol. 1, Gostekhizdat, Moscow, 1937, 332–348 (Russian)

[20] Chaplygin S. A., “Some cases of motion of a rigid body in a fluid: 1, 2”, Collected Works: Vol. 1, Gostekhizdat, Moscow, 1948, 136–311 (Russian)