Abstract:
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, limit cycles.
Citation:
Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hess–Appelrot Case and Quantization of the Rotation Number”, Regul. Chaotic Dyn., 22:2 (2017), 180–196
\Bibitem{BizBorMam17}
\by Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev
\paper The Hess–Appelrot Case and Quantization of the Rotation Number
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 2
\pages 180--196
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\crossref{https://doi.org/10.1134/S156035471702006X}
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Linking options:
https://www.mathnet.ru/eng/rcd250
https://www.mathnet.ru/eng/rcd/v22/i2/p180
This publication is cited in the following 10 articles:
Alexey Glutsyuk, “On germs of constriction curves in model of overdamped Josephson junction, dynamical isomonodromic foliation and Painlevé 3 equation”, Mosc. Math. J., 23:4 (2023), 479–513
Y Bibilo, A A Glutsyuk, “On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation*”, Nonlinearity, 35:10 (2022), 5427
Alexander A. Burov, Anna D. Guerman, Vasily I. Nikonov, “Asymptotic Invariant Surfaces for Non-Autonomous Pendulum-Type Systems”, Regul. Chaotic Dyn., 25:1 (2020), 121–130
O. V. Kholostova, “On the Dynamics of a Rigid Body in the Hess Case at High-Frequency Vibrations of a Suspension Point”, Rus. J. Nonlin. Dyn., 16:1 (2020), 59–84
I. A. Bizyaev, I. S. Mamaev, “Dynamics of the nonholonomic Suslov problem under periodic control: unbounded speedup and strange attractors”, J. Phys. A-Math. Theor., 53:18 (2020), 185701
Vyacheslav P. Kruglov, Sergey P. Kuznetsov, “Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators”, Regul. Chaotic Dyn., 24:6 (2019), 725–738
H. Zoladek, “Perturbations of the Hess-Appelrot and the Lagrange cases in the rigid body dynamics”, J. Geom. Phys., 142 (2019), 121–136
A. Borisov, A. Kilin, I. Mamaev, “Invariant submanifolds of genus 5 and a Cantor staircase in the nonholonomic model of a snakeboard”, Int. J. Bifurcation Chaos, 29:3 (2019), 1930008
A. Borisov, I. Mamaev, “Rigid body dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520
Ol'shanskii V.Yu., “Partial Linear Integrals of the Poincaré-Zhukovskii Equations (the General Case)”, Pmm-J. Appl. Math. Mech., 81:4 (2017), 270–285
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