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:
I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “The Hess–Appelrot case and quantization of the rotation number”, Nelin. Dinam., 13:3 (2017), 433–452; Regular and Chaotic Dynamics, 22:2 (2017), 180–196
\Bibitem{BizBorMam17}
\by I.~A.~Bizyaev, A.~V.~Borisov, I.~S.~Mamaev
\paper The Hess–Appelrot case and quantization of the rotation number
\jour Nelin. Dinam.
\yr 2017
\vol 13
\issue 3
\pages 433--452
\mathnet{http://mi.mathnet.ru/nd576}
\crossref{https://doi.org/10.20537/nd1703010}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3658423}
\elib{https://elibrary.ru/item.asp?id=29993271}
\transl
\jour Regular and Chaotic Dynamics
\yr 2017
\vol 22
\issue 2
\pages 180--196
\crossref{https://doi.org/10.1134/S156035471702006X}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85031088531}
Linking options:
https://www.mathnet.ru/eng/nd576
https://www.mathnet.ru/eng/nd/v13/i3/p433
This publication is cited in the following 8 articles:
Y Bibilo, A A Glutsyuk, “On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation*”, Nonlinearity, 35:10 (2022), 5427
A. A. Burov, “Linear invariant relations in the problem of the motion of a bundle of two bodies”, Dokl. Phys., 65:4 (2020), 147–148
Ivan A Bizyaev, Ivan 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
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
Alexey Borisov, Alexander Kilin, Ivan Mamaev, “Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard”, Int. J. Bifurcation Chaos, 29:03 (2019), 1930008
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
Henryk Żoła̧dek, “Perturbations of the Hess–Appelrot and the Lagrange cases in the rigid body dynamics”, Journal of Geometry and Physics, 142 (2019), 121
V. Yu. Ol'shanskii, “Partial linear integrals of the Poincaré–Zhukovskii equations (the general case)”, Journal of Applied Mathematics and Mechanics, 81:4 (2017), 270
Citation in format AMSBIB
Contact us: