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Regular and Chaotic Dynamics, 2019, Volume 24, Issue 3, Pages 329–352
DOI: https://doi.org/10.1134/S1560354719030067 (Mi rcd481) 		 
This article is cited in 1 scientific paper (total in 1 paper)

A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness

Alexey V. Borisovab, Alexander A. Kilinc, Ivan S. Mamaevde

a Institute of Mathematics and Mechanics of the Ural Branch of RAS, ul. S.Kovalevskoi 16, Ekaterinburg, 620990 Russia
b A.A. Blagonravov Mechanical Engineering Research Institute of RAS, ul. Bardina 4, Moscow, 117334 Russia
c Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
d Center for Technologies in Robotics and Mechatronics Components, Innopolis University, ul. Universitetskaya 1, Innopolis, 420500 Russia
e Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia
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DOI: https://doi.org/10.1134/S1560354719030067
Abstract: This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both well-known results and a number of new results.
We consider the two most widespread friction (resistance) models: dry (Coulomb) friction and viscous friction. It is shown that the addition of external damping (air drag) can lead to stability of equilibrium at the saddle point and hence to preservation of the region of bounded motion in a neighborhood of the saddle point. Analysis of three-dimensional Poincaré sections shows that limit cycles can arise in this case in the neighborhood of the saddle point.
Keywords: parabolic pendulum, Paul trap, rotating paraboloid, internal damping, external damping, friction, resistance, linear stability, Hill’s region, bifurcational diagram, Poincaré section, bounded trajectory, chaos, integrability, nonintegrability, sepa.
Funding agency Grant number
Russian Science Foundation 15-12-00235
Ministry of Education and Science of the Russian Federation 5–100
The work of A.V.Borisov is supported by the program of the Presidium of the Russian Academy of Sciences no. 01 “Fundamental Mathematics and its Applications”. The work of A.A.Kilin (Section 3.2 and Appendix 1) is supported by the RSF grant no. 15-12-00235. The work of I. S.Mamaev is carried out at MIPT under project 5–100 for state support for leading universities of the Russian Federation.
Received: 28.03.2019
Accepted: 06.05.2019
Bibliographic databases:
Document Type: Article
MSC: 37J25, 37J05
Language: English
Citation: Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness”, Regul. Chaotic Dyn., 24:3 (2019), 329–352
Citation in format AMSBIB
\Bibitem{BorKilMam19}
\by Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev
\paper A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 3
\pages 329--352
\mathnet{http://mi.mathnet.ru/rcd481}
\crossref{https://doi.org/10.1134/S1560354719030067}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85066605240}
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    • This publication is cited in the following 1 articles:
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