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.
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.
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
\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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Linking options:
https://www.mathnet.ru/eng/rcd481
https://www.mathnet.ru/eng/rcd/v24/i3/p329
This publication is cited in the following 2 articles:
Ivan A. Bizyaev, Ivan S. Mamaev, “Nonlinear Dynamics of a Roller Bicycle”, Regul. Chaotic Dyn., 29:5 (2024), 728–750
D. D. Kulminskiy, M. V. Malyshev, “Experimental Study of the Accuracy of a Pendulum Clock with a Vibrating Pivot Point”, Rus. J. Nonlin. Dyn., 20:4 (2024), 553–563
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