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Regul. Chaotic Dyn., 2020, Volume 25, Issue 4, Pages 401–410 (Mi rcd1073)  

The Method of Averaging for the Kapitza – Whitney Pendulum

Ivan Yu. Polekhin

Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, 119991 Moscow, Russia

Abstract: A generalization of the classical Kapitza pendulum is considered: an inverted planar mathematical pendulum with a vertically vibrating pivot point in a time-periodic horizontal force field. We study the existence of forced oscillations in the system. It is shown that there always exists a periodic solution along which the rod of the pendulum never becomes horizontal, i.e., the pendulum never falls, provided the period of vibration and the period of horizontal force are commensurable. We also present a sufficient condition for the existence of at least two different periodic solutions without falling. We show numerically that there exist stable periodic solutions without falling.

Keywords: averaging, Kapitza’s pendulum, Whitney’s pendulum, forced oscillations, averaging on an infinite interval

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation MK-1826.2020.1
This work has been supported by the Grant of the President of the Russian Federation (Project MK-1826.2020.1).


DOI: https://doi.org/10.1134/S1560354720040073

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MSC: 34C29, 70K40
Received: 04.06.2020
Accepted:09.07.2020
Language:

Citation: Ivan Yu. Polekhin, “The Method of Averaging for the Kapitza – Whitney Pendulum”, Regul. Chaotic Dyn., 25:4 (2020), 401–410

Citation in format AMSBIB
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\by Ivan Yu. Polekhin
\paper The Method of Averaging for the Kapitza – Whitney Pendulum
\jour Regul. Chaotic Dyn.
\yr 2020
\vol 25
\issue 4
\pages 401--410
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\crossref{https://doi.org/10.1134/S1560354720040073}
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