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On the existence of periodic motions of the excited inverted pendulum by elementary methods

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Abstract

Using purely elementary methods, necessary and sufficient conditions are given for the existence of 2T-periodic and 4T-periodic solutions around the upper equilibrium of the mathematical pendulum when the suspension point is vibrating with period 2T. The equation of the motion is of the form

$$\ddot{\theta}-\frac{1}{l}(g+a(t)) \theta=0,$$

where l, g are constants and

$$a(t) := \begin{cases} A &\text{if } 2kT\leq t < (2k+1)T,\\ -A &\text{if } (2k+1)T\leq t < (2k+2)T,\end{cases}\quad (k=0,1,\dots);$$

A, T are positive constants. The exact stability zones for the upper equilibrium are presented.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720, Szeged, Hungary

    L. Csizmadia & L. Hatvani

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  1. L. Csizmadia
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  2. L. Hatvani
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Correspondence to L. Hatvani.

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Supported by the Hungarian National Foundation for Scientific Research (OTKA) K109782.

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Csizmadia, L., Hatvani, L. On the existence of periodic motions of the excited inverted pendulum by elementary methods. Acta Math. Hungar. 155, 298–312 (2018). https://doi.org/10.1007/s10474-018-0835-6

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  • DOI: https://doi.org/10.1007/s10474-018-0835-6

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