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Nelin. Dinam., 2019, Volume 15, Number 3, Pages 351–363 (Mi nd665)  

Mathematical problems of nonlinearity

Vibrational Stability of Periodic Solutions of the Liouville Equations

E. V. Vetchanina, E. A. Mikishaninab

a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
b Chuvash State University, Moskovskii prosp. 15, Cheboksary, 428015 Russia

Abstract: The dynamics of a body with a fixed point, variable moments of inertia and internal rotors are considered. A stability analysis of permanent rotations and periodic solutions of the system is carried out. In some simplest cases the stability analysis is reduced to investigating the stability of the zero solution of Hills equation. It is shown that by periodically changing the moments of inertia it is possible to stabilize unstable permanent rotations of the system. In addition, stable dynamical regimes can lose stability due to a parametric resonance. It is shown that, as the oscillation frequency of the moments of inertia increases, the dynamics of the system becomes close to an integrable one.

Keywords: Liouville equations, Euler Poisson equations, Hills equation, Mathieu equation, parametric resonance, vibrostabilization, Euler Poinsot case, Joukowski Volterra case

Funding Agency Grant Number
Russian Science Foundation 18-71-00111
This work was supported by the Russian Science Foundation under grant 18-71-00111.


DOI: https://doi.org/10.20537/nd190312

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MSC: 70E17, 70J40
Received: 17.07.2019
Accepted:23.09.2019

Citation: E. V. Vetchanin, E. A. Mikishanina, “Vibrational Stability of Periodic Solutions of the Liouville Equations”, Nelin. Dinam., 15:3 (2019), 351–363

Citation in format AMSBIB
\Bibitem{VetMik19}
\by E. V. Vetchanin, E. A. Mikishanina
\paper Vibrational Stability of Periodic Solutions of the Liouville Equations
\jour Nelin. Dinam.
\yr 2019
\vol 15
\issue 3
\pages 351--363
\mathnet{http://mi.mathnet.ru/nd665}
\crossref{https://doi.org/10.20537/nd190312}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4021375}


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