Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics]
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 Nelin. Dinam., 2014, Volume 10, Number 4, Pages 465–472 (Mi nd457)

Examples of topological approach to the problem of inverted pendulum with moving pivot point

Ivan Yu. Polekhin

Lomonosov Moscow State University, GSP-1, Leninskie Gory 1, Moscow, 119991, Russia

Abstract: Two examples concerning application of topology in study of dynamics of inverted plain mathematical pendulum with pivot point moving along horizontal straight line are considered. The first example is an application of the Ważewski principle to the problem of existence of solution without falling. The second example is a proof of existence of periodic solution in the same system when law of motion is periodic as well. Moreover, in the second case it is also shown that along obtained periodic solution pendulum never becomes horizontal (falls).

Keywords: inverted pendulum, Lefschetz–Hopf theorem, Ważewski principle, periodic solution.

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Bibliographic databases:
UDC: 51-72
MSC: 70K40, 70G40, 37B55
Revised: 19.11.2014

Citation: Ivan Yu. Polekhin, “Examples of topological approach to the problem of inverted pendulum with moving pivot point”, Nelin. Dinam., 10:4 (2014), 465–472

Citation in format AMSBIB
\Bibitem{Pol14} \by Ivan~Yu.~Polekhin \paper Examples of topological approach to the problem of inverted pendulum with moving pivot point \jour Nelin. Dinam. \yr 2014 \vol 10 \issue 4 \pages 465--472 \mathnet{http://mi.mathnet.ru/nd457} \zmath{https://zbmath.org/?q=an:06430346} 

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This publication is cited in the following articles:
1. S. V. Bolotin, V. V. Kozlov, “Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney's inverted pendulum problem”, Izv. Math., 79:5 (2015), 894–901
2. I. Polekhin, “On motions without falling of an inverted pendulum with dry friction”, J. Geom. Mech., 10:4 (2018), 411–417
3. I. Polekhin, “On topological obstructions to global stabilization of an inverted pendulum”, Syst. Control Lett., 113 (2018), 31–35
4. R. Srzednicki, “On periodic solutions in the Whitney's inverted pendulum problem”, Discret. Contin. Dyn. Syst.-Ser. S, 12:7 (2019), 2127–2141
5. I. Yu. Polekhin, “Remarks on Forced Oscillations in Some Systems with Gyroscopic Forces”, Rus. J. Nonlin. Dyn., 16:2 (2020), 343–353
6. Ivan Yu. Polekhin, “The Method of Averaging for the Kapitza – Whitney Pendulum”, Regul. Chaotic Dyn., 25:4 (2020), 401–410
7. Ivan Yu. Polekhin, “Some Results on the Existence of Forced Oscillations in Mechanical Systems”, Proc. Steklov Inst. Math., 310 (2020), 250–261
8. N. A. Stepanov, M. A. Skvortsov, “Lyapunov exponent for Whitney's problem with random drive”, JETP Letters, 112:6 (2020), 376–382
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