Geodesic
On periodic solutions of one second-order differential equation
Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 535-548 
G. V. Demidenko; A. V. Dulepova. On periodic solutions of one second-order differential equation. Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 535-548. http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a8/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In this paper, we investigate the movement of an inverted pendulum, the suspension point of which performs high-frequency oscillations along a line making a small angle with the vertical. We prove that under certain conditions on the function describing the oscillations of the suspension point of the pendulum, a periodic motion of the pendulum arises, and it is asymptotically stable.

[1] Bogolyubov N. N., O nekotorykh statisticheskikh metodakh v matematicheskoi fizike, Izd-vo AN USSR, Kiev, 1945

[2] Bogolyubov N. N., “Teoriya vozmuschenii v nelineinoi mekhanike”, Sb. tr. In-ta stroitelnoi mekhaniki AN USSR, 14, 1950, 9–34

[3] Bogolyubov N. N., Mitropolskii Yu. A., Asimptoticheskie metody teorii nelineinykh kolebanii, Fizmatlit, M., 1963

[4] Burd V. Sh., Metod usredneniya na beskonechnom promezhutke i nekotorye zadachi teorii kolebanii, YarGU, Yaroslavl, 2013

[5] Daletskii Yu. L., Krein M. G., Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970

[6] Demidenko G. V., Matveeva I. I., “Ob ustoichivosti reshenii lineinykh sistem s periodicheskimi koeffitsientami”, Sib. mat. zh., 42:2 (2001), 332–348 | Zbl

[7] Demidenko G. V., Matveeva I. I., “Ob ustoichivosti reshenii kvazilineinykh periodicheskikh sistem differentsialnykh uravnenii”, Sib. mat. zh., 45:6 (2004), 1271–1284 | Zbl

[8] Mitropolskii Yu. A., Khoma G. P., Matematicheskoe obosnovanie asimptoticheskikh metodov nelineinoi mekhaniki, Naukova Dumka, Kiev, 1983