On the stability of the state of equilibrium of an oscillator with infinitely great frequency of proper oscillations

Authors

  • Yuri N. Bibikov
  • Veronika R. Bukaty

Abstract

A problem of the stability of the state of equilibrium of an oscillator with infinitely great frequency of proper oscillations under periodic in time perturbations, is considered. It is shown that in general case the problem can be solved by consideration of the linear approximation of the perturbation only. In singular case a procedure of the construction of a constant whose sign defines the character of the stability or instability, is presented. Namely, if this constant is negative then the state of equilibrium is asymptotically stable; in the opposite case it is unstable.

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References

1. Пуанкаре А. О кривых, определяемых дифференциальными уравнениями. М.; Л.: ГИТТЛ, 1947.

2. Ляпунов А.М. Общая задача об устойчивости движения. Собр. соч. в пяти томах. Т. 2. М.; Л.: Изд-во АН СССР, 1956. С. 7-267.

3. Ляпунов А.М. Исследование одного из особенных случаев задачи об устойчивости движения. Собр. соч. в пяти томах. Т. 2. М.; Л.: Изд-во АН СССР, 1956. С. 272-331.

4. Бибиков Ю.Н., Букаты В. Р., Трушина Н.В. Об устойчивости положения равновесия при периодических возмущениях осциллятора со степенной восстанавливающей силой с рациональным показателем // Прикладная математика и механика. 2016. Т. 80. Вып. 6. С. 626-636.

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Published

2020-08-16

How to Cite

Bibikov, Y. N., & Bukaty, V. R. (2020). On the stability of the state of equilibrium of an oscillator with infinitely great frequency of proper oscillations. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 6(3), 394–398. Retrieved from https://math-mech-astr-journal.spbu.ru/article/view/8395

Issue

Vol. 6 No. 3 (2019)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.