On Lyapunov stability/instability of equilibria of free damped pendulum with periodically oscillating suspension point

Jiří Šremr

Applications of Mathematics (2025)

  • Issue: 1, page 11-45
  • ISSN: 0862-7940

Abstract

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We discuss Lyapunov stability/instability of both lower and upper equilibria of free damped pendulum with periodically oscillating suspension point. We recall the results of Bogolyubov and Kapitza, provide new effective criteria of stability/instability of the equilibria of pendulum equation, and give the exact and complete proofs. The criteria obtained are formulated in terms of positivity/negativity of Green's functions of the periodic boundary value problems for linearized equations. Furthermore, we show that if both lower and upper equilibria are stable, then the pendulum considered may possess a periodic motion that corresponds to the ``quasistatic solution'' of Bogolyubov as well as to the ``quasistatic balance'' of Kapitza.

How to cite

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Šremr, Jiří. "On Lyapunov stability/instability of equilibria of free damped pendulum with periodically oscillating suspension point." Applications of Mathematics (2025): 11-45. <http://eudml.org/doc/299929>.

@article{Šremr2025,
abstract = {We discuss Lyapunov stability/instability of both lower and upper equilibria of free damped pendulum with periodically oscillating suspension point. We recall the results of Bogolyubov and Kapitza, provide new effective criteria of stability/instability of the equilibria of pendulum equation, and give the exact and complete proofs. The criteria obtained are formulated in terms of positivity/negativity of Green's functions of the periodic boundary value problems for linearized equations. Furthermore, we show that if both lower and upper equilibria are stable, then the pendulum considered may possess a periodic motion that corresponds to the ``quasistatic solution'' of Bogolyubov as well as to the ``quasistatic balance'' of Kapitza.},
author = {Šremr, Jiří},
journal = {Applications of Mathematics},
keywords = {second-order nonlinear differential equation; stability; instability; Floquet multiplier; Lyapunov exponent; periodic solution},
language = {eng},
number = {1},
pages = {11-45},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Lyapunov stability/instability of equilibria of free damped pendulum with periodically oscillating suspension point},
url = {http://eudml.org/doc/299929},
year = {2025},
}

TY - JOUR
AU - Šremr, Jiří
TI - On Lyapunov stability/instability of equilibria of free damped pendulum with periodically oscillating suspension point
JO - Applications of Mathematics
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 11
EP - 45
AB - We discuss Lyapunov stability/instability of both lower and upper equilibria of free damped pendulum with periodically oscillating suspension point. We recall the results of Bogolyubov and Kapitza, provide new effective criteria of stability/instability of the equilibria of pendulum equation, and give the exact and complete proofs. The criteria obtained are formulated in terms of positivity/negativity of Green's functions of the periodic boundary value problems for linearized equations. Furthermore, we show that if both lower and upper equilibria are stable, then the pendulum considered may possess a periodic motion that corresponds to the ``quasistatic solution'' of Bogolyubov as well as to the ``quasistatic balance'' of Kapitza.
LA - eng
KW - second-order nonlinear differential equation; stability; instability; Floquet multiplier; Lyapunov exponent; periodic solution
UR - http://eudml.org/doc/299929
ER -

References

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