Publication result detail

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

ŠREMR, J.

Original Title

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

English Title

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

Type

WoS Article

Original 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.

English 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.

Keywords

second-order nonlinear differential equation; stability; instability; Floquet multiplier; Lyapunov exponent; periodic solution

Key words in English

second-order nonlinear differential equation; stability; instability; Floquet multiplier; Lyapunov exponent; periodic solution

Authors

ŠREMR, J.

Released

03.02.2025

Publisher

SPRINGERNATURE

Location

LONDON

ISBN

0862-7940

Periodical

Applications of Mathematics

Volume

70

Number

1

State

Czech Republic

Pages from

11

Pages to

45

Pages count

35

URL

https://link.springer.com/article/10.21136/AM.2025.0206-24

BibTex

@article{BUT197361,
  author="Jiří {Šremr}",
  title="On Lyapunov stability/instability of equilibria of free damped pendulum with periodically oscillating suspension point",
  journal="Applications of Mathematics",
  year="2025",
  volume="70",
  number="1",
  pages="11--45",
  doi="10.21136/AM.2025.0206-24",
  issn="0862-7940",
  url="https://link.springer.com/article/10.21136/AM.2025.0206-24"
}