Publication detail

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

ŠREMR, J.

English title

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

Type

journal article in Web of Science

Language

en

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 in English

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

Released

03.02.2025

Publisher

SPRINGERNATURE

Location

LONDON

ISSN

0862-7940

Volume

70

Number

1

Pages from–to

11–45

Pages count

35

BIBTEX


@article{BUT197361,
  author="Jiří {Šremr},
  title="On Lyapunov stability/instability of equilibria of free damped pendulum with periodically oscillating suspension point",
  year="2025",
  volume="70",
  number="1",
  month="February",
  pages="11--45",
  publisher="SPRINGERNATURE",
  address="LONDON",
  issn="0862-7940"
}