A computer-assisted proof of Σ3-chaos in the forced damped pendulum equation

Balázs Bánhelyi, T. Csendes, Barnabas M. Garay, László Hatvani

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

The present paper is devoted to studying Hubbard's pendulum equation ẍ + 10-1 ẋ + sin(x) = cos(t). Using rigorous/interval methods of computation, the main assertion of Hubbard on chaos properties of the induced dynamics is raised from the level of experimentally observed facts to the level of a theorem completely proved. A special family of solutions is shown to be chaotic in the sense that, on consecutive time intervals (2kπ, 2(k + 1)π) (k ∈ ℤ), individual members of the family can freely "choose" between the following possibilities: the pendulum crosses the bottom position exactly once clockwise or does not cross the bottom position at all or crosses the bottom position exactly once counterclockwise. The proof follows the topological index/degree approach by Mischaikow, Mrozek, and Zgliczynski. The new feature of this paper is a definition of the transition graph for which the periodic orbit lemma - the key technical result of the approach mentioned above - turns out to be a consequence of Brouwer's fixed point theorem. The role of wholly automatic versus "trial-and-error with human overheads" computer procedures in detecting chaos is also discussed.

Original languageEnglish
Pages (from-to)843-867
Number of pages25
JournalSIAM Journal on Applied Dynamical Systems
Volume7
Issue number3
DOIs
Publication statusPublished - 2008

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Computer-assisted Proof
Pendulum
Pendulums
Chaos theory
Damped
Chaos
Coversine
sin(-x)
Brouwer Fixed Point Theorem
Anticlockwise
Clockwise
Topological Index
Interval Methods
Orbits
Trial and error
Assertion
Periodic Orbits
Consecutive
Lemma
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Keywords

  • Σ-chaos
  • Computer-assisted proof
  • Forced damped pendulum
  • Interval arithmetic
  • Transition graph

ASJC Scopus subject areas

  • Analysis
  • Modelling and Simulation

Cite this

A computer-assisted proof of Σ3-chaos in the forced damped pendulum equation. / Bánhelyi, Balázs; Csendes, T.; Garay, Barnabas M.; Hatvani, László.

In: SIAM Journal on Applied Dynamical Systems, Vol. 7, No. 3, 2008, p. 843-867.

Research output: Contribution to journalArticle

Bánhelyi, Balázs ; Csendes, T. ; Garay, Barnabas M. ; Hatvani, László. / A computer-assisted proof of Σ3-chaos in the forced damped pendulum equation. In: SIAM Journal on Applied Dynamical Systems. 2008 ; Vol. 7, No. 3. pp. 843-867.
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