A Computer-Assisted Proof of Σ3-Chaos in the Forced Damped Pendulum Equation

نویسندگان

  • Balázs Bánhelyi
  • Tibor Csendes
  • Barnabas M. Garay
  • László Hatvani
چکیده

The present paper is devoted to studying Hubbard’s pendulum equation ẍ+ 10ẋ+ sin(x) = cos(t) . By rigorous/interval methods of computation, the main assertion of Hubbard on chaos properties of the induced dynamics is lifted from the level of experimentally observed facts to the level of a theorem completely proved. A distinguished 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 either 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 novelty is a definition of the transition graph for which the periodic orbit lemma, the key technical result of the approach aforementioned, turns out to be a consequence of Brouwer’s fixed point theorem. The role of wholly automatic versus ‘trial and error with human overhead’ computer procedures in detecting chaos is also discussed.

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عنوان ژورنال:
  • SIAM J. Applied Dynamical Systems

دوره 7  شماره 

صفحات  -

تاریخ انتشار 2008