### 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 language | English |
---|---|

Pages (from-to) | 843-867 |

Number of pages | 25 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 7 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2008 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Modelling and Simulation

### Cite this

_{3}-chaos in the forced damped pendulum equation.

*SIAM Journal on Applied Dynamical Systems*,

*7*(3), 843-867. https://doi.org/10.1137/070695599

**A computer-assisted proof of Σ _{3}-chaos in the forced damped pendulum equation.** / Bánhelyi, Balázs; Csendes, T.; Garay, Barnabas M.; Hatvani, László.

Research output: Contribution to journal › Article

_{3}-chaos in the forced damped pendulum equation',

*SIAM Journal on Applied Dynamical Systems*, vol. 7, no. 3, pp. 843-867. https://doi.org/10.1137/070695599

_{3}-chaos in the forced damped pendulum equation. SIAM Journal on Applied Dynamical Systems. 2008;7(3):843-867. https://doi.org/10.1137/070695599

}

TY - JOUR

T1 - A computer-assisted proof of Σ3-chaos in the forced damped pendulum equation

AU - Bánhelyi, Balázs

AU - Csendes, T.

AU - Garay, Barnabas M.

AU - Hatvani, László

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Σ-chaos

KW - Computer-assisted proof

KW - Forced damped pendulum

KW - Interval arithmetic

KW - Transition graph

UR - http://www.scopus.com/inward/record.url?scp=55849114347&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=55849114347&partnerID=8YFLogxK

U2 - 10.1137/070695599

DO - 10.1137/070695599

M3 - Article

AN - SCOPUS:55849114347

VL - 7

SP - 843

EP - 867

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 3

ER -