Open AccessEstimating Attractor Dimension on the Nonlinear Pendulum Time Series
Author(s) -
Luiz Franca,
Marcelo A. Savi
Publication year - 2001
Publication title -
revista brasileira de ciências mecânicas
Language(s) - English
Resource type - Journals
ISSN - 0100-7386
DOI - 10.1590/s0100-73862001000400004
Subject(s) - attractor , correlation dimension , chaotic , dimension (graph theory) , series (stratigraphy) , phase space , dynamical systems theory , time series , pendulum , mathematics , nonlinear system , state space , noise (video) , statistical physics , computer science , mathematical analysis , fractal dimension , physics , statistics , artificial intelligence , fractal , image (mathematics) , paleontology , quantum mechanics , pure mathematics , biology , thermodynamics
Chaotic dynamical systems exhibit trajectories in their phase space that converges to a strange attractor. The strangeness of the chaotic attractor is associated with its dimension in which instance it is described by a noninteger dimension. This contribution presents an overview of the main definitions of dimension discussing their evaluation from time series employing the correlation and the generalized dimension. The investigation is applied to the nonlinear pendulum where signals are generated by numerical integration of the mathematical model, selecting a single variable of the system as a time series. In order to simulate experimental data sets, a random noise is introduced in the time series. State space reconstruction and the determination of attractor dimensions are carried out regarding periodic and chaotic signals. Results obtained from time series analyses are compared with a reference value obtained from the analysis of mathematical model, estimating noise sensitivity. This procedure allows one to identify the best techniques to be applied in the analysis of experimental data