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What is Fractal Dimension

Handbook of Research on Advanced Intelligent Control Engineering and Automation
Is the most basic property of a chaotic (strange) attractor. The dimension of a strange attractor is a measure of its geometric scaling properties or its “complexity”.
Published in Chapter:
Further Investigation of the Period-Three Route to Chaos in the Passive Compass-Gait Biped Model
Hassène Gritli (Institut Supérieur des Etudes Technologiques de Kélibia, Tunisia), Nahla Khraief (Ecole Supérieure de Technologie et d'Informatique, Tunisia), and Safya Belghith (Ecole Nationale d'Ingénieurs de Tunis, Tunisia)
DOI: 10.4018/978-1-4666-7248-2.ch010
Abstract
This chapter presents further investigations into the period-three route to chaos exhibited in the passive dynamic walking of the compass-gait biped robot as it goes down an inclined surface. This discovered kind of route in the passive bipedal locomotion was found to coexist with the conventional period-one passive hybrid limit cycle. The further analysis on the period-three route chaos is realized by means of the Lyapunov exponents and the fractal Lyapunov dimension. Numerical computation method of these two tools is presented. The first return Poincaré map of the chaotic attractor and its basin of attraction are presented. Furthermore, the further study of the period-three passive gait is realized. The analysis of the period-three hybrid limit cycle is given. The balance between the potential energy and the kinetic energy of the biped robot is illustrated. In addition, the basin of attraction of the period-three passive gait is also presented.
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More Results
Entropy, Chaos, and Language
Ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured.
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The Interpretation of Vernacular Architecture through Fractal Models: In the Case of Iran, Masouleh Village
Special ratio for evaluating the density and visual compactness, which is based on mathematical algorithms.
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