Multifractal properties of elementary cellular automata in a discrete wavelet approach of MF-DFA

¹  Departamento de Físico Matemáticas, Universidad Autónoma de San Luis Potosí - Alvaro Obregón 64, 78000 San Luis Potosí, S.L.P., Mexico
²  IPICYT - Instituto Potosino de Investigación Científica y Tecnológica - Camino a la presa San José 2055, 78216, San Luis Potosí, SLP, Mexico

Corresponding author: hcr@ipicyt.edu.mx

Received: 28 March 2009
Accepted: 13 July 2009

Abstract

In 2005, Nagler and Claussen (Phys. Rev. E, 71 (2005) 067103) investigated the time series of the elementary cellular automata (ECA) for possible (multi)fractal behavior. They eliminated the polynomial background at^b through the direct fitting of the polynomial coefficients a and b. We here reconsider their work eliminating the polynomial trend by means of the multifractal-based detrended fluctuation analysis (MF-DFA) in which the wavelet multiresolution property is employed to filter out the trend in a more speedy way than the direct polynomial fitting and also with respect to the wavelet transform modulus maxima (WTMM) procedure. In the algorithm, the discrete fast wavelet transform is used to calculate the trend as a local feature that enters the so-called details signal. We illustrate our result for three representative ECA rules: 90, 105, and 150. We confirm their multifractal behavior and provide our results for the scaling parameters.