2013/12/23 by Vladimir Garcı́a-Morales, Garcia-Morales, Vladimir
Computer Science · Mathematics · Physics and Astronomy · #Cellular Automata and Applications #Cellular Automata and Lattice Gases (nlin.CG) #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.1312.6534
openalex publication_date 2013/12/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A digit function is presented which provides the ith-digit in base p of any real number x. By means of this function, formulated within B-calculus, the local, nonlocal and global dynamical behaviors of cellular automata (CAs) are systematically explored and universal maps are derived for the three levels of description. None of the maps contain any freely adjustable parameter and they are valid for any number of symbols in the alphabet p and neighborhood range ρ. A discrete general method to approximate any real continuous map in the unit interval by a CA on the rational numbers ℚ (Diophantine approximation) is presented. This result leads to establish a correspondence between the qualitative behavior found in bifurcation diagrams of real nonlinear maps and the Wolfram classes of CAs. The method is applied to the logistic map, for which a logistic CA is derived. The period doubling cascade into chaos is interpreted as a sequence of global cellular automata of Wolfram's class 2 leading to Class 3 aperiodic behavior. Class 4 behavior is also found close to the period-3 orbits.