Regensburg 2025 – wissenschaftliches Programm
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DY: Fachverband Dynamik und Statistische Physik
DY 18: Pattern Formation
DY 18.6: Vortrag
Dienstag, 18. März 2025, 15:15–15:30, H47
Self-similarity in 1 and 2-dimensional cellular automata — •Jens Christian Claussen — University of Birmingham, UK
Cellular automata with a localized single seed initial condition can exhibit deterministic time series with power-law scaling, which led us numerically to the identification of two universality (sub)classes within the Wolfram class IV cellular automata [1], where rule 90 (Sierpinski) and rule 150 are representatives of these classes. The generated time series can be analytically described by a tensorial Fibonacci iteration [2]. An exploration of 2-dimensional outer-totalistic cellular automata showed that fractals with more general one- or two-step self-similarity may exist, including a rule providing a triple replication, and generating a 2-dim spatial Sierpinski pattern.
Here we also consider the more general question what variety of universality classes can be found, eventually extending the dynamics to more general algebraic structures. We show that in the 1-dimensional ECA case of a mod 2 dynamics indeed only the two self-similarity cases represented by rule 90 and rule 150 exist.
[1] J. Nagler and J.C.Claussen (2005) 1/fα spectra in elementary cellular automata and fractal signals, Phys. Rev. E 71, 067103 (2005)
[2] Time evolution of the rule 150 cellular automaton activity from a Fibonacci iteration, J. Math. Phys 49, 062701 (2008)
Keywords: Cellular Autiomata; 1/f noise; Integer Series; Fractals; Universality Classes