Time evolution of the rule 150 cellular automaton activity from a Fibonacci iteration

Research output: Contribution to journalArticle

Abstract

The rule 150 cellular automaton is a remarkable discrete dynamical system, as it shows 1 fα spectra if started from a single seed [J. Nagler and J. C. Claussen, Phys. Rev. E 71, 067103 (2005)]. Despite its simplicity, a feasible solution for its time behavior is not obvious. Its self-similarity does not follow a one-step iteration like other elementary cellular automata. Here it is shown how its time behavior can be solved as a two-step vectorial, or string, iteration, which can be viewed as a generalization of Fibonacci iteration generating the time series from a sequence of vectors of increasing length. This allows us to compute the total activity time series more efficiently than by simulating the whole spatiotemporal process or even by using the closed expression. The results are further extended to the generalization of rule 150 to the two-dimensional case and to Bethe lattices and the relation to corresponding integer sequences is discussed.

Original languageEnglish
Article number062701
JournalJournal of Mathematical Physics
Volume49
Issue number6
DOIs
Publication statusPublished - 10 Jul 2008

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cellular automata
Cellular Automata
iteration
Iteration
Time series
Spatio-temporal Process
Integer Sequences
Bethe Lattice
Discrete Dynamical Systems
Self-similarity
Simplicity
Strings
dynamical systems
integers
seeds
Closed
strings
Generalization

Bibliographical note

© 2008 American Institute of Physics. Journal of Mathematical Physics 49, 062701 (2008); doi: 10.1063/1.2939398

Cite this

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abstract = "The rule 150 cellular automaton is a remarkable discrete dynamical system, as it shows 1 fα spectra if started from a single seed [J. Nagler and J. C. Claussen, Phys. Rev. E 71, 067103 (2005)]. Despite its simplicity, a feasible solution for its time behavior is not obvious. Its self-similarity does not follow a one-step iteration like other elementary cellular automata. Here it is shown how its time behavior can be solved as a two-step vectorial, or string, iteration, which can be viewed as a generalization of Fibonacci iteration generating the time series from a sequence of vectors of increasing length. This allows us to compute the total activity time series more efficiently than by simulating the whole spatiotemporal process or even by using the closed expression. The results are further extended to the generalization of rule 150 to the two-dimensional case and to Bethe lattices and the relation to corresponding integer sequences is discussed.",
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Time evolution of the rule 150 cellular automaton activity from a Fibonacci iteration. / Claussen, Jens Christian.

In: Journal of Mathematical Physics, Vol. 49, No. 6, 062701, 10.07.2008.

Research output: Contribution to journalArticle

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