### 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 language | English |
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Article number | 062701 |

Journal | Journal of Mathematical Physics |

Volume | 49 |

Issue number | 6 |

DOIs | |

Publication status | Published - 10 Jul 2008 |

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### 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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**Time evolution of the rule 150 cellular automaton activity from a Fibonacci iteration.** / Claussen, Jens Christian.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Time evolution of the rule 150 cellular automaton activity from a Fibonacci iteration

AU - Claussen, Jens Christian

N1 - © 2008 American Institute of Physics. Journal of Mathematical Physics 49, 062701 (2008); doi: 10.1063/1.2939398

PY - 2008/7/10

Y1 - 2008/7/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=46449105145&partnerID=8YFLogxK

UR - https://aip.scitation.org/doi/10.1063/1.2939398

U2 - 10.1063/1.2939398

DO - 10.1063/1.2939398

M3 - Article

AN - SCOPUS:46449105145

VL - 49

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 6

M1 - 062701

ER -