### Abstract

A rule-dynamical system is constructed using a simple one-dimensional cellular automata (CA), and a method to construct a generalized rule-dynamical system is also introduced. The various synchronization phenomena in one-dimensional elementary CA we observe are systematically studied. In the case of 'autonomous rule dynamics', rule changes are coupled to the temporal variation of the density, and it is shown that global statistical aspects of the dynamical attractor can be determined uniquely, independent of the initial conditions. Next, the response of CA to forced rule variation is studied. This is called 'forced rule dynamics'. Under periodic forcing, density variations exhibit a wide variety of synchronization phenomena which depend not only on the class of rules employed, but also on the temporal ordering of the rules. If all rules are class 3, density synchronization occurs in all cases. For all density-synchronized cases, the density variation can be divided into two components. One oscillatory-type component can be characterized according to the constituent forcing rules and their periodic structure, while the other fluctuation-type component exhibits a standard deviation which has 1/√N system size dependence. This suggests that the rule variation completely determines the density variation as N→∞. The biological significance of the density synchronization is briefly discussed in relation to information processing in central nervous systems.

Original language | English |
---|---|

Pages (from-to) | 729-748 |

Number of pages | 20 |

Journal | Progress of Theoretical Physics |

Volume | 102 |

Issue number | 4 |

Publication status | Published - 1999 Oct |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Progress of Theoretical Physics*,

*102*(4), 729-748.

**Synchronization phenomena in rule dynamical systems.** / Kim, Song Ju; Aizawa, Yoji.

Research output: Contribution to journal › Article

*Progress of Theoretical Physics*, vol. 102, no. 4, pp. 729-748.

}

TY - JOUR

T1 - Synchronization phenomena in rule dynamical systems

AU - Kim, Song Ju

AU - Aizawa, Yoji

PY - 1999/10

Y1 - 1999/10

N2 - A rule-dynamical system is constructed using a simple one-dimensional cellular automata (CA), and a method to construct a generalized rule-dynamical system is also introduced. The various synchronization phenomena in one-dimensional elementary CA we observe are systematically studied. In the case of 'autonomous rule dynamics', rule changes are coupled to the temporal variation of the density, and it is shown that global statistical aspects of the dynamical attractor can be determined uniquely, independent of the initial conditions. Next, the response of CA to forced rule variation is studied. This is called 'forced rule dynamics'. Under periodic forcing, density variations exhibit a wide variety of synchronization phenomena which depend not only on the class of rules employed, but also on the temporal ordering of the rules. If all rules are class 3, density synchronization occurs in all cases. For all density-synchronized cases, the density variation can be divided into two components. One oscillatory-type component can be characterized according to the constituent forcing rules and their periodic structure, while the other fluctuation-type component exhibits a standard deviation which has 1/√N system size dependence. This suggests that the rule variation completely determines the density variation as N→∞. The biological significance of the density synchronization is briefly discussed in relation to information processing in central nervous systems.

AB - A rule-dynamical system is constructed using a simple one-dimensional cellular automata (CA), and a method to construct a generalized rule-dynamical system is also introduced. The various synchronization phenomena in one-dimensional elementary CA we observe are systematically studied. In the case of 'autonomous rule dynamics', rule changes are coupled to the temporal variation of the density, and it is shown that global statistical aspects of the dynamical attractor can be determined uniquely, independent of the initial conditions. Next, the response of CA to forced rule variation is studied. This is called 'forced rule dynamics'. Under periodic forcing, density variations exhibit a wide variety of synchronization phenomena which depend not only on the class of rules employed, but also on the temporal ordering of the rules. If all rules are class 3, density synchronization occurs in all cases. For all density-synchronized cases, the density variation can be divided into two components. One oscillatory-type component can be characterized according to the constituent forcing rules and their periodic structure, while the other fluctuation-type component exhibits a standard deviation which has 1/√N system size dependence. This suggests that the rule variation completely determines the density variation as N→∞. The biological significance of the density synchronization is briefly discussed in relation to information processing in central nervous systems.

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UR - http://www.scopus.com/inward/citedby.url?scp=0033234855&partnerID=8YFLogxK

M3 - Article

VL - 102

SP - 729

EP - 748

JO - Progress of Theoretical Physics

JF - Progress of Theoretical Physics

SN - 0033-068X

IS - 4

ER -