### Abstract

We show the appearance of emergent hierarchical structure by a generalized system based on hyper-dilation (P.-Y. Gunji, S. Toyoda, Biosystem 38 (1996) 127-133), that is a new approach for complex systems. Hyper-dilation comprises a measurement problem hidden in any formal system. In hyper-dilation, the measurement process is expressed as the interface between an infinite lattice, which is an infinite loop of a rule, and a finite lattice, which is a tree structure of a rule. Hyper-dilation shows two intrinsic properties, one is degeneracy of a Cantor set in return map, and the other is generation of 1/f noise. We present various versions of hyper-dilation and suggest that to generate 1/f noise, it is required that the rules of the system are changed through a coherent process and that the system is closed for information. From these two properties of hyper-dilation, we discuss that the degeneracy is regarded as the model of emergence of a new hierarchy, and in hyper-dilation, actually we show the motion of a system that can be regarded as the appearance of a new hierarchy.

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

Pages (from-to) | 153-178 |

Number of pages | 26 |

Journal | Applied Mathematics and Computation |

Volume | 104 |

Issue number | 2-3 |

Publication status | Published - 1999 Sep |

Externally published | Yes |

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### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Numerical Analysis

### Cite this

*Applied Mathematics and Computation*,

*104*(2-3), 153-178.

**Appearance of hierarchical structure in hyper-dilation model : Model of generalized measurement process.** / Toyoda, Shin'ichi; Gunji, Yukio.

Research output: Contribution to journal › Article

*Applied Mathematics and Computation*, vol. 104, no. 2-3, pp. 153-178.

}

TY - JOUR

T1 - Appearance of hierarchical structure in hyper-dilation model

T2 - Model of generalized measurement process

AU - Toyoda, Shin'ichi

AU - Gunji, Yukio

PY - 1999/9

Y1 - 1999/9

N2 - We show the appearance of emergent hierarchical structure by a generalized system based on hyper-dilation (P.-Y. Gunji, S. Toyoda, Biosystem 38 (1996) 127-133), that is a new approach for complex systems. Hyper-dilation comprises a measurement problem hidden in any formal system. In hyper-dilation, the measurement process is expressed as the interface between an infinite lattice, which is an infinite loop of a rule, and a finite lattice, which is a tree structure of a rule. Hyper-dilation shows two intrinsic properties, one is degeneracy of a Cantor set in return map, and the other is generation of 1/f noise. We present various versions of hyper-dilation and suggest that to generate 1/f noise, it is required that the rules of the system are changed through a coherent process and that the system is closed for information. From these two properties of hyper-dilation, we discuss that the degeneracy is regarded as the model of emergence of a new hierarchy, and in hyper-dilation, actually we show the motion of a system that can be regarded as the appearance of a new hierarchy.

AB - We show the appearance of emergent hierarchical structure by a generalized system based on hyper-dilation (P.-Y. Gunji, S. Toyoda, Biosystem 38 (1996) 127-133), that is a new approach for complex systems. Hyper-dilation comprises a measurement problem hidden in any formal system. In hyper-dilation, the measurement process is expressed as the interface between an infinite lattice, which is an infinite loop of a rule, and a finite lattice, which is a tree structure of a rule. Hyper-dilation shows two intrinsic properties, one is degeneracy of a Cantor set in return map, and the other is generation of 1/f noise. We present various versions of hyper-dilation and suggest that to generate 1/f noise, it is required that the rules of the system are changed through a coherent process and that the system is closed for information. From these two properties of hyper-dilation, we discuss that the degeneracy is regarded as the model of emergence of a new hierarchy, and in hyper-dilation, actually we show the motion of a system that can be regarded as the appearance of a new hierarchy.

UR - http://www.scopus.com/inward/record.url?scp=0347596681&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347596681&partnerID=8YFLogxK

M3 - Article

VL - 104

SP - 153

EP - 178

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 2-3

ER -