### Abstract

The large deviation properties of the Lempel-Ziv complexity are studied using a one-dimensional non-hyperbolic chaos map called the "modified Bernoulli map", where the transition between stationary and non-stationary chaos is clearly observed. The upper limit of the Lempel-Ziv complexity in the non-stationary regime is theoretically evaluated, and the relationship between the algorithmic complexity and the Lempel-Ziv complexity is discussed. Non-stationary processes are universal phenomena in non-hyperbolic systems, and they are usually characterized by an infinite ergodic measure and intrinsic long time tails, such as 1/f^{ν} spectral fluctuations. It is shown that the Lempel-Ziv complexity obeys universal scaling laws and that the Lempel-Ziv complexity has the L^{1}-function property, which guarantees the Darling-Kac-Aaronson theorem for an infinite ergodic system. The most striking result is that the maximum diversity appears at the transition point from stationary chaos to non-stationary chaos where the exact 1/ f spectral process is generated.

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

Pages (from-to) | 503-515 |

Number of pages | 13 |

Journal | Progress of Theoretical Physics |

Volume | 116 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2006 Sep |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Progress of Theoretical Physics*,

*116*(3), 503-515. https://doi.org/10.1143/PTP.116.503

**The Lempel-Ziv complexity of non-stationary chaos in infinite ergodic cases.** / Shinkai, Soya; Aizawa, Yoji.

Research output: Contribution to journal › Article

*Progress of Theoretical Physics*, vol. 116, no. 3, pp. 503-515. https://doi.org/10.1143/PTP.116.503

}

TY - JOUR

T1 - The Lempel-Ziv complexity of non-stationary chaos in infinite ergodic cases

AU - Shinkai, Soya

AU - Aizawa, Yoji

PY - 2006/9

Y1 - 2006/9

N2 - The large deviation properties of the Lempel-Ziv complexity are studied using a one-dimensional non-hyperbolic chaos map called the "modified Bernoulli map", where the transition between stationary and non-stationary chaos is clearly observed. The upper limit of the Lempel-Ziv complexity in the non-stationary regime is theoretically evaluated, and the relationship between the algorithmic complexity and the Lempel-Ziv complexity is discussed. Non-stationary processes are universal phenomena in non-hyperbolic systems, and they are usually characterized by an infinite ergodic measure and intrinsic long time tails, such as 1/fν spectral fluctuations. It is shown that the Lempel-Ziv complexity obeys universal scaling laws and that the Lempel-Ziv complexity has the L1-function property, which guarantees the Darling-Kac-Aaronson theorem for an infinite ergodic system. The most striking result is that the maximum diversity appears at the transition point from stationary chaos to non-stationary chaos where the exact 1/ f spectral process is generated.

AB - The large deviation properties of the Lempel-Ziv complexity are studied using a one-dimensional non-hyperbolic chaos map called the "modified Bernoulli map", where the transition between stationary and non-stationary chaos is clearly observed. The upper limit of the Lempel-Ziv complexity in the non-stationary regime is theoretically evaluated, and the relationship between the algorithmic complexity and the Lempel-Ziv complexity is discussed. Non-stationary processes are universal phenomena in non-hyperbolic systems, and they are usually characterized by an infinite ergodic measure and intrinsic long time tails, such as 1/fν spectral fluctuations. It is shown that the Lempel-Ziv complexity obeys universal scaling laws and that the Lempel-Ziv complexity has the L1-function property, which guarantees the Darling-Kac-Aaronson theorem for an infinite ergodic system. The most striking result is that the maximum diversity appears at the transition point from stationary chaos to non-stationary chaos where the exact 1/ f spectral process is generated.

UR - http://www.scopus.com/inward/record.url?scp=33846267348&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846267348&partnerID=8YFLogxK

U2 - 10.1143/PTP.116.503

DO - 10.1143/PTP.116.503

M3 - Article

AN - SCOPUS:33846267348

VL - 116

SP - 503

EP - 515

JO - Progress of Theoretical Physics

JF - Progress of Theoretical Physics

SN - 0033-068X

IS - 3

ER -