### Abstract

Ge, Rusjan, and Zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one-dimensional unimodal maps. We consider the scaling behavior in a modified tree which takes into account the self-similarity of the window structure. A nonuniversal geometric convergence of the associated superstable parameter values towards a Misiurewicz point is observed for almost all binary sequences with periodic tails. For these sequences the window period grows arithmetically down the binary tree. There are an infinite number of exceptional sequences, however, for which the growth of the window period is faster. Numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences.

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

Pages (from-to) | 643-668 |

Number of pages | 26 |

Journal | Journal of Statistical Physics |

Volume | 75 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 1 May 1994 |

MoE publication type | Not Eligible |

### Fingerprint

### Keywords

- binary tree
- chaos
- Misiurewicz point
- periodic window
- scaling
- Unimodal map

### Cite this

*Journal of Statistical Physics*,

*75*(3-4), 643-668. https://doi.org/10.1007/BF02186875

}

*Journal of Statistical Physics*, vol. 75, no. 3-4, pp. 643-668. https://doi.org/10.1007/BF02186875

**Binary tree approach to scaling in unimodal maps.** / Ketoja, Jukka A.; Kurkijärvi, Juhani.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Binary tree approach to scaling in unimodal maps

AU - Ketoja, Jukka A.

AU - Kurkijärvi, Juhani

PY - 1994/5/1

Y1 - 1994/5/1

N2 - Ge, Rusjan, and Zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one-dimensional unimodal maps. We consider the scaling behavior in a modified tree which takes into account the self-similarity of the window structure. A nonuniversal geometric convergence of the associated superstable parameter values towards a Misiurewicz point is observed for almost all binary sequences with periodic tails. For these sequences the window period grows arithmetically down the binary tree. There are an infinite number of exceptional sequences, however, for which the growth of the window period is faster. Numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences.

AB - Ge, Rusjan, and Zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one-dimensional unimodal maps. We consider the scaling behavior in a modified tree which takes into account the self-similarity of the window structure. A nonuniversal geometric convergence of the associated superstable parameter values towards a Misiurewicz point is observed for almost all binary sequences with periodic tails. For these sequences the window period grows arithmetically down the binary tree. There are an infinite number of exceptional sequences, however, for which the growth of the window period is faster. Numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences.

KW - binary tree

KW - chaos

KW - Misiurewicz point

KW - periodic window

KW - scaling

KW - Unimodal map

UR - http://www.scopus.com/inward/record.url?scp=34249767598&partnerID=8YFLogxK

U2 - 10.1007/BF02186875

DO - 10.1007/BF02186875

M3 - Article

AN - SCOPUS:34249767598

VL - 75

SP - 643

EP - 668

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -