Binary tree approach to scaling in unimodal maps

Jukka A. Ketoja, Juhani Kurkijärvi

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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 languageEnglish
Pages (from-to)643-668
Number of pages26
JournalJournal of Statistical Physics
Volume75
Issue number3-4
DOIs
Publication statusPublished - 1 May 1994
MoE publication typeNot Eligible

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Keywords

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

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