Fractal embedded boxes of bifurcations-Reference-Cited by-同舟云学术

Fractal embedded boxes of bifurcations

Published:2024-02-02 Issue:1 Volume:76 Page:75-91
ISSN:1027-3190
Container-title:Ukrains’kyi Matematychnyi Zhurnal
language:
Short-container-title:Ukr. Mat. Zhurn.

Author:

Mira Christian

Abstract

UDC 517.9 This descriptive text is essentially based on the Sharkovsky's and Myrberg's publications on the ordering of periodic solutions (cycles) generated by a D i m 1 unimodal smooth map f ( x , λ ) . Taking as an example f ( x , λ ) = x 2 - λ , it was shown in a paper published in1975 that the bifurcations are organized in the form of a sequence of well-defined fractal embedded ``boxes'' (parameter λ intervals), each of which is associated with a basic cycle of period k and a symbol j permitting to distinguish cycles with the same period k . Without using the denominations Intermittency (1980) and Attractors in Crisis (1982), this new text shows that the notion of fractal embedded ``boxes'' describes the properties of each of these two situations as the limit of a sequence of well-defined boxes ( k , j ) as k → ∞ .

Publisher

SIGMA (Symmetry, Integrability and Geometry: Methods and Application)

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