Fractal embedded boxes of bifurcations
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 ${\rm Dim\,}1$ unimodal smooth map $f(x,\lambda).$  Taking as an example $f(x,\lambda)=x^{2}-\...
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| Date: | 2024 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2024
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7661 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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 ${\rm Dim\,}1$ unimodal smooth map $f(x,\lambda).$  Taking as an example $f(x,\lambda)=x^{2}-\lambda,$  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 $\lambda$ 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 $k\rightarrow\infty.$ |
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| DOI: | 10.3842/umzh.v76i1.7661 |