Fractal embedded boxes of bifurcations - Mira, Christian; Mira, Christian (Institute of Mathematics, NAS of Ukraine, 2024)
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 $f(x,λ).$ 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 $k→∞.$