TY - JOUR
AU - Christian Mira
PY - 2024/02/02
Y2 - 2024/09/12
TI - Fractal embedded boxes of bifurcations
JF - Ukrainsâ€™kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 76
IS - 1
SE - Research articles
DO - 10.3842/umzh.v76i1.7661
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/7661
AB - UDC 517.9This 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 $k\rightarrow\infty.$
ER -