@article{Mira_2024, title={Fractal embedded boxes of bifurcations}, volume={76}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/7661}, DOI={10.3842/umzh.v76i1.7661}, abstractNote={<p>UDC 517.9</p> <p>This descriptive text is essentially based on the Sharkovsky’s and Myrberg’s publications on the ordering of periodic solutions (<em>cycles</em>) generated by a ${\rm Dim\,}1$ unimodal smooth map $f(x,\lambda).$<span class="Apple-converted-space"> </span>Taking as an example $f(x,\lambda)=x^{2}-\lambda,$<span class="Apple-converted-space"> </span>it was shown in a paper published in1975 that the bifurcations are organized in the form of a sequence of <em>well-defined fractal embedded</em> ``<em>boxes</em>’’ (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 $k.$<span class="Apple-converted-space"> </span>Without using the denominations <em>Intermittency</em> (1980) and <em>Attractors in Crisis</em> (1982), this new text shows that the notion of <em>fractal embedded</em> ``<em>boxes</em>’’ describes the properties of each of these two situations as the <em>limit of a sequence of well-defined boxes</em> $(k, j)$ as $k\rightarrow\infty.$</p>}, number={1}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Mira, Christian}, year={2024}, month={Feb.}, pages={75 - 91} }