Fractal embedded boxes of bifurcations

Christian Mira

doi:10.3842/umzh.v76i1.7661

Christian Mira Groupe d'Etude des Systèmes Non Linéaire et Applications, INSA Toulouse, France

DOI: https://doi.org/10.3842/umzh.v76i1.7661

Keywords: EMBEDDED BOXE, BIFURCATION

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 ${\rm Dim\,}1$ 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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