Fractal Embedded Boxes of Bifurcations

This descriptive text is essentially based on Sharkovsky’s and Myrberg’s publications on the ordering of periodic solutions (cycles) generated by a Dim 1 unimodal smooth map f(x, λ). Taking f(x, λ) = x²−λ as an example, it was shown in a paper published in 1975 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 → ∞.