CHAOTIC HIERARCHY FROM THE ONE-DIMENSIONAL BOX-WITHIN-A-BOX BIFURCATIONS STRUCTURE PROPERTIES

The purpose of this paper is to show that a chaotic hierarchy appears also by consideration of the simplest one dimensional map T(x→x² –λ), x being a real variable, λ a real parameter. This hierarchy results from the fractal properties of the box-within-a-box bifurcations structure of T, inducing different levels of self similarities on the x-axis, when repulsive cycles merge into critical points, and accumulations of such situations.

By embedding T into a two-dimensional diffeomorphism, one obtains a transfer of this hierarchy into a wider class of problems …