The partial visibility curve of the Feigenbaum cascade to chaos

Abstract

A family of classical mathematical problems considers the visibility properties of geometric figures in the plane, e.g. curves or polygons. In particular, the {\it domination problem} tries to find the minimum number of points that are able to dominate the whole set, the so called, {\it domination number}. Alternatively, other problems try to determine the subsets of points with a given cardinality, that maximize the basin of domination, the {\it partial dominating set}. Since a discrete time series can be viewed as an ordered set of points in the plane, the dominating number and the partial dominating set can be used to obtain additional information about the visibility properties of the series; in particular, the total visibility number and the partial visibility set. In this paper, we apply these two concepts to study times series that are generated from the logistic map. More specifically, we focus this work on the description of the Feigenbaum cascade to the onset of chaos. We show that the whole cascade has the same total visibility number, $v_T=1/4$. However, a different distribution of the partial visibility sets and the corresponding partial visibility curves can be obtained inside both periodic and chaotic regimes. We prove that the partial visibility curve at the Feigenbaum accumulation point $r_{\infty} \approx 3.5699$ is the limit curve of the partial visibility curves ($n+1$-polygonals) that correspond to the periods $T=2^{n}$ for $n=1,2,\ldots$. We analytically calculate the length of these $n+1$-polygonals and, as a limit, we obtain the length of the partial visibility curve at the onset of chaos, $L_{\infty} = L(r_{\infty}) \approx 1.0414387863$. Finally, we compare these results with those obtained from the period 3-cascade, and with the partial visibility curve of the chaotic series at the crossing point $r_c \approx 3.679$.