The partial visibility curve of the Feigenbaum cascade to chaos

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 $r_c \approx 3.679$.

💡 Research Summary

The paper investigates the visibility properties of time‑series generated by the logistic map, focusing on the Feigenbaum period‑doubling cascade that leads to chaos. By treating a discrete time series as an ordered set of points (t_i, x_i) in the plane, the authors adopt the concept of horizontal visibility: two points see each other if every intermediate point lies below the lower of the two. From this definition they introduce several quantitative measures: the visibility of a single point (the number of points it can see), the visibility of a set of points (the size of the union of their basins), the total visibility number v_T (the cardinality of the smallest set that can see the whole series), and the partial visibility v_m (the maximal visibility achievable with exactly m points).

Using the logistic map x_{n+1}=r x_n(1−x_n), they vary the growth parameter r from the onset of the Feigenbaum cascade (r≈1) up to the accumulation point r_∞≈3.569945, where the period‑doubling sequence T=2^k (k=1,2,…) converges to chaos. For each period T=2^k they derive the exact distribution of point visibilities. The series splits into k+1 classes; the fraction of points belonging to class j is 2^{−j} (j=0,…,k) and points in class j see 2^{k+j}+3 points (including themselves). This hierarchical distribution yields a piecewise‑linear “partial visibility curve” ν_k(μ), where μ=m/N is the fraction of points selected and ν_k(μ) is the fraction of the whole series that becomes visible. The curve consists of k+1 linear segments, each with decreasing slope, and is given analytically by formula (6) in the manuscript.

A striking universal result is that the total visibility number is constant across the entire cascade: v_T=1/4. In other words, the minimal dominating set always contains exactly one quarter of the points, regardless of the period. The authors compute the length L_k of each polygonal ν_k(μ) by integrating the absolute slope over μ∈