Relative-stability interpretation of subset-shaped curves

Establish that the family of subset-shaped curves formed by amplitude bins with low betweenness centrality in the amplitude transition network of the logistic map depicts the relative stability of points in the map’s phase space.

Background

The paper constructs an amplitude transition network from logistic-map time series by treating uniformly discretized amplitude bins as nodes and transitions between bins as directed weighted links. It then examines node-wise betweenness centrality across the control-parameter range r from 3.5 to 4. Low-betweenness regions form a hierarchy of visually distinct subset-shaped curves in the orbit diagram.

The authors hypothesize that these curves encode the relative stability of periodic points and compare them with the density of unstable periodic points obtained by solving F^p(x)=x for periods up to 20. Although the comparison is presented as confirming that betweenness centrality delineates the unstable-periodic-point skeleton, the explicit conjecture identifies the broader interpretation of the entire family of curves as an unresolved claim.