Pascal (Yang Hui) triangles and power laws in the logistic map

TL;DR

This paper reveals the simultaneous presence of Pascal triangle patterns and power-law scaling in the logistic map, linking combinatorial structures with complex dynamical behaviors.

Contribution

It demonstrates the occurrence of Pascal triangle patterns and power-law scaling in unimodal maps, especially in bifurcation cascades and the onset of chaos.

Findings

Pascal triangle patterns appear in supercycle diameters and band widths.

Power-law scaling is observed at bifurcation points and chaotic band splitting.

The onset of chaos exhibits both Gaussian and power-law distributions.

Abstract

We point out the joint occurrence of Pascal triangle patterns and power-law scaling in the standard logistic map, or more generally, in unimodal maps. It is known that these features are present in its two types of bifurcation cascades: period and chaotic- band doubling of attractors. Approximate Pascal triangles are exhibited by the sets of lengths of supercycle diameters and by the sets of widths of opening bands. Additionally, power-law scaling manifests along periodic attractor supercycle positions and chaotic band splitting points. Consequently, the attractor at the mutual accumulation point of the doubling cascades, the onset of chaos, displays both Gaussian and power-law distributions. Their combined existence implies both ordinary and exceptional statistical-mechanical descriptions of dynamical properties.