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

4th International Workshop on Statistical Physics and Mathematics for Complex Systems

物理学;数学

Velarde, Carlos^1 ; Robledo, Alberto^2

Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas, Universidad Nacional Autónoma de México, Mexico City, Mexico^1

Instituto de Física, Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Mexico City, Mexico^2

关键词: Accumulation points;    Dynamical properties;    Logistic maps;    Onset of chaos;    Pascal triangle;    Periodic attractor;    Power law distribution;    Power law scalings;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/604/1/012018/pdf
DOI  :  10.1088/1742-6596/604/1/012018

来源: IOP

【 摘 要 】

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.