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

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 chaoticband 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.

A zodiac of studies on complex systems

“…More details about these sets of studies are given in the following sections while future directions are delineated in the concluding section. Additional related reading and commentaries can be found in [57][58][59][60][61][62][63][64][65][66]. The significance of the results obtained so far rests on the observation that our low dimensional model systems capture the main behaviors of high dimensional complex systems.…”

The Pseudo-Pascal Triangle of Maximum Deng Entropy

“…Pascal triangle (known as the Yang Hui Triangle in China) is a triangular arrangement of the binomial coefficients and that it contains many remarkable numerical relations [19,64]. Pascal-Triangle has also become a suitable model system for the exploration of statistical mechanical structures [53,64].…”

“…Pascal triangle (known as the Yang Hui Triangle in China) is a triangular arrangement of the binomial coefficients and that it contains many remarkable numerical relations [19,64]. Pascal-Triangle has also become a suitable model system for the exploration of statistical mechanical structures [53,64]. Pascal Triangle has attracted many people attention and been used in many fields [45,46], such as network [28], cellular automata [79] and so on [5,6,43].…”