Spatial and Temporal Taylor's Law in 1-Dim Chaotic Maps

TL;DR

This paper investigates Taylor's Law in 1-D chaotic maps, analyzing the differences between spatial and temporal ensembles, and explaining their relationships through distribution skewness and temporal correlations.

Contribution

It clarifies the relationship between spatial and temporal Taylor's Law in chaotic maps, providing analytical insights and explaining mechanisms behind observed power laws.

Findings

STL explained by distribution skewness

TTL influenced by temporal correlations

Analytical derivation of Bartlett's law for logistic and tent maps

Abstract

By using low-dimensional chaos maps, the power law relationship established between the sample mean and variance called Taylor's Law (TL) is studied. In particular, we aim to clarify the relationship between TL from the spatial ensemble (STL) and the temporal ensemble (TTL). Since the spatial ensemble corresponds to independent sampling from a stationary distribution, we confirm that STL is explained by the skewness of the distribution. The difference between TTL and STL is shown to be originated in the temporal correlation of a dynamics. In case of logistic and tent maps, the quadratic relationship in the mean and variance, called Bartlett's law, is found analytically. On the other hand, TTL in the Hassell model can be well explained by the chunk structure of the trajectory, whereas the TTL of the Ricker model have a different mechanism originated from the specific form of the map.