Sums of variables at the onset of chaos, replenished

TL;DR

This paper explores the distributions of sums of positions at chaos onset in the logistic map, revealing how multiscale and Gaussian distributions emerge from the interplay of regular and chaotic dynamics, especially near band-splitting points.

Contribution

It extends previous work by analyzing distributions at chaotic band-splitting points, showing multiscale features and their relation to the hierarchical structure of the system.

Findings

Multiscale distributions observed at the Feigenbaum point.

Finite sums reveal multiscale features linked to repellor preimages.

Truncated q-Gaussian distributions resemble t-Student distributions before Gaussian limit.

Abstract

As a counterpart to our previous study of the stationary distribution formed by sums of positions at the Feigenbaum point via the period-doubling cascade in the logistic map (Eur. Phys. J. B 87 32, (2014)), we determine the family of related distributions for the accompanying cascade of chaotic band-splitting points in the same system. By doing this we rationalize how the interplay of regular and chaotic dynamics gives rise to either multiscale or gaussian limit distributions. As demonstrated before (J. Stat. Mech. P01001 (2010)), sums of trajectory positions associated with the chaotic-band attractors of the logistic map lead only to a gaussian limit distribution, but, as we show here, the features of the stationary multiscale distribution at the Feigenbaum point can be observed in the distributions obtained from finite sums with sufficiently small number of terms. The multiscale features are acquired from the repellor preimage structure that dominates the dynamics toward the chaotic attractors. When the number of chaotic bands increases this hierarchical structure with multiscale and discrete scale-invariant properties develops. Also, we suggest that the occurrence of truncated q-gaussian-shaped distributions for specially prescribed sums are t-Student distributions premonitory of the gaussian limit distribution.