Sums of variables at the onset of chaos, replenished

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.

arXiv commentTo be published in European Physical Journal Special Topics: Temporal and Spatio-Temporal Dynamic Instabilities: Novel Computational and Experimental Approaches

References in corpus (7)

• Generalized entropies and the transformation group of superstatistics
• Central limit behavior of deterministic dynamical systems
• A closer look at time averages of the logistic map at the edge of chaos
• Generalized Huberman-Rudnick scaling law and robustness of $q$-Gaussian probability distributions
• Renormalization group structure for sums of variables generated by incipiently chaotic maps
• Sums of variables at the onset of chaos
• Scaling of distributions of sums of positions for chaotic dynamics at band-splitting points

Cited by in corpus (1)

• Logistic map trajectory distributions: Renormalization-group, entropy and criticality at the transition to chaos