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

Sums of variables at the onset of chaos, replenished