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.

A zodiac of studies on complex systems

“…Sums. The distributions for sums of successive positions of trajectories, as in random walks, for families of chaotic attractors of the quadratic map were found to conform to a renormalization group scheme such that its trivial fixed-point matches the central limit theorem [21][22][23][24][25]. Rankings.…”

“…In our work [21][22][23][24][25] we considered sums of positions from a single trajectory and also from an ensemble of them, in the latter case started from a set of uniformly-distributed initial conditions along the interval of definition of the map. The chaotic-band attractors of the quadratic map are ergodic and therefore single and ensemble sums lead to the same limit distribution albeit the former sum takes more terms than the latter in resembling the final form [21,22].…”

“…When µ > µ ∞ the flow towards the trivial fixed point displays a crossover behavior [21,22], the fine details of which and other relevant issues were resolved numerically [24] by considering the family of attractors at chaotic-band-splitting or Misiurewics points. As it turned out [25] the distribution at the crossover is related to incomplete sampling of data and therefore resembles the socalled T-Student distribution, that can in turn be rewritten into the form of a q-Gaussian distribution [73,74]. At the present date, this research line focuses on a specific model of correlated walks that displays anomalous diffusion and arrest.…”

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

“…The band-splitting sequence for µ > µ ∞ is the chaotic equivalent to the perioddoubling supercycles for µ < µ ∞ . For recent developments assisted through their use, see for example [12,13]. The determination of the band widths at the nth Misiurewicz point is facilitated by the circumstance that the set of band edge points, that we denote by…”

“…The evolution via sequential gap formation of uniformly distributed ensemble of trajectories towards supercycle and Misiurewicz point attractors display a 'recapitulation' property [13,15,16], i.e. progression towards 2 n -periodic or 2 n -band chaotic attractors repeats successively that towards those attractors with 2 k , k = 0, 1, 2, ..., n − 1.…”