Scaling of distributions of sums of positions for chaotic dynamics at band-splitting points

Abstract

The stationary distributions of sums of positions of trajectories generated by the logistic map have been found to follow a basic renormalization group (RG) structure: a nontrivial fixed-point multi-scale distribution at the period-doubling onset of chaos and a Gaussian trivial fixed-point distribution for all chaotic attractors. Here we describe in detail the crossover distributions that can be generated at chaotic band-splitting points that mediate between the aforementioned fixed-point distributions. Self affinity in the chaotic region imprints scaling features to the crossover distributions along the sequence of band splitting points. The trajectories that give rise to these distributions are governed first by the sequential formation of phase-space gaps when, initially uniformly-distributed, sets of trajectories evolve towards the chaotic band attractors. Subsequently, the summation of positions of trajectories already within the chaotic bands closes those gaps. The possible shapes of the resultant distributions depend crucially on the disposal of sets of early positions in the sums and the stoppage of the number of terms retained in them. PACS 5.45.Ac Low-dimensional chaos PACS 05.45.Pq Numerical simulations of chaotic systems PACS 05.10.Cc Renormalization group methods

A zodiac of studies on complex systems

“…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].…”

“…The nontrivial corresponds to the multifractal distribution at µ = µ ∞ while the trivial one is that prescribed by the ordinary central limit theorem [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].…”

“…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.…”

Sums of variables at the onset of chaos, replenished

“…This ladder organization was elucidated [3] through consideration of the family of periodic attractors, conveniently, the super-stable attractors called supercycles [4], along the period-doubling cascade. Here we complement this study by considering instead the cascade of chaotic band-splitting attractors, or Misiurewicz (M n , n = 0, 1, 2, ...) points [5]. This gives us the opportunity of analyzing the transformation of the developing multiscale distributions into the gaussian limit distribution.…”

“…On the other hand, as the number of summands N increases the distribution for each value of n develops a symmetrical shape and eventually approaches the gaussian limit distribution for N → ∞, as anticipated for all chaotic attractors. We discuss the occurrence of so-called q-gaussian distributions for special sums of positions at, or around, the M n points [5], [8], [9] in terms of distributions for finite-size data sets drawn from gaussian variables such as it is the case of the t-Student distribution.…”