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.