Central Limit Behavior at the Edge of Chaos in the z-Logistic Map

TL;DR

This paper investigates the statistical behavior of sums of iterates at the edge of chaos in the z-logistic map, showing convergence to q-Gaussian distributions and deriving a predictive formula for the entropic index based on nonlinearity.

Contribution

It extends the understanding of central limit behavior at the edge of chaos to a broader class of unimodal maps with varying nonlinearity order, providing a predictive law for the limiting distribution.

Findings

Sums of iterates converge to q-Gaussian distributions at the edge of chaos.

Derived a closed-form formula for the entropic index based on the nonlinearity parameter z.

Finite variance for z > 2 and divergent variance for 1 < z < 2.

Abstract

We focus on the FeigenbaumCoulletTresser point of the dissipative one-dimensional z logistic map. We show that sums of iterates converge to q Gaussian distributions, which optimize the nonadditive entropic functional Sq under simple constraints. We derive a closedform prediction for the entropic index, and validate it numerically via data collapse for typical z values. The formula captures how the limiting law depends on the nonlinearity order and implies finite variance for z larger than 2 and divergent variance for z in between 1 and 2. These results extend edge of chaos central limit behavior beyond the standard case and provide a simple predictive law for unimodal maps with varying maximum order.