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

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.