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

Abstract

We focus on the Feigenbaum-Coullet-Tresser point of the dissipative one-dimensional z-logistic map x(t+1) = 1-a|x(t )|(z) (z 1). We show that sums of iterates converge to q-Gaussian distributions P-q(y) = P-q(0) exp(q)(-beta(q)y(2)) = P-q(0 )[1 + (q-1)beta(q) y(2)](1/(1-q))(q >= 1; beta(q) > 0), which optimize the nonadditive entropic functional Sq under simple constraints. We propose and justify heuristically a closed-form prediction for the entropic index, q(z) = 1 + 2/(z + 1), 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 > 2 and divergent variance for 1 <= z <= 2. These results extend edge-of-chaos central limit behavior beyond the standard (z = 2) case and provide a simple predictive law for unimodal maps with varying maximum order.