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}(-β_{q}y^{2})=P_{q}(0)[1+(q-1)β_{q}y^{2}]^{1/(1-q)}(q≥1;β_{q}>0), which optimize the nonadditive entropic functional S_{q} 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.