Universality of the scaling q(z)=1+2/(z+1) for the z-logistic map at the Feigenbaum–Coullet–Tresser point

Establish that, for the one-dimensional z-logistic map x_{t+1}=1−a|x_t|^z (z≥1) evaluated at the Feigenbaum–Coullet–Tresser accumulation point a=a_c(z), the central-limit attractor of properly rescaled sums of iterates converges to a q-Gaussian with entropic index q(z)=1+2/(z+1) for all z≥1, thereby proving the conjectured universal scaling relation derived from the tail–moment criterion.

Background

The paper studies the central-limit behavior of sums of iterates of the generalized z-logistic map at the edge of chaos, where the Lyapunov exponent vanishes and classical central-limit theorem does not apply. In this regime, q-Gaussian attractors are expected under generalized central-limit frameworks.

Using a tail–moment divergence argument tied to the nonlinearity order z, the authors propose a closed-form scaling relation for the entropic index of the limiting distribution: q(z)=1+2/(z+1). They provide numerical support near z≈2 via data collapse under Huberman–Rudnick scaling, but explicitly present the universality of this law across all z≥1 as a conjecture.

References

It might well be that this scaling applies to {\it all values of} $z$. Therefore, the CL behavior of the probability distribution at the $z$-generalized Feigenbaum-Coullet-Tresser point would be expected to approach a $q$-Gaussian with the index $q$ {\it a priori} known from the equation given as indicated in Eq. (\ref{qzz}). Consequently, $q$ monotonically decreases from 2 to 1 when $z$ increases from 1 to infinity. In what follows, we provide numerical support to this conjecture.

— Central Limit Behavior at the Edge of Chaos in the z-Logistic Map (2508.13170 - Saberi et al., 8 Aug 2025) in Section “Relation between q and z” (Eq. (qzz))