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We generalize Huberman-Rudnick universal scaling law for all periodic windowsof the logistic map and show the robustness of $q$-Gaussian probabilitydistributions in the vicinity of chaos threshold. Our scaling relation isuniversal for the self-similar windows of the map which exhibit period-doublingsubharmonic bifurcations. Using this generalized scaling argument, for allperiodic windows, as chaos threshold is approached, a developing convergence to$q$-Gaussian is numerically obtained both in the central regions and tails ofthe probability distributions of sums of iterates.

Generalized Huberman-Rudnick scaling law and robustness of $q$-Gaussian probability distributions