A rigorous lower bound for the stability regions of the quadratic map

We establish a lower bound on the measure of the set of stable parameters aa for the quadratic map Qa(x)=ax(1−x)Qa(x)=ax(1−x). For these parameters, we prove that QaQa either has a single stable periodic orbit or a period-doubling bifurcation. From this result, we also obtain a non-trivial upper bound on the set of stochastic parameters for QaQa.