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 a for the quadratic map Q a ( x ) = a x ( 1 − x ) . For these parameters, we prove that Q a 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 Q a .