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Two of the functions cross over in a non chaotic region!  The extrapolaations of the functions lead to crossing points of later functions!  Also there are features which appear symmetric about these functions.  [Simon] difference of two lines is measure of chaos? Hitting the resolution barrier, the primary intersect was found to be at the position $3.67857$,$10.7144$, with a $\pm0.0001$ uncertainty on $x$ and a $\pm0.0002$ uncertainty on $y$, when $\beta=1/2$, i.e a scaling of $16$ on the regular logistic map. We can verify if this point is found on the original function $4(r-1)$. Using the x measurment would give $10.71428$ which is fairly close. With the $x$ axis error, we have the range $10.71424$ to $10.71432$.