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\end{equation}  In any range $a_i$ to $a_{i+1}$ there are $2^i$ solutions.  Where $\delta$ appears to be the Feigenbaum constant 4.669201609102990671853203821578...  The $\beta^*$ is the Embree–Trefethen constant, constant $\approx$0.70258,  which is related also to fixed iterations and the limit between exponential growth. Both of these constants therefore make sense.The point 3.4494897 is the second bifurcation point.