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\end{equation}  Provided the first iteration does not equal zero, the scheme can be coded nicely.  Then one may vary $\alpha$ and $\beta$. Due to initial reasons $\beta$ was set to $0.5$, and $\alpha$ varied from $1$ till $4$, at low values of $\alpha$ the integral converges nicely, at values less than $3$ there is a point where the scheme hops between two convergent values. Luckily, it was noticed at $3.5$ four convergent values appeared, and from prior knowledge (a guess at Lyapunov/Feigenbaum behaviour) I took a sweep for suspected bifurcations. Amazingly, the solution to $I_\alpha(0.5)$ for any $\alpha$ is looks incredibly similar  the logistic map! The tradiditional bifurcation diagram when taking the iteration \begin{equation} x_{n+1}=rx_n(1+x_n)  \end{equation}  and plotting convergent values with changing $r$. Looking back this is now obvious, as when solving the integral and changing $\alpha$ to $r$ we have the iterative form \begin{equation}  [I_r(\beta)]_{i+1}= r[I_r(\beta)]_{i}(1 - \frac{\beta}{2r} [I_r(\beta)]_{i})  \end{equation}