deletions | additions       diff --git a/main.tex b/main.tex  index d9fd92d..89e5e41 100644  --- a/main.tex  +++ b/main.tex 
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Provided the first iteration does not equal zero, the scheme can be coded nicely.  Then one may vary $\alpha$ and $\beta$. In the initial investigation $\beta$ was set to $0.5$, and $\alpha$ varied from $1$ till $4$. At low values of $\alpha$ (between $1$ and $3$) the integral converges nicely, there is a point where the scheme hops between two convergent values (between $3$ and $~3.5$). Luckily, it was noticed four convergent values appeared after $3.5$, and from this inference a sweep for suspected bifurcations was made. Amazingly, the solution to $I_\alpha(0.5)$ for any $\alpha$ looked similar the logistic map, but scaled by a factor $16$. The traditional bifurcation diagram is obtained when taking the iteration \begin{equation} 
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\end{equation}  The single valued region between $1$ and $3$ is believable, as the integral converges. We can justify the result analytically, the integral[Equation Num] is an example of an integral we can state the result of. We wish for the class of integral where we know (rather than impose) \begin{equation}  \int_0^m r-\beta x \; \mathrm{dx} = m  \end{equation}  Which would mean \begin{equation}  \big[ rx - \frac{\beta x^2}{2} \big]^m_0=m  \end{equation}  Which rearranges to $m=2(r-1)/\beta$, which is the form of the first of the equations! So it is justified.  Long numbers here are based on the bifurcation points $a_n$ and will be replaced with appropriate symbols soon. We have \begin{equation}  \begin{array}{|c|c|}