deletions | additions       diff --git a/main.tex b/main.tex  index c737350..24f7830 100644  --- a/main.tex  +++ b/main.tex 
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I_r(\beta)=\int_0^{I_r(\beta)} r - \beta x \; \mathrm{dx} = \frac{2}{\beta}(r-a_0), \;\; a_0\le r \le a_1 \\  I_r(0.5)=\int_0^{I_r(0.5)} r - \beta x \; \mathrm{dx} = \Bigg\{  \begin{matrix}  8 I_{a_1}(\beta)  + 4\sqrt{r-a_1} + \frac{\delta}{2}(r-a_1), \;\; a_1\le r \le a_2 \\  8 \frac{\delta}{2}(r-a_1)\\  I_{a_1}(\beta)  - 4\sqrt{r-a_1} + \frac{\beta^*\delta}{2}(r-a_1), \;\; a_1\le r \le a_2 \frac{\beta^*\delta}{2}(r-a_1)  \end{matrix}  , \;\; a_1\le r \le a_2  \\  I_r(0.5)=\int_0^{I_r(0.5)} r - \beta x \; \mathrm{dx} = \Bigg\{  \begin{matrix} \Bigg\{ \\ \\ \Bigg\{ \end{matrix}  \begin{matrix}  11.7276...+2\sqrt{r-a_2}+\frac{\beta^*\delta}{2}(r-a_2),\;\; a_2\le r \le a_3\\  11.7276...-2\sqrt{r-a_2}+\frac{\delta^2}{4}(r-a_2),\;\; a_2\le r \le a_3\\  6.07...+4\sqrt{r-a_2}+\frac{\beta^*\delta}{2}(r-a_2),\;\; a_2\le r \le a_3\\  6.07...-4\sqrt{r-a_2}+\frac{\beta^*\delta^2}{4}(r-a_2),\;\; a_2\le r \le a_3 I_{a_2}^>(\beta)+2\sqrt{r-a_2}+\frac{\beta^*\delta}{2}(r-a_2)\\  I_{a_2}^>(\beta)-2\sqrt{r-a_2}+\frac{\delta^2}{4}(r-a_2)\\  I_{a_2}^<(\beta)+4\sqrt{r-a_2}+\frac{\beta^*\delta}{2}(r-a_2)\\  I_{a_2}^<(\beta)-4\sqrt{r-a_2}+\frac{\beta^*\delta^2}{4}(r-a_2)  \end{matrix}  ,\;\; a_2\le r \le a_3  \end{equation}  Long numbers here are based on the bifurcation points $a_n$ and will be replaced with appropriate symbols soon. We have \begin{equation} 
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\end{array}  \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, 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.