Summary

1. Period 1 solutions
   • $x=0$, unstable for $r > 0$.

   • $x=1$, stable for $0 < r < 2 = r_{0}$.

2. Period two solutions satisfy $x_{n+2} = x_{n}$, $x_{n+3} = x_{n+1}$ for $2 < r < \sqrt{6} = r_{1}$ and are given by
   
   $$rx = 1 +{r\over 2} \pm {1\over 2}\sqrt{r^{2} - 4}.$$
   
   

3. Period doubling occurs at $r_{0}, r_{1}, r_{2}, \cdots$

4. Feigenbaum discovered the sequence of $r$ values at which bifurcations occur satisfy the relation
   
   $${r_{n+1} - r_{n}\over r_{n} -r_{n-1}} = {1\over \delta} = 0.214169$$