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Summary

  1. Period 1 solutions

  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

    \begin{displaymath}
rx = 1 +{r\over 2} \pm {1\over 2}\sqrt{r^{2} - 4}.
\end{displaymath}

  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

    \begin{displaymath}
{r_{n+1} - r_{n}\over r_{n} -r_{n-1}} = {1\over \delta} = 0.214169
\end{displaymath}



Prof. Alan Hood
2000-02-01