Chapter 7.6, Problem 8P

For certain $r$ intervals, or windows, the Lorenz equations exhibit a period-doubling property. Careful calculations may reveal this phenomenon.

(a) One period-doubling window contains the value $r=100$. Let $r=100$ and plot the trajectory starting at $\left(5,5,5\right)$ or some other initial point of your choice. Does the solution appear to be periodic? What is the period?

(b) Repeat the calculation in part (a) for slightly smaller values of $r$. When $r\cong 99.98$, you may be able to observe that the period of the solution doubles. Try to observe this result by performing calculations with nearby values of $r$.

(c) As $r$ decreases further, the period of the solution doubles repeatedly. The next period doubling occurs at about $r\cong 99.629$. Try to observe this by plotting trajectories for nearby values of $r$.