Chapter 7.6, Problem 7P

In each of Problems 5 through 7, carry out the indicated investigations of the Lorenz equations.

(a) For $r=21$, plot $x$ versus $t$ for the solutions starting at the initial points $\left(3,8,0\right),\left(5,5,5\right)$, and $\left(5,5,10\right)$. Use a $t$ interval of at least $0\le t\le 30$. Compare your graphs with those in Figure 7.6.4.

(b) Repeat the calculation in part (a) for $r=22,r=23$, and $r=24$. Increase the $t$ interval as necessary so that you can determine when each solution begins to converge to one of the critical points. Record the approximate duration of the chaotic transient in each case. Describe how this quantity depends on the value of $r$.

(c) Repeat the calculation on parts (a) and (b) for values of $r$ slightly greater than $24$. Try to estimate the value of $r$ for which the duration of the chaotic transient approaches infinity.