Compute the Feigenbaum delta from the logistic map. The logistic map is given by

$,$

and the Feigenbaum delta is defined as

where

and where is the value of for which is in the orbit of the period$-$ cycle with $.$

Here is a resonable outline:

Loop $1$ Start at period$-$ with $,$ and increment with each iteration

Compute initial guess for using $,$ and $.$

Loop $2$ Iterate Newton's method, either a fixed number of times or until convergence

Initialize logistic map

Loop $3$ Iterate the logistic map times

Compute and

Loop $3($end$)$

One step of Newton's method

Loop $2($end$)$

Save and compute

Loop $1($end$)$

Grading will be done on the converged values of up to $.$ Set $.\%$ Compute the Feigenbaum delta

$\%$ Store approximate values in the row vector delta for assessment, where length$($delta$)=$ num$\_$doublings and

$\%$ delta$(2$:num$\_$doublings$)$ are computed from the algorithm described in Lectures $21-23.$

num$\_$doublings$=11$; delta$=$zeros$(1,$num$\_$doublings$)$; delta$(1)=5$;$\%$ Output your results

fprintf$(\text{'}$n delta$($n$)$

$\text{'})$;

for n$=1$:num$\_$doublings

fprintf$(\text{'}\%2$g $\%18\mathrm{.}15$f

$\text{'},$n$,$delta$($n$))$;

end