Consider the tilt$-$tent map: $x_{n+1}=f(x_n),$ where

$f(x)=\{\begin{array}{cc}x,& 0x\frac{1}{4},\\ \frac{}{3}(1-x),& \frac{1}{4}x1.\end{array}$

r$..$

Fig. $1$ The tilt$-$tent map.

a$)$ Write a small computer program and use computer to generate the cobweb graphs for this tilt$-$tent map with the following values of $=0\mathrm{.}8,1\mathrm{.}2,2\mathrm{.}0,3\mathrm{.}0,3\mathrm{.}01,3\mathrm{.}5.$ Try to find the critical value of $,$ leading to chaos. $($Use Matlab or Python$)$

b$)$ Is it possible to find such critical value of $$ analvtically?