Calculate period doubling bifurcation points

Question

For a given logistic family $f_{\mu}(x)= \mu*x*(1-x),$ where $\mu \in [0, 4]$ and $x \in [0,1].$ This family undergoes the period doubling bifurcation. Let $\mu_{n}$ denote the value of $\mu$ where a $2^n$-cycle first appears. It is quite difficult to calculate $\mu_{n}$ for large $n$. Can someone please tell me how the values of $\mu_{n}$ will be evaluated by Matlab or Mathematica. P.S.: If anyone has a Code/program to calculate "Bifurcation points", please provide the same. Thank you!!!

Answer 1

Follows a MATHEMATICA script to plot the bifurcations.

Logistic = Compile[{{\[Mu], _Real}}, ({\[Mu], #} &) /@Union[Drop[NestList[\[Mu] # (1 - #) &, .2, 300], 100]]];
f = Table[Logistic[\[Mu]], {\[Mu], 0, 4, .0006}];
ListPlot[Flatten[f, 1], PlotStyle -> {Black,Opacity[0.02], AbsolutePointSize[.001]}]