Graphs of f and f^2

Clear[r];
f[r_,x_] = r x (1-x);
Manipulate[Plot[{x, f[r,x], f[r, f[r,x]]}, {x, 0, 1}, PlotRange -> {0, 1}, PlotStyle -> {Black, {Blue, Thick}, Red}], {r, 0, 4}]

Graphs of f, f^2 and f^4

Clear[r];
f[r_,x_] = r x (1-x);
Manipulate[Plot[{x, f[r,x], f[r, f[r,x]], f[r,f[r, f[r, f[r,x]]]] }, {x, 0, 1}, PlotRange -> {0, 1}, PlotStyle -> {Black, {Blue, Thick}, Red, Green}], {r, 3, 4}]

Orbit of the critical point :  50 iterates

Manipulate[ListPlot[{NestList[N[Function[r # (1 - #)]], 1/2,50], NestList[N[Function[r # (1 - #)]], 1/2, 50]}, PlotRange -> {0, 1}, Joined -> {True, False}, PlotStyle -> {{Blue},{Red, PointSize[Medium]}} ], {r, 0,4}]

Orbit of the critical point for 500 iterates

Manipulate[ListPlot[{NestList[N[Function[r # (1 - #)]], 1/2,500], NestList[N[Function[r # (1 - #)]], 1/2, 500]}, PlotRange -> {0, 1}, Joined -> {False, False}, PlotStyle -> {{Blue},{Red, PointSize[Medium]}} ], {r, 0,4}]