bifurcation[
x0_,mumin_,mumax_,del_,nmax_,opts___Rule] := (
ListPlot[
Flatten[
Table[
Thread[
List[
mu,
Drop[
NestList[(If[#<0.5,2mu #,2mu(1-#)])&, x0, nmax],
200
]
]
],
{mu,mumin,mumax,del}
],
1
],
opts
];
)

bifurcation[
0.2,0.01,0.99,0.01,500,
AxesOrigin -> {0, 0},
PlotRange -> {{0,1},{0,1}},
AxesLabel -> {α, (Subscript[x, n])}
]