Manipulate[ListPlot[With[{init=0.4,nsteps=50000},Module[{func},
func[x_]:=r x(1-x);
NestList[func[#]&,init,nsteps]
]
],
PlotStyle->PointSize->0.001,
Frame->True,PlotRange->{All,{0,1}},
PlotRangePadding->None,
FrameTicksStyle->Directive[FontSize->14],
PlotLabel->Style["Evolution of (Subscript[x, n])   r = "<>ToString[r],FontSize->20],
FrameLabel->(Style[#,FontSize->16]&/@{"No. of steps n","Value of x"})],{{r,1.1,Style["Parameter r",FontSize->14]},1,4}]