x0=0.1;l=1.01;
NestList[l#&,x0,20]
ListPlot[NestList[l#&,x0,200]]

ListLogPlot[NestList[l#&,x0,200]]

x0=0.1;l=1.01;
NestList[l#(1-#)&,x0,20]

Manipulate[ListPlot[NestList[l#(1-#)&,x0,30],PlotRange->{0,1},Joined->True],{{l,3},0,4},{{x0,0.234},0,1}]

Manipulate[Plot[{x,l x(1-x)},{x,0,1},PlotRange->{0,1},AspectRatio->1],{l,0,4}]

Manipulate[
GraphicsRow@{Plot[{x,l x(1-x),Nest[l#(1-#)&,x,4]},{x,0,1},PlotRange->{0,1},AspectRatio->1],
ListPlot[NestList[l#(1-#)&,x0,50],PlotRange->{0,1},Joined->True,AspectRatio->1]},
{{l,3},0,4},{{x0,0.234},0,1}]

l=0.9;x0=0.234;
NestList[l#(1-#)&,x0,6]

x0=0.234;
ListPlot[Flatten[Table[Partition[Riffle[NestList[l#(1-#)&,x0,100][[-30;;-1]],l],2],{l,1,4,0.1}],1]]

r=Range[20]

r[[13;;18]]

r[[2;;-2]]

r[[-10;;-1]]

r[[{1,2,3,4,5}]]

x0=0.24;
ListPlot[Reverse/@Flatten[Table[Partition[Riffle[NestList[l#(1-#)&,x0,500][[-50;;-1]],l],2],{l,1,4,0.005}],1],Frame->True,GridLines->Automatic,FrameStyle->Directive[Thick,14],PlotStyle->Pink]