Funkcjanaiwna[x_,λ_]:=λ x
Fn=#1 #2&

NestList[Fn[#,1.01]&,0.2,30]

ListPlot[NestList[Fn[#,1.01]&,0.2,300]]

x0=0.2;
l=2;
ListPlot[NestList[l#(1-#)&,x0,30]]

Fu[xx_,ll_]:=ll xx(1-xx)

x0=0.2;
l=2;

ListPlot[NestList[Fu[#,l]&,x0,30]]

Manipulate[ListPlot[NestList[Fu[#,λ]&,x0,30],PlotRange->{0,1}],{{λ,0.5},0,4},{{x0,0.2},0,1}]

Manipulate[Plot[{x,λ x(1-x)},{x,0,1}],{λ,0,4}]

Manipulate[

{{Plot[{x,λ x(1-x)},{x,0,1}],

ListPlot[NestList[Fu[#,λ]&,x0,30],PlotRange->{0,1}]}},


{{λ,0.5},0,4},{{x0,0.2},0,1}]

Manipulate[

GraphicsGrid@@{{{Plot[{x,λ x(1-x),Nest[λ#(1-#)&,x,4]},{x,0,1}],

ListPlot[NestList[Fu[#,λ]&,x0,300],PlotRange->{0,1},Joined->True,GridLines->Automatic]}}},


{{λ,3},1,4},{{x0,0.2},0,1}]

NestList[l#(1-#)&,x0,nn]

Table[NestList[l#(1-#)&,x0,nn],{l,1,1.5,0.1}]

Table[{l,NestList[l#(1-#)&,x0,nn]},{l,1,1.5,0.1}]

Table[
Riffle[NestList[l#(1-#)&,x0,nn],l]
,{l,1,1.5,0.1}]

Flatten[Table[Partition[Riffle[NestList[l#(1-#)&,x0,nn],l],2],{l,1,1.5,0.1}],1]

nn=100;x0=0.231;
ListPlot[Flatten[Table[Partition[Riffle[NestList[l#(1-#)&,x0,nn][[-20;;-1]],l],2],{l,1,4,0.1}],1]]

nn=200;x0=0.231;
ListPlot[Reverse/@Flatten[Table[Partition[Riffle[NestList[l#(1-#)&,x0,nn][[-100;;-1]],l],2],{l,1,4,0.01}],1]]