f[λ_, x_]:= λ x(1-x)

NestList[f[1.1,#]&, 0.3, 29]

ListPlot[NestList[f[1.1,#]&, 0.3, 29], PlotRange -> {{0,All}, {0,1}}, Joined-> True]

Manipulate[
ListPlot[NestList[f[λ,#]&, x, 100], PlotRange -> {{0,All}, {0,1}}],
{λ,1,4},{x,0,1}
]

Plot[Nest[f[3.5,#]&, x,2],{x,0,1}]

Manipulate[Plot[{Nest[f[λ,#]&, x,n],x},{x,0,1}], {n,10,100,1}, {λ,1,4}]

ntaiteracja [λ_, x_, n_] := Nest[f[λ, #]&, x, n]

Manipulate[Plot[{ntaiteracja[λ, x,n],x}, {x,0,1}, PlotRange -> {{0,All}, {0,1}}], {λ, 1,4}, {n,1,20,1}]

x0 = RandomReal[];

asymptot1[λ_, n_] := NestList[f[λ, #]&, x0, n-1]

?Take

asymptot2[λ_, n_, n0_] := Take[NestList[f[λ, #]&, x0, n-1], -(n-n0)]

asymptot2[1,10,8]

?Riffle

asymptot3[λ_, n_, n0_]:= Riffle[ConstantArray[λ, n-n0], Take[NestList[f[λ, #]&, x0, n-1], -(n-n0)]]

asymptot3[1,10,8]

Clear[λ]

Table[asymptot3[λ, 10,8], {λ, ,1,4,1}]

Flatten[Table[asymptot3[λ, 10,8], {λ, ,1,4,1}]]

Partition[Flatten[Table[asymptot3[λ, 10,8], {λ, ,1,4,1}]],2]

diagrambif[x0_, n_, n0_, krok_] := ListPlot[Partition[Flatten[Table[asymptot3[λ, n, n0] , {λ,1,4,krok}]],2]]

diagrambif[x0, 400, 390, 0.01]