Discrete logistic model

Definition of the function

fn[x_]:=r(1-x)x

Definition of a procedure to draw cob-webbing

pairWise[x_List]:=Block[{len = Length[x],points={},start,end},
start = {x[[1]],0};
AppendTo[points, start];
AppendTo[points,{x[[1]],x[[1]]}];

Do[
	AppendTo[points,{x[[i]],x[[i+1]]}];
	AppendTo[points,{x[[i+1]],x[[i+1]]}],{i,1,len-1}
];

Return[points]
]

Now you set up the parameter "r" to numerically iterate the logistic map.

next[x_]:=fn[x]/.r->3.4
seq = NestList[next,0.1,100]

ListPlot[seq,Joined->True,PlotRange->{0,1}]

gsim=ListPlot[pairWise[seq],Joined->True,PlotStyle->RGBColor[1,0,0]]

g = Plot[{next[x],x},{x,0,1}]

Show[gsim,g,PlotRange->All,AxesOrigin->{0,0},AspectRatio->1]

g2 = Plot[{next[next[x]],x},{x,0,1}]

Show[gsim,g,g2,PlotRange->All,AxesOrigin->{0,0},AspectRatio->1]